[ { "id": "authors:n4v7p-6jw96", "collection": "authors", "collection_id": "n4v7p-6jw96", "cite_using_url": "https://authors.library.caltech.edu/records/n4v7p-6jw96", "type": "article", "title": "Formal languages and TQFTs with defects", "author": [ { "family_name": "Boateng", "given_name": "Luisa" }, { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" } ], "abstract": "

A construction that assigns a Boolean 1D TQFT with defects to a finite state automaton was recently developed by Gustafson, Im, Kaldawy, Khovanov, and Lihn. We show that the construction is functorial with respect to the category of finite state automata with transducers as morphisms. Certain classes of subregular languages correspond to additional cohomological structures on the associated TQFTs. We also show that the construction generalizes to context-free grammars through a categorical version of the Chomsky–Schützenberger representation theorem, due to Melliès and Zeilberger. The corresponding TQFTs are then described as morphisms of colored operads on an operad of cobordisms with defects.

", "doi": "10.1016/j.geomphys.2026.105771", "issn": "0393-0440", "publisher": "Elsevier", "publication": "Journal of Geometry and Physics", "publication_date": "2026-04", "volume": "222", "pages": "105771" }, { "id": "authors:90s0t-khx62", "collection": "authors", "collection_id": "90s0t-khx62", "cite_using_url": "https://authors.library.caltech.edu/records/90s0t-khx62", "type": "article", "title": "Formal languages, spin systems, and quasicrystals", "author": [ { "family_name": "Fernandes", "given_name": "Francesca", "clpid": "Fernandes-Francesca" }, { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" } ], "abstract": "We present a categorical formalism for context-free languages with morphisms given by correspondences obtained from rational transductions. We show that D0L-systems are a special case of the correspondences that define morphisms in this category. We construct a functorial mapping to aperiodic spin chains. We then generalize this construction to a class of mildly context sensitive grammars, the multiple-context-free grammars (MCFG), with a similar functorial mapping to spin systems in higher dimensions, with Boltzmann weights describing interacting spins on vertices of hypercubes. We show that a particular motivating example for this general construction is provided by the Korepin completely integrable model on the icosahedral quasicrystal, which we construct as the spin system associated to a multiple-context-free grammar describing the geometry of the Ammann planes quasilattice. We review the main properties of this spin system, including solvability, bulk free energy, and criticality, based on results of Baxter and the known relation to the Zamolodchikov tetrahedron equation, and we show that the latter has a generalization for the Boltzmann weights on hypercubes of the spin systems associated to more general MCFGs in terms of two dual cubulations of the n-simplex. We formulate analogous questions about bulk free energy and criticality for our construction of spin systems.", "doi": "10.1016/j.geomphys.2025.105568", "issn": "0393-0440", "publisher": "Elsevier", "publication": "Journal of Geometry and Physics", "publication_date": "2025-10", "volume": "216", "pages": "105568" }, { "id": "authors:mmvj8-7gt83", "collection": "authors", "collection_id": "mmvj8-7gt83", "cite_using_url": "https://authors.library.caltech.edu/records/mmvj8-7gt83", "type": "article", "title": "Log concavity of the Grothendieck class of M\u2080,n", "author": [ { "family_name": "Aluffi", "given_name": "Paolo", "orcid": "0000-0002-8249-685X" }, { "family_name": "Chen", "given_name": "Stephanie" }, { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" } ], "abstract": "

Using a known recursive formula for the Grothendieck classes of the moduli spaces M\u2080,n, we prove that they satisfy an asymptotic form of ultra-log-concavity as polynomials in the Lefschetz class. We also observe that these polynomials are γ-positive. Both properties, along with numerical evidence, support the conjecture that these polynomials only have real zeros. This conjecture may be viewed as a particular case of a possible extension of a conjecture of Ferroni-Schröter and Huh on Hilbert series of Chow rings of matroids. We prove asymptotic ultra-log-concavity by studying differential equations obtained from the recursion, whose solutions are the generating functions of the individual Betti numbers of M\u2080,n. We obtain a rather complete description of these generating functions, determining their asymptotic behavior; their dominant term is controlled by the coefficients of the Lambert W function. The γ-positivity property follows directly from the recursion, extending the argument of Ferroni et al. proving γ-positivity for the Hilbert series of the Chow ring of matroids.

", "doi": "10.1007/s10801-025-01428-0", "issn": "0925-9899", "publisher": "Springer Science and Business Media LLC", "publication": "Journal of Algebraic Combinatorics", "publication_date": "2025-06", "series_number": "4", "volume": "61", "issue": "4", "pages": "47" }, { "id": "authors:fjd86-gk764", "collection": "authors", "collection_id": "fjd86-gk764", "cite_using_url": "https://authors.library.caltech.edu/records/fjd86-gk764", "type": "article", "title": "Homotopy Theoretic and Categorical Models of Neural Information Networks", "author": [ { "family_name": "Manin", "given_name": "Yuri I.", "clpid": "Manin-Yuri-I" }, { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" } ], "abstract": "

In this paper we develop a novel mathematical formalism for the modeling of neural information networks endowed with additional structure in the form of assignments of resources, either computational or metabolic or informational. The starting point for this construction is the notion of summing functors and of Segal's Gamma-spaces in homotopy theory. The main results in this paper include functorial assignments of concurrent/distributed computing architectures and associated binary codes to networks and their subsystems, a categorical form of the Hopfield network dynamics, which recovers the usual Hopfield equations when applied to a suitable category of weighted codes, a functorial assignment to networks of corresponding information structures and information cohomology, and a cohomological version of integrated information.

", "doi": "10.46298/compositionality-6-4", "issn": "2631-4444", "publisher": "Centre pour la Communication Scientifique Directe (CCSD)", "publication": "Compositionality", "publication_date": "2024-09-06", "series_number": "4", "volume": "6", "issue": "4", "pages": "14135" }, { "id": "authors:gqw1x-yeh18", "collection": "authors", "collection_id": "gqw1x-yeh18", "cite_using_url": "https://authors.library.caltech.edu/records/gqw1x-yeh18", "type": "article", "title": "Quantum statistical mechanics and the boundary of modular curves", "author": [ { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" }, { "family_name": "Panangaden", "given_name": "Jane", "orcid": "0000-0002-6214-2130", "clpid": "Panangaden-Jane" } ], "abstract": "
\n\n\n

The theory of limiting modular symbols provides a noncommutative geometric model of the boundary of modular curves that includes irrational points in addition to cusps. A noncommutative space associated to this boundary is constructed, as part of a family of noncommutative spaces associated to different continued fractions algorithms, endowed with the structure of a quantum statistical mechanical system. Two special cases of this family of quantum systems can be interpreted as a boundary of the system associated to the Shimura variety of GL2 and an analog for SL2. The structure of equilibrium states for this family of systems is discussed. In the geometric cases, the ground states evaluated on boundary arithmetic elements are given by pairings of cusp forms and limiting modular symbols.

\n\n
", "doi": "10.1063/5.0179805", "issn": "0022-2488", "publisher": "American Institute of Physics", "publication": "Journal of Mathematical Physics", "publication_date": "2024-04", "series_number": "4", "volume": "65", "issue": "4", "pages": "042104" }, { "id": "authors:pdjw5-v8b56", "collection": "authors", "collection_id": "pdjw5-v8b56", "cite_using_url": "https://authors.library.caltech.edu/records/pdjw5-v8b56", "type": "article", "title": "Motives of melonic graphs", "author": [ { "family_name": "Aluffi", "given_name": "Paolo", "orcid": "0000-0002-8249-685X", "clpid": "Aluffi-Paolo" }, { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" }, { "family_name": "Qaisar", "given_name": "Waleed", "clpid": "Qaisar-Waleed" } ], "abstract": "

We investigate recursive relations for the Grothendieck classes of the affine graph hypersurface complements of melonic graphs. We compute these classes explicitly for several families of melonic graphs, focusing on the case of graphs with valence-4 internal vertices, relevant to CTKT tensor models. The results hint at a complex and interesting structure in terms of divisibility relations or nontrivial relations between classes of graphs in different families. Using the recursive relations, we prove that the Grothendieck classes of all melonic graphs are positive as polynomials in the class of the moduli space M_(0,4). We also conjecture that the corresponding polynomials are log-concave, on the basis of hundreds of explicit computations.

", "doi": "10.4171/AIHPD/156", "issn": "2308-5835", "publisher": "European Mathematical Society", "publication": "Annales de l'Institut Henri Poincar\u00e9. D. Combinatorics, physics and their interactions", "publication_date": "2023-07-25", "series_number": "3", "volume": "10", "issue": "3", "pages": "503-554" }, { "id": "authors:7wfn5-g5k36", "collection": "authors", "collection_id": "7wfn5-g5k36", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20230526-436673000.11", "type": "article", "title": "Moufang patterns and geometry of information", "author": [ { "family_name": "Combe", "given_name": "No\u00e9mie", "orcid": "0000-0002-8038-3683", "clpid": "Combe-No\u00e9mie" }, { "family_name": "Manin", "given_name": "Yuri I.", "clpid": "Manin-Yuri-I" }, { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" } ], "abstract": "Technology of data collection and information transmission is based on various mathematical models of encoding. The words \"Geometry of information\" refer to such models, whereas the words \"Moufang patterns\" refer to various sophisticated symmetries appearing naturally in such models.\n\nIn this paper, we show that the symmetries of spaces of probability distributions, endowed with their canonical Riemannian metric of information geometry, have the structure of a commutative Moufang loop. We also show that the F-manifold structure on the space of probability distribution can be described in terms of differential \n-webs and Malcev algebras. We then present a new construction of (non-commutative) Moufang loops associated to almost-symplectic structures over finite fields, and use them to construct a new class of code loops with associated quantum error-correcting codes and networks of perfect tensors.", "doi": "10.4310/pamq.2023.v19.n1.a7", "issn": "1558-8599", "publisher": "International Press of Boston", "publication": "Pure and Applied Mathematics Quarterly", "publication_date": "2023-04-03", "series_number": "1", "volume": "19", "issue": "1", "pages": "149-189" }, { "id": "authors:fk795-afh19", "collection": "authors", "collection_id": "fk795-afh19", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20220705-671988000", "type": "article", "title": "Fractality in cosmic topology models with spectral action gravity", "author": [ { "family_name": "Guicardi", "given_name": "Pedro", "clpid": "Guicardi-Pedro" }, { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" } ], "abstract": "We consider cosmological models based on the spectral action formulation of (modified) gravity. We analyze the coupled effects, in this model, of the presence of nontrivial cosmic topology and of fractality in the large scale structure of spacetime. We show that the topology constrains the possible fractal structures, and in turn the correction terms to the spectral action due to fractality distinguish the various cosmic topology candidates, with effects detectable in a slow-roll inflation scenario, through the power spectra of the scalar and tensor fluctuations. We also discuss explicit effects of the presence of fractal structures on the gravitational waves equations.", "doi": "10.1088/1361-6382/ac7d8c", "issn": "0264-9381", "publisher": "IOP", "publication": "Classical and Quantum Gravity", "publication_date": "2022-08-18", "series_number": "16", "volume": "39", "issue": "16", "pages": "Art. No. 165007" }, { "id": "authors:979vv-4rp38", "collection": "authors", "collection_id": "979vv-4rp38", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20210825-184625013", "type": "article", "title": "Geometry of information: Classical and quantum aspects", "author": [ { "family_name": "Combe", "given_name": "No\u00e9mie", "clpid": "Combe-No\u00e9mie" }, { "family_name": "Manin", "given_name": "Yuri I.", "clpid": "Manin-Yuri-I" }, { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" } ], "abstract": "In this article, we describe various aspects of categorification of the structures, appearing in information theory. These aspects include probabilistic models both of classical and quantum physics, emergence of F\u2013manifolds, and motivic enrichments.", "doi": "10.1016/j.tcs.2021.10.020", "issn": "0304-3975", "publisher": "Elsevier", "publication": "Theoretical Computer Science", "publication_date": "2022-03-24", "volume": "908", "pages": "2-27" }, { "id": "authors:rdzzb-z5f98", "collection": "authors", "collection_id": "rdzzb-z5f98", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20190115-160601117", "type": "article", "title": "Nonarchimedean holographic entropy from networks of perfect tensors", "author": [ { "family_name": "Heydeman", "given_name": "Matthew", "orcid": "0000-0001-7033-9075", "clpid": "Heydeman-Matthew" }, { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" }, { "family_name": "Parikh", "given_name": "Sarthak", "orcid": "0000-0002-5831-3873", "clpid": "Parikh-Sarthak" }, { "family_name": "Saberi", "given_name": "Ingmar", "clpid": "Saberi-Ingmar-A" } ], "abstract": "We consider a class of holographic quantum error-correcting codes, built from perfect tensors in network configurations dual to Bruhat\u2013Tits trees and their quotients by Schottky groups corresponding to BTZ black holes. The resulting holographic states can be constructed in the limit of infinite network size. We obtain a p\u2011adic version of entropy which obeys a Ryu\u2013Takayanagi like formula for bipartite entanglement of connected or disconnected regions, in both genus-zero and genus-one p\u2011adic backgrounds, along with a Bekenstein\u2013Hawking-type formula for black hole entropy.We prove entropy inequalities obeyed by such tensor networks, such as subadditivity, strong subadditivity, and monogamy of mutual information (which is always saturated). In addition, we construct infinite classes of perfect tensors directly from semiclassical states in phase spaces over finite fields, generalizing the CRSS algorithm, and give Hamiltonians exhibiting these as vacua.", "doi": "10.4310/ATMP.2021.v25.n3.a2", "issn": "1095-0761", "publisher": "International Press", "publication": "Advances in Theoretical and Mathematical Physics", "publication_date": "2022-03-21", "series_number": "3", "volume": "25", "issue": "3", "pages": "591-721" }, { "id": "authors:yevya-b5221", "collection": "authors", "collection_id": "yevya-b5221", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20220909-227096000", "type": "article", "title": "Topological Analysis of Syntactic Structures", "author": [ { "family_name": "Port", "given_name": "Alexander", "orcid": "0000-0002-2862-6895", "clpid": "Port-Alexander" }, { "family_name": "Karidi", "given_name": "Taelin", "clpid": "Karidi-Taelin" }, { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" } ], "abstract": "We use the persistent homology method of topological data analysis and dimensional analysis techniques to study data of syntactic structures of world languages. We analyze relations between syntactic parameters in terms of dimensionality, of hierarchical clustering structures, and of non-trivial loops. We show there are relations that hold across language families and additional relations that are family-specific. We then analyze the trees describing the merging structure of persistent connected components for languages in different language families and we show that they partly correlate to historical phylogenetic trees but with significant differences. We also show the existence of interesting non-trivial persistent first homology groups in various language families. We give examples where explicit generators for the persistent first homology can be identified, some of which appear to correspond to homoplasy phenomena, while others may have an explanation in terms of historical linguistics, corresponding to known cases of syntactic borrowing across different language subfamilies.", "doi": "10.1007/s11786-021-00520-5", "issn": "1661-8270", "publisher": "Springer Verlag", "publication": "Mathematics in Computer Science", "publication_date": "2022-03", "series_number": "1", "volume": "16", "issue": "1", "pages": "Art. No. 2" }, { "id": "authors:4ygnq-sbn35", "collection": "authors", "collection_id": "4ygnq-sbn35", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20210415-115930614", "type": "article", "title": "Phylogenetics of Indo-European Language Families via an Algebro-Geometric Analysis of Their Syntactic Structures", "author": [ { "family_name": "Shu", "given_name": "Kevin", "clpid": "Shu-Kevin" }, { "family_name": "Ortegaray", "given_name": "Andrew", "clpid": "Ortegaray-Andrew" }, { "family_name": "Berwick", "given_name": "Robert C.", "orcid": "0000-0002-1061-1871", "clpid": "Berwick-Robert-C" }, { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" } ], "abstract": "Using Phylogenetic Algebraic Geometry, we analyze computationally the phylogenetic tree of subfamilies of the Indo-European language family, using data of syntactic structures. The two main sources of syntactic data are the SSWL database and Longobardi's recent data of syntactic parameters. We compute phylogenetic invariants and estimates of the Euclidean distance functions for two sets of Germanic languages, a set of Romance languages, a set of Slavic languages and a set of early Indo-European languages, and we compare the results with what is known through historical linguistics.", "doi": "10.1007/s11786-021-00507-2", "issn": "1661-8270", "publisher": "Springer", "publication": "Mathematics in Computer Science", "publication_date": "2021-12", "series_number": "4", "volume": "15", "issue": "4", "pages": "803-857" }, { "id": "authors:zjqcj-8vr56", "collection": "authors", "collection_id": "zjqcj-8vr56", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20210908-171121986", "type": "article", "title": "Functor of points and height functions for noncommutative Arakelov geometry", "author": [ { "family_name": "Lima", "given_name": "Alicia", "clpid": "Lima-Alicia" }, { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" } ], "abstract": "We propose a notion of functor of points for noncommutative spaces, valued in categories of bimodules, and endowed with an action functional determined by a notion of hermitian structures and height functions, modeled on an interpretation of the classical functor of points as a physical sigma model. We discuss different choices of such height functions, based on different notions of volumes and traces, including one based on the Hattori-Stallings rank. We show that the height function determines a dynamical time evolution on an algebra of observables associated to our functor of points. We focus in particular the case of noncommutative arithmetic curves, where the relevant algebras are sums of matrix algebras over division algebras over number fields, and we discuss a more general notion of noncommutative arithmetic spaces in higher dimensions, where our approach suggests an interpretation of the Jones index as a height function.", "doi": "10.1016/j.geomphys.2021.104337", "issn": "0393-0440", "publisher": "Elsevier", "publication": "Journal of Geometry and Physics", "publication_date": "2021-11", "volume": "169", "pages": "Art. No. 104337" }, { "id": "authors:a1a3t-zxy51", "collection": "authors", "collection_id": "a1a3t-zxy51", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20210818-171944955", "type": "article", "title": "Gluing Non-commutative Twistor Spaces", "author": [ { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" }, { "family_name": "Penrose", "given_name": "Roger", "clpid": "Penrose-Roger" } ], "abstract": "We describe a general procedure, based on Gerstenhaber\u2013Schack complexes, for extending to quantized twistor spaces the Donaldson\u2013Friedman gluing of twistor spaces via deformation theory of singular spaces. We consider in particular various possible quantizations of twistor spaces that leave the underlying spacetime manifold classical, including the geometric quantization of twistor spaces originally constructed by the second author, as well as some variants based on non-commutative geometry. We discuss specific aspects of the gluing construction for these different quantization procedures.", "doi": "10.1093/qmath/haab024", "issn": "0033-5606", "publisher": "Oxford University Press", "publication": "Quarterly Journal of Mathematics", "publication_date": "2021-06", "series_number": "1-2", "volume": "72", "issue": "1-2", "pages": "417-454" }, { "id": "authors:8gctx-1rp35", "collection": "authors", "collection_id": "8gctx-1rp35", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20210818-171348862", "type": "article", "title": "Anyon Networks from Geometric Models of Matter", "author": [ { "family_name": "Atiyah", "given_name": "Michael", "clpid": "Atiyah-Michael" }, { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" } ], "abstract": "This paper, completed in its present form by the second author after the first author passed away in 2019, describes an intended continuation of the previous joint work on anyons in geometric models of matter. This part outlines a construction of anyon tensor networks based on four-dimensional orbifold geometries and braid representations associated with surface-braids defined by multisections of the orbifold normal bundle of the surface of orbifold points.", "doi": "10.1093/qmath/haab004", "issn": "0033-5606", "publisher": "Oxford University Press", "publication": "Quarterly Journal of Mathematics", "publication_date": "2021-06", "series_number": "1-2", "volume": "72", "issue": "1-2", "pages": "717-733" }, { "id": "authors:zk5t7-7fb65", "collection": "authors", "collection_id": "zk5t7-7fb65", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20200518-092511147", "type": "article", "title": "Quantum Statistical Mechanics of the Absolute Galois Group", "author": [ { "family_name": "Manin", "given_name": "Yuri I.", "clpid": "Manin-Y-I" }, { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" } ], "abstract": "We present possible extensions of the quantum statistical mechanical formulation of class field theory to the non-abelian case, based on the action of the absolute Galois group on Grothendieck's dessins d'enfant, the embedding in the Grothendieck-Teichm\u00fcller group, and the Drinfeld-Ihara involution.", "doi": "10.3842/sigma.2020.038", "issn": "1815-0659", "publisher": "Institute of Mathematics of National Academy of Sciences of Ukraine", "publication": "Symmetry, Integrability and Geometry, Methods and Applications (SIGMA)", "publication_date": "2020-05-05", "volume": "16", "pages": "Art. No. 038" }, { "id": "authors:53wvv-vm244", "collection": "authors", "collection_id": "53wvv-vm244", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20200211-084519645", "type": "article", "title": "Bell Polynomials and Brownian Bridge in Spectral Gravity Models on Multifractal Robertson\u2013Walker Cosmologies", "author": [ { "family_name": "Fathizadeh", "given_name": "Farzad", "orcid": "0000-0002-7863-4009", "clpid": "Fathizadeh-Farzad" }, { "family_name": "Kafkoulis", "given_name": "Yeorgia", "clpid": "Kafkoulis-Yeorgia" }, { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" } ], "abstract": "We obtain an explicit formula for the full expansion of the spectral action on Robertson\u2013Walker spacetimes, expressed in terms of Bell polynomials, using Brownian bridge integrals and the Feynman\u2013Kac formula. We then apply this result to the case of multifractal Packed Swiss Cheese Cosmology models obtained from an arrangement of Robertson\u2013Walker spacetimes along an Apollonian sphere packing. Using Mellin transforms, we show that the asymptotic expansion of the spectral action contains the same terms as in the case of a single Robertson\u2013Walker spacetime, but with zeta-regularized coefficients, given by values at integers of the zeta function of the fractal string of the radii of the sphere packing, as well as additional log-periodic correction terms arising from the poles (off the real line) of this zeta function.", "doi": "10.1007/s00023-020-00894-5", "issn": "1424-0637", "publisher": "Springer", "publication": "Annales Henri Poincar\u00e9", "publication_date": "2020-04", "series_number": "4", "volume": "21", "issue": "4", "pages": "1329-1382" }, { "id": "authors:7003j-mxf28", "collection": "authors", "collection_id": "7003j-mxf28", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20191205-092818459", "type": "article", "title": "Nori Diagrams and Persistent Homology", "author": [ { "family_name": "Manin", "given_name": "Yuri I.", "clpid": "Manin-Y-I" }, { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" } ], "abstract": "Recently, it was found that there is a remarkable intuitive similarity between studies in theoretical computer science dealing with large data sets on the one hand, and categorical methods of topology and geometry in pure mathematics, on the other. In this article, we treat the key notion of persistency from computer science in the algebraic geometric context involving Nori motivic constructions and related methods. We also discuss model structures for persistent topology.", "doi": "10.1007/s11786-019-00422-7", "issn": "1661-8270", "publisher": "Springer", "publication": "Mathematics in Computer Science", "publication_date": "2020-03", "volume": "14", "pages": "77-102" }, { "id": "authors:rtzsw-gpe56", "collection": "authors", "collection_id": "rtzsw-gpe56", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20200518-094042414", "type": "article", "title": "Holographic Codes on Bruhat-Tits buildings and Drinfeld Symmetric Spaces", "author": [ { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" } ], "abstract": "This paper is based on the author's talk at the Arbeitstagung 2017. It discusses some general approaches to the construction of classical and quantum holographic codes on Bruhat\u2013Tits trees and buildings and on Drinfeld symmetric spaces, in the context of the p-adic AdS/CFT correspondence.", "doi": "10.4310/PAMQ.2020.v16.n1.a1", "issn": "1558-8602", "publisher": "International Press", "publication": "Pure and Applied Mathematics Quarterly", "publication_date": "2020-02-06", "series_number": "1", "volume": "16", "issue": "1", "pages": "1-33" }, { "id": "authors:dehe7-2w237", "collection": "authors", "collection_id": "dehe7-2w237", "cite_using_url": "https://authors.library.caltech.edu/records/dehe7-2w237", "type": "article", "title": "Adinkras, Dessins, Origami, and Supersymmetry Spectral Triples", "author": [ { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" }, { "family_name": "Zolman", "given_name": "Nick", "clpid": "Zolman-N" } ], "abstract": "
\n
\n

We investigate the spectral geometry and spectral action functionals associated to 1D Supersymmetry Algebras, using the classification of these superalgebras in terms of Adinkra graphs and the construction of associated dessin d’enfant and origami curves. The resulting spectral action functionals are computed in terms of the Selberg (super) trace formula and of a Poisson summation formula, respectively.

\n
\n
", "doi": "10.1134/S2070046619030051", "issn": "2070-0474", "publisher": "Springer", "publication": "p-Adic Numbers, Ultrametric Analysis and Applications", "publication_date": "2019-07", "series_number": "3", "volume": "11", "issue": "3", "pages": "223-247" }, { "id": "authors:mqmcj-qeh56", "collection": "authors", "collection_id": "mqmcj-qeh56", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20190711-104459099", "type": "article", "title": "Asymptotic bounds for spherical codes", "author": [ { "family_name": "Manin", "given_name": "Yu. I.", "clpid": "Manin-Yu-I" }, { "family_name": "Marcolli", "given_name": "M.", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" } ], "abstract": "The set of all error-correcting codes C over a fixed finite alphabet F of cardinality q determines the set of code points in the unit square [0,1]^2 with coordinates (R(C), \u03b4(C)):= (relative transmission rate, relative minimal distance). The central problem of the theory of such codes consists in maximising simultaneously the transmission rate of the code and the relative minimum Hamming distance between two different code words. The classical approach to this problem explored in vast literature consists in inventing explicit constructions of \"good codes\" and comparing new classes of codes with earlier ones.\nA less classical approach studies the geometry of the whole set of code points (R, \u03b4) (with q fixed), at first independently of its computability properties, and only afterwards turning to problems of computability, analogies with statistical physics, and so on.\nThe main purpose of this article consists in extending this latter strategy to the domain of spherical codes.", "doi": "10.1070/im8739", "issn": "1064-5632", "publisher": "IOP Publishing", "publication": "Izvestiya: Mathematics", "publication_date": "2019-06", "series_number": "3", "volume": "83", "issue": "3", "pages": "540-564" }, { "id": "authors:pdq5s-axw36", "collection": "authors", "collection_id": "pdq5s-axw36", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20190222-112740050", "type": "article", "title": "Gamma spaces and information", "author": [ { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" } ], "abstract": "We investigate the role of Segal's Gamma-spaces in the context of classical and quantum information, based on categories of finite probabilities with stochastic maps and density matrices with quantum channels. The information loss functional extends to the setting of probabilistic Gamma-spaces considered here. The Segal construction of connective spectra from Gamma-spaces can be used in this setting to obtain spectra associated to certain categories of gapped systems.", "doi": "10.1016/j.geomphys.2019.02.004", "issn": "0393-0440", "publisher": "Elsevier", "publication": "Journal of Geometry and Physics", "publication_date": "2019-06", "volume": "140", "pages": "26-55" }, { "id": "authors:7gg35-x8b57", "collection": "authors", "collection_id": "7gg35-x8b57", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20170712-082102987", "type": "article", "title": "Reconstructing global fields from dynamics in the abelianized Galois group", "author": [ { "family_name": "Cornelissen", "given_name": "Gunther", "clpid": "Cornelissen-G" }, { "family_name": "Li", "given_name": "Xin", "clpid": "Li-Xin" }, { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" }, { "family_name": "Smit", "given_name": "Harry", "orcid": "0000-0001-9547-8282", "clpid": "Smit-H" } ], "abstract": "We study a dynamical system induced by the Artin reciprocity map for a global field. We translate the conjugacy of such dynamical systems into various arithmetical properties that are equivalent to field isomorphism, relating it to anabelian geometry.", "doi": "10.1007/s00029-019-0469-8", "issn": "1022-1824", "publisher": "Springer", "publication": "Selecta Mathematica - New Series", "publication_date": "2019-06", "series_number": "6", "volume": "25", "issue": "6", "pages": "Art. No. 24" }, { "id": "authors:wtpdn-47391", "collection": "authors", "collection_id": "wtpdn-47391", "cite_using_url": "https://authors.library.caltech.edu/records/wtpdn-47391", "type": "article", "title": "Motivic Information", "author": [ { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" } ], "abstract": "
\n
\n

We introduce notions of information/entropy and information loss associated to exponentiable motivic measures. We show that they satisfy appropriate analogs to the Khinchin-type properties that characterize information loss in the context of measures on finite sets.

\n
\n
", "doi": "10.1007/s40574-018-0167-z", "issn": "1972-6724", "publisher": "Springer", "publication": "Bollettino dell'Unione Matematica Italiana", "publication_date": "2019-03-10", "series_number": "1-2", "volume": "12", "issue": "1-2", "pages": "19-41" }, { "id": "authors:tyfa7-r1013", "collection": "authors", "collection_id": "tyfa7-r1013", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20170712-082510174", "type": "article", "title": "Characterization of global fields by Dirichlet L-series", "author": [ { "family_name": "Cornelissen", "given_name": "Gunther", "clpid": "Cornelissen-Gunther" }, { "family_name": "de Smit", "given_name": "Bart", "clpid": "de-Smit-Bart" }, { "family_name": "Li", "given_name": "Xin", "clpid": "Li-Xin" }, { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" }, { "family_name": "Smit", "given_name": "Harry", "orcid": "0000-0001-9547-8282", "clpid": "Smit-Harry" } ], "abstract": "We prove that two global fields are isomorphic if and only if there is an isomorphism of groups of Dirichlet characters that preserves L-series.", "doi": "10.1007/s40993-018-0143-9", "issn": "2522-0160", "publisher": "Springer Nature", "publication": "Research in Number Theory", "publication_date": "2019-03", "series_number": "1", "volume": "5", "issue": "1", "pages": "Art. No. 7" }, { "id": "authors:j57ra-ax211", "collection": "authors", "collection_id": "j57ra-ax211", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20170712-083350725", "type": "article", "title": "Feynman quadrics-motive of the massive sunset graph", "author": [ { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" }, { "family_name": "Tabuada", "given_name": "Gon\u00e7alo", "clpid": "Tabuada-G" } ], "abstract": "We prove that the Feynman quadrics-motive of the massive sunset graph is \"generically\" not mixed-Tate. Moreover, we explicitly describe its \"extra\" complexity in terms of a Prym variety.", "doi": "10.1016/j.jnt.2018.06.001", "issn": "0022-314X", "publisher": "Elsevier", "publication": "Journal of Number Theory", "publication_date": "2019-02", "volume": "195", "pages": "159-183" }, { "id": "authors:xcfv5-3k826", "collection": "authors", "collection_id": "xcfv5-3k826", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20170712-105237127", "type": "article", "title": "Modular forms in the spectral action of Bianchi IX gravitational instantons", "author": [ { "family_name": "Fan", "given_name": "Wentao", "clpid": "Fan-Wentao" }, { "family_name": "Fathizadeh", "given_name": "Farzad", "orcid": "0000-0002-7863-4009", "clpid": "Fathizadeh-F" }, { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" } ], "abstract": "We prove a modularity property for the heat kernel and the Seeley-deWitt coefficients of the heat kernel expansion for the Dirac-Laplacian on the Bianchi IX gravitational instantons. We prove, via an isospectrality result for the Dirac operators, that each term in the expansion is a vector-valued modular form, with an associated ordinary (meromorphic) modular form of weight 2. We discuss explicit examples related to well known modular forms. Our results show the existence of arithmetic structures in Euclidean gravity models based on the spectral action functional.", "doi": "10.1007/JHEP01(2019)234", "issn": "1126-6708", "publisher": "Springer", "publication": "Journal of High Energy Physics", "publication_date": "2019-01", "series_number": "1", "volume": "2019", "issue": "1", "pages": "Art. No. 234" }, { "id": "authors:53g70-x7774", "collection": "authors", "collection_id": "53g70-x7774", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20181101-083114372", "type": "article", "title": "Motives and periods in Bianchi IX gravity models", "author": [ { "family_name": "Fan", "given_name": "Wentao", "clpid": "Fan-Wentao" }, { "family_name": "Fathizadeh", "given_name": "Farzad", "orcid": "0000-0002-7863-4009", "clpid": "Fathizadeh-F" }, { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" } ], "abstract": "We show that, when considering the anisotropic scaling factors and their derivatives as affine variables, the coefficients of the heat-kernel expansion of the Dirac\u2013Laplacian on SU(2) Bianchi IX metrics are algebro-geometric periods of motives of complements in affine spaces of unions of quadrics and hyperplanes. We show that the motives are mixed Tate and we provide an explicit computation of their Grothendieck classes.", "doi": "10.1007/s11005-018-1096-6", "issn": "0377-9017", "publisher": "Springer", "publication": "Letters in Mathematical Physics", "publication_date": "2018-12", "series_number": "12", "volume": "108", "issue": "12", "pages": "2729-2747" }, { "id": "authors:z0v9j-1z604", "collection": "authors", "collection_id": "z0v9j-1z604", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20170712-145442721", "type": "article", "title": "Some remarks concerning Voevodsky's nilpotence conjecture", "author": [ { "family_name": "Bernardara", "given_name": "Marcello", "clpid": "Bernardara-Marcello" }, { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" }, { "family_name": "Tabuada", "given_name": "Gon\u00e7alo", "clpid": "Tabuada-Gon\u00e7alo" } ], "abstract": "In this article we extend Voevodsky's nilpotence conjecture from smooth projective schemes to the broader setting of smooth proper dg categories. Making use of this noncommutative generalization, we then address Voevodsky's original conjecture in the following cases: quadric fibrations, intersection of quadrics, linear sections of Grassmannians, linear sections of determinantal varieties, homological projective duals, and Moishezon manifolds.", "doi": "10.1515/crelle-2015-0068", "issn": "0075-4102", "publisher": "De Gruyter", "publication": "Journal f\u00fcr die Reine und Angewandte Mathematik", "publication_date": "2018-05", "volume": "738", "pages": "299-312" }, { "id": "authors:q2w3t-6td28", "collection": "authors", "collection_id": "q2w3t-6td28", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20170712-111147439", "type": "article", "title": "Persistent Topology of Syntax", "author": [ { "family_name": "Port", "given_name": "Alexander", "clpid": "Port-A" }, { "family_name": "Gheorghita", "given_name": "Iulia", "clpid": "Gheorghita-I" }, { "family_name": "Guth", "given_name": "Daniel", "clpid": "Guth-D" }, { "family_name": "Clark", "given_name": "John M.", "clpid": "Clark-J-M" }, { "family_name": "Liang", "given_name": "Crystal", "clpid": "Liang-Crystal" }, { "family_name": "Dasu", "given_name": "Shival", "clpid": "Dasu-S" }, { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" } ], "abstract": "We study the persistent homology of a data set of syntactic parameters of world languages. We show that, while homology generators behave erratically over the whole data set, non-trivial persistent homology appears when one restricts to specific language families. Different families exhibit different persistent homology. We focus on the cases of the Indo-European and the Niger\u2013Congo families, for which we compare persistent homology over different cluster filtering values. The persistent components appear to correspond to linguistic subfamilies, while the meaning, in historical linguistic terms, of the presence of persistent generators of the first homology is more mysterious. We investigate the possible significance of the persistent first homology generator that we find in the Indo-European family. We show that it is not due to the Anglo-Norman bridge (which is a lexical, not syntactic phenomenon), but is related instead to the position of Ancient Greek and the Hellenic branch within the Indo-European phylogenetic network.", "doi": "10.1007/s11786-017-0329-x", "issn": "1661-8270", "publisher": "Springer Verlag", "publication": "Mathematics in Computer Science", "publication_date": "2018-03", "series_number": "1", "volume": "12", "issue": "1", "pages": "33-50" }, { "id": "authors:pqpa9-chj11", "collection": "authors", "collection_id": "pqpa9-chj11", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20160615-160129721", "type": "article", "title": "Tensor networks, p-adic fields, and algebraic curves: arithmetic and the AdS_3/CFT_2 correspondence", "author": [ { "family_name": "Heydeman", "given_name": "Matthew", "orcid": "0000-0001-7033-9075", "clpid": "Heydeman-Matthew" }, { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" }, { "family_name": "Saberi", "given_name": "Ingmar A.", "clpid": "Saberi-Ingmar-A" }, { "family_name": "Stoica", "given_name": "Bogdan", "clpid": "Stoica-B" } ], "abstract": "One of the many remarkable properties of conformal field theory in two dimensions is its connection to algebraic geometry. Since every compact Riemann surface is a projective algebraic curve, many constructions of interest in physics (which a priori depend on the analytic structure of the spacetime) can be formulated in purely algebraic language. This opens the door to interesting generalizations, obtained by taking another choice of field: for instance, the p-adics. We generalize the AdS/CFT correspondence according to this principle; the result is a formulation of holography in which the bulk geometry is discrete\u2014the Bruhat\u2013Tits tree for PGL(2,Qp)\u2014but the group of bulk isometries nonetheless agrees with that of boundary conformal transformations and is not broken by discretization. We suggest that this forms the natural geometric setting for tensor networks that have been proposed as models of bulk reconstruction via quantum error correcting codes; in certain cases, geodesics in the Bruhat\u2013Tits tree reproduce those constructed using quantum error correction. Other aspects of holography also hold: Standard holographic results for massive free scalar fields in a fixed background carry over to the tree, whose vertical direction can be interpreted as a renormalization-group scale for modes in the boundary CFT. Higher-genus bulk geometries (the BTZ black hole and its generalizations) can be understood straightforwardly in our setting, and the Ryu\u2013Takayanagi formula for the entanglement entropy appears naturally.", "doi": "10.4310/ATMP.2018.v22.n1.a4", "issn": "1095-0761", "publisher": "International Press", "publication": "Advances in Theoretical and Mathematical Physics", "publication_date": "2018", "series_number": "1", "volume": "22", "issue": "1", "pages": "93-176" }, { "id": "authors:204g5-emb23", "collection": "authors", "collection_id": "204g5-emb23", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20170712-091141446", "type": "article", "title": "Periods and motives in the spectral action of Robertson-Walker spacetimes", "author": [ { "family_name": "Fathizadeh", "given_name": "Farzad", "orcid": "0000-0002-7863-4009", "clpid": "Fathizadeh-F" }, { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" } ], "abstract": "We show that, when considering the scaling factor as an affine variable, the coefficients of the asymptotic expansion of the spectral action on a (Euclidean) Robertson\u2013Walker spacetime are periods of mixed Tate motives, involving relative motives of complements of unions of hyperplanes and quadric hypersurfaces and divisors given by unions of coordinate hyperplanes.", "doi": "10.1007/s00220-017-2991-x", "issn": "0010-3616", "publisher": "Springer", "publication": "Communications in Mathematical Physics", "publication_date": "2017-12", "series_number": "2", "volume": "356", "issue": "2", "pages": "641-671" }, { "id": "authors:pnnpa-h6826", "collection": "authors", "collection_id": "pnnpa-h6826", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20170712-084525536", "type": "article", "title": "q-deformations of statistical mechanical systems and motives over finite fields", "author": [ { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" }, { "family_name": "Zhi", "given_name": "Ren", "clpid": "Zhi-Ren" } ], "abstract": "We consider q-deformations of Witt rings, based on geometric operations on zeta functions of motives over finite fields, and we use these deformations to construct q-analogs of the Bost-Connes quantum statistical mechanical system. We show that the q-deformations obtained in this way can be related to Habiro ring constructions of analytic functions over F_1 and to categorifications of Bost-Connes systems.", "doi": "10.1134/S2070046617030049", "issn": "2070-0466", "publisher": "Springer", "publication": "p-Adic Numbers, Ultrametric Analysis and Applications", "publication_date": "2017-08-03", "series_number": "3", "volume": "9", "issue": "3", "pages": "204-227" }, { "id": "authors:j71yf-a4p48", "collection": "authors", "collection_id": "j71yf-a4p48", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20170712-085827096", "type": "article", "title": "Anyons in Geometric Models of Matter", "author": [ { "family_name": "Atiyah", "given_name": "Michael", "clpid": "Atiyah-M" }, { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" } ], "abstract": "We show that the \"geometric models of matter\" approach proposed by the first author can be used to construct models of anyon quasiparticles with fractional quantum numbers, using 4-dimensional edge-cone orbifold geometries with orbifold singularities along embedded 2-dimensional surfaces. The anyon states arise through the braid representation of surface braids wrapped around the orbifold singularities, coming from multisections of the orbifold normal bundle of the embedded surface. We show that the resulting braid representations can give rise to a universal quantum computer.", "doi": "10.1007/JHEP07(2017)076", "issn": "1126-6708", "publisher": "Springer", "publication": "Journal of High Energy Physics", "publication_date": "2017-07", "series_number": "07", "volume": "2017", "issue": "07", "pages": "Art. No. 076" }, { "id": "authors:6j17p-vfb40", "collection": "authors", "collection_id": "6j17p-vfb40", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20170707-065452085", "type": "article", "title": "Edge length dynamics on graphs with applications to p-adic AdS/CFT", "author": [ { "family_name": "Gubser", "given_name": "Steven S.", "clpid": "Gubser-S-S" }, { "family_name": "Heydeman", "given_name": "Matthew", "orcid": "0000-0001-7033-9075", "clpid": "Heydeman-Matthew" }, { "family_name": "Jepsen", "given_name": "Christian", "orcid": "0000-0002-1159-0574", "clpid": "Jepsen-C-B" }, { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" }, { "family_name": "Parikh", "given_name": "Sarthak", "orcid": "0000-0002-5831-3873", "clpid": "Parikh-Sarthak" }, { "family_name": "Saberi", "given_name": "Ingmar", "clpid": "Saberi-Ingmar-A" }, { "family_name": "Stoica", "given_name": "Bogdan", "clpid": "Stoica-B" }, { "family_name": "Trundy", "given_name": "Brian", "clpid": "Trundy-B" } ], "abstract": "We formulate a Euclidean theory of edge length dynamics based on a notion of Ricci curvature on graphs with variable edge lengths. In order to write an explicit form for the discrete analog of the Einstein-Hilbert action, we require that the graph should either be a tree or that all its cycles should be sufficiently long. The infinite regular tree with all edge lengths equal is an example of a graph with constant negative curvature, providing a connection with p-adic AdS/CFT, where such a tree takes the place of anti-de Sitter space. We compute simple correlators of the operator holographically dual to edge length fluctuations. This operator has dimension equal to the dimension of the boundary, and it has some features in common with the stress tensor.", "doi": "10.1007/JHEP06(2017)157", "issn": "1029-8479", "publisher": "Springer", "publication": "Journal of High Energy Physics", "publication_date": "2017-06-30", "series_number": "6", "volume": "2017", "issue": "6", "pages": "1-35" }, { "id": "authors:5tgcv-gc327", "collection": "authors", "collection_id": "5tgcv-gc327", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20170417-110402207", "type": "article", "title": "Quantum statistical mechanics in arithmetic topology", "author": [ { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" }, { "family_name": "Xu", "given_name": "Yujie", "clpid": "Xu-Yujie" } ], "abstract": "This paper provides a construction of a quantum statistical mechanical system associated to knots in the 33-sphere and cyclic branched coverings of the 33-sphere, which is an analog, in the sense of arithmetic topology, of the Bost\u2013Connes system, with knots replacing primes, and cyclic branched coverings of the 33-sphere replacing abelian extensions of the field of rational numbers. The operator algebraic properties of this system differ significantly from the Bost\u2013Connes case, due to the properties of the action of the semigroup of knots on a direct limit of knot groups. The resulting algebra of observables is a noncommutative Bernoulli product. We describe the main properties of the associated quantum statistical mechanical system and of the relevant partition functions, which are obtained from simple knot invariants like genus and crossing number.", "doi": "10.1016/j.geomphys.2016.11.029", "issn": "0393-0440", "publisher": "Elsevier", "publication": "Journal of Geometry and Physics", "publication_date": "2017-04", "volume": "114", "pages": "153-183" }, { "id": "authors:g8752-7y730", "collection": "authors", "collection_id": "g8752-7y730", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20170531-104221597", "type": "article", "title": "Spectral action gravity and cosmological models", "author": [ { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" } ], "abstract": "This paper surveys recent work of the author and collaborators on cosmological models based on the spectral action functional of gravity. A more detailed presentation of the topics surveyed here will be available in a forthcoming book [1].", "doi": "10.1016/j.crhy.2017.03.001", "issn": "1631-0705", "publisher": "Elsevier Masson SAS", "publication": "Comptes Rendus Physique", "publication_date": "2017-03", "series_number": "3-4", "volume": "18", "issue": "3-4", "pages": "226-234" }, { "id": "authors:n97fn-5hz17", "collection": "authors", "collection_id": "n97fn-5hz17", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20170614-110933652", "type": "article", "title": "Syntactic Structures and Code Parameters", "author": [ { "family_name": "Shu", "given_name": "Kevin", "clpid": "Shu-Kevin" }, { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" } ], "abstract": "We assign binary and ternary error-correcting codes to the data of syntactic structures of world languages and we study the distribution of code points in the space of code parameters. We show that, while most codes populate the lower region approximating a superposition of Thomae functions, there is a substantial presence of codes above the Gilbert\u2013Varshamov bound and even above the asymptotic bound and the Plotkin bound. We investigate the dynamics induced on the space of code parameters by spin glass models of language change, and show that, in the presence of entailment relations between syntactic parameters the dynamics can sometimes improve the code. For large sets of languages and syntactic data, one can gain information on the spin glass dynamics from the induced dynamics in the space of code parameters.", "doi": "10.1007/s11786-017-0298-0", "issn": "1661-8270", "publisher": "Springer Verlag", "publication": "Mathematics in Computer Science", "publication_date": "2017-03", "series_number": "1", "volume": "11", "issue": "1", "pages": "79-90" }, { "id": "authors:b4p7m-8qn09", "collection": "authors", "collection_id": "b4p7m-8qn09", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20170913-081309497", "type": "article", "title": "Noncommutative geometry and particle physics [Book Review]", "author": [ { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" } ], "abstract": "The use of noncommutative geometry (NCG) as a tool for constructing particle physics models originated in the 1990s [9, 11]. The main idea can be heuristically\nregarded as similar to the idea of \"extra dimensions\" in String Theory, except for the fact that the nature and scope of these extra dimensions is quite different. In the\nNCG model one considers an \"almost commutative geometry\", which is a product (or locally a product in a more refined and more recent version [4]) of a four-dimensional\nspacetime manifold and a space of inner degrees of freedom, which is a \"finite\" noncommutative space, whose ring of functions is a sum of matrix algebras. According to the choice of this finite geometry, one obtains different\npossible particle contents for the resulting physics model. The physical content is expressed through an action functional, the spectral action [5], which is defined for\nmore general noncommutative spaces, in terms of the spectrum of a Dirac operator.", "doi": "10.1090/bull/1555", "issn": "0273-0979", "publisher": "American Mathematical Society", "publication": "Bulletin of the American Mathematical Society", "publication_date": "2017-01", "series_number": "1", "volume": "54", "issue": "1", "pages": "167-169" }, { "id": "authors:xfdda-wzy15", "collection": "authors", "collection_id": "xfdda-wzy15", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20170330-165516108", "type": "article", "title": "Bost\u2013Connes systems, categorification, quantum statistical mechanics, and Weil numbers", "author": [ { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" }, { "family_name": "Tabuada", "given_name": "Gon\u00e7alo", "clpid": "Tabuada-G" } ], "abstract": "In this article we develop a broad generalization of the classical Bost\u2013Connes system, where roots of unity are replaced by an algebraic datum consisting of an abelian group and a semi-group of endomorphisms. Examples include roots of unity, Weil restriction, algebraic numbers,Weil numbers, CM fields, germs, completion ofWeil numbers, etc. Making use of the Tannakian formalism, we categorify these algebraic data. For example, the categorification of roots of unity is given by a limit of orbit categories of Tate motives while the categorification of Weil numbers is given by Grothendieck's category of numerical motives over a finite field. To some of these algebraic data (e.g. roots of unity, algebraic numbers, Weil numbers, etc), we associate also a quantum statistical mechanical system with several remarkable properties, which generalize those of the classical Bost\u2013Connes system. The associated partition function, low temperature Gibbs states, and Galois action on zero-temperature states are then studied in detail. For example, we show that in the particular case of the Weil numbers the partition function and the low temperature Gibbs states can be described as series of polylogarithms.", "doi": "10.4171/JNCG/11-1-1", "issn": "1661-6952", "publisher": "European Mathematical Society", "publication": "Journal of Noncommutative Geometry", "publication_date": "2017", "series_number": "1", "volume": "11", "issue": "1", "pages": "1-49" }, { "id": "authors:p80nw-3tg76", "collection": "authors", "collection_id": "p80nw-3tg76", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20170712-130729557", "type": "article", "title": "Twisted index theory on orbifold symmetric products and the fractional quantum Hall effect", "author": [ { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" }, { "family_name": "Seipp", "given_name": "Kyle", "clpid": "Seipp-K" } ], "abstract": "We extend the noncommutative geometry model of the fractional quantum Hall effect, previously developed by Mathai and the first author, to orbifold symmetric products. It retains the same properties of quantization of the Hall conductance at integer multiples of the fractional Satake orbifold Euler characteristics. We show that it also allows for interesting composite fermions and anyon representations, and possibly for Laughlin type wave functions.", "doi": "10.4310/ATMP.2017.v21.n2.a3", "issn": "1095-0761", "publisher": "International Press", "publication": "Advances in Theoretical and Mathematical Physics", "publication_date": "2017", "series_number": "2", "volume": "21", "issue": "2", "pages": "451-501" }, { "id": "authors:j7bhm-3q764", "collection": "authors", "collection_id": "j7bhm-3q764", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20170124-100433482", "type": "article", "title": "Semantic Spaces", "author": [ { "family_name": "Manin", "given_name": "Yuri I.", "clpid": "Manin-Y-I" }, { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" } ], "abstract": "Any natural language can be considered as a tool for producing large databases (consisting of texts, written, or discursive). This tool for its description in turn requires other large databases (dictionaries, grammars etc.). Nowadays, the notion of database is associated with computer processing and computer memory. However, a natural language resides also in human brains and functions in human communication, from interpersonal to intergenerational one. We discuss in this survey/research paper mathematical, in particular geometric, constructions, which help to bridge these two worlds. In particular, in this paper we consider the Vector Space Model of semantics based on frequency matrices, as used in Natural Language Processing. We investigate underlying geometries, formulated in terms of Grassmannians, projective spaces, and flag varieties. We formulate the relation between vector space models and semantic spaces based on semic axes in terms of projectability of subvarieties in Grassmannians and projective spaces. We interpret Latent Semantics as a geometric flow on Grassmannians. We also discuss how to formulate G\u00e4rdenfors' notion of \"meeting of minds\" in our geometric setting.", "doi": "10.1007/s11786-016-0278-9", "issn": "1661-8270", "publisher": "Springer Verlag", "publication": "Mathematics in Computer Science", "publication_date": "2016-12", "series_number": "4", "volume": "10", "issue": "4", "pages": "459-477" }, { "id": "authors:sgp63-mg084", "collection": "authors", "collection_id": "sgp63-mg084", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20160105-102322403", "type": "article", "title": "Spectral Action Models of Gravity on Packed Swiss Cheese Cosmology", "author": [ { "family_name": "Ball", "given_name": "Adam", "clpid": "Ball-A" }, { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" } ], "abstract": "We present a model of (modified) gravity on spacetimes with fractal structure based on packing of spheres, which are (Euclidean) variants of the packed swiss cheese cosmology models. As the action functional for gravity we consider the spectral action of noncommutative geometry, and we compute its expansion on a space obtained as an Apollonian packing of three-dimensional spheres inside a four-dimensional ball. Using information from the zeta function of the Dirac operator of the spectral triple, we compute the leading terms in the asymptotic expansion of the spectral action. They consist of a zeta regularization of the divergent sum of the leading terms of the spectral actions of the individual spheres in the packing. This accounts for the contribution of points 1 and 3 in the dimension spectrum (as in the case of a 3-sphere). There is an additional term coming from the residue at the additional point in the real dimension spectrum that corresponds to the packing constant, as well as a series of fluctuations coming from log-periodic oscillations, created by the points of the dimension spectrum that are off the real line. These terms detect the fractality of the residue set of the sphere packing. We show that the presence of fractality influences the shape of the slow-roll potential for inflation, obtained from the spectral action. We also discuss the effect of truncating the fractal structure at a certain scale related to the energy scale in the spectral action.", "doi": "10.1088/0264-9381/33/11/115018", "issn": "0264-9381", "publisher": "IOP", "publication": "Classical and Quantum Gravity", "publication_date": "2016-06-09", "series_number": "11", "volume": "33", "issue": "11", "pages": "Art. No. 115018" }, { "id": "authors:ajrer-hfd63", "collection": "authors", "collection_id": "ajrer-hfd63", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20160420-165543640", "type": "article", "title": "Syntactic Parameters and a Coding Theory Perspective on Entropy and Complexity of Language Families", "author": [ { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" } ], "abstract": "We present a simple computational approach to assigning a measure of complexity and information/entropy to families of natural languages, based on syntactic parameters and the theory of error correcting codes. We associate to each language a binary string of syntactic parameters and to a language family a binary code, with code words the binary string associated to each language. We then evaluate the code parameters (rate and relative minimum distance) and the position of the parameters with respect to the asymptotic bound of error correcting codes and the Gilbert\u2013Varshamov bound. These bounds are, respectively, related to the Kolmogorov complexity and the Shannon entropy of the code and this gives us a computationally simple way to obtain estimates on the complexity and information, not of individual languages but of language families. This notion of complexity is related, from the linguistic point of view to the degree of variability of syntactic parameter across languages belonging to the same (historical) family.", "doi": "10.3390/e18040110", "issn": "1099-4300", "publisher": "MDPI", "publication": "Entropy", "publication_date": "2016-04-07", "series_number": "4", "volume": "18", "issue": "4", "pages": "Art. No. 110" }, { "id": "authors:msyc3-sq590", "collection": "authors", "collection_id": "msyc3-sq590", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20160505-082023887", "type": "article", "title": "KMS weights on higher rank buildings", "author": [ { "family_name": "Marcinek", "given_name": "J.", "clpid": "Marcinek-J" }, { "family_name": "Marcolli", "given_name": "M.", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" } ], "abstract": "We extend some of the results of Carey-Marcolli-Rennie on modular index invariants of Mumford curves to the case of higher rank buildings. We discuss notions of KMS weights on buildings, that generalize the construction of graph weights over graph C*-algebras.", "doi": "10.1134/S2070046616010040", "issn": "2070-0466", "publisher": "Springer", "publication": "P-Adic Numbers, Ultrametric Analysis, and Applications", "publication_date": "2016-01", "series_number": "1", "volume": "8", "issue": "1", "pages": "45-67" }, { "id": "authors:jtjgb-ezj24", "collection": "authors", "collection_id": "jtjgb-ezj24", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20160408-132805755", "type": "article", "title": "Noncommutative numerical motives, Tannakian structures, and motivic Galois groups", "author": [ { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" }, { "family_name": "Tabuada", "given_name": "Gon\u00e7alo", "clpid": "Tabuada-G" } ], "abstract": "In this article we further the study of noncommutative numerical motives, initiated in [30, 31]. By exploring the change-of-coefficients mechanism, we start by improving some of the main results of [30]. Then, making use of the notion of Schur-finiteness, we prove that the category NNum(k)_F of noncommutative numerical motives is (neutral) super-Tannakian. As in the commutative world, NNum(k)_F is not Tannakian. In order to solve this problem we promote periodic cyclic homology to a well-defined symmetric monoidal functor HP* on the category of noncommutative Chow motives. This allows us to introduce the correct noncommutative analogues C_(NC) and D_(NC) of Grothendieck's standard conjectures C and D. Assuming C_(NC), we prove that NNum(k)_F can be made into a Tannakian category NNum (k)_F by modifying its symmetry isomorphism constraints. By further assuming D_(NC), we neutralize the Tannakian category Num (k)_F using HP*. Via the (super-)Tannakian formalism, we then obtain well-defined noncommutative motivic Galois (super-)groups. Finally, making use of Deligne-Milne's theory of Tate triples, we construct explicit morphisms relating these noncommutative motivic Galois (super-)groups with the classical ones as suggested by Kontsevich.", "doi": "10.4171/JEMS/598", "issn": "1435-9855", "publisher": "European Mathematical Society", "publication": "Journal of the European Mathematical Society", "publication_date": "2016", "series_number": "3", "volume": "18", "issue": "3", "pages": "623-655" }, { "id": "authors:sdhnc-fgy26", "collection": "authors", "collection_id": "sdhnc-fgy26", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20170712-142940847", "type": "article", "title": "Rota-Baxter algebras, singular hypersurfaces, and renormalization on Kausz compactifications", "author": [ { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" }, { "family_name": "Ni", "given_name": "Xiang", "clpid": "Ni-Xiang" } ], "abstract": "We consider Rota-Baxter algebras of meromorphic forms with poles along a (singular) hypersurface in a smooth projective variety and the associated Birkhoff factorization for algebra homomorphisms from a commutative Hopf algebra. In the case of a normal crossings divisor, the Rota-Baxter structure simplifies considerably and the factorization becomes a simple pole subtraction. We apply this formalism to the unrenormalized momentum space Feynman amplitudes, viewed as (divergent) integrals in the complement of the determinant hypersurface. We lift the integral to the Kausz compactification of the general linear group, whose boundary divisor is normal crossings. We show that the Kausz compactification is a Tate motive and that the boundary divisor and the divisor that contains the boundary of the chain of integration are mixed Tate configurations. The regularization of the integrals that we obtain differs from the usual renormalization of physical Feynman amplitudes, and in particular it may give mixed Tate periods in some cases that have non-mixed Tate contributions when computed with other renormalization methods.", "doi": "10.5427/jsing.2016.15e", "issn": "1949-2006", "publisher": "Worldwide Center of Mathematics", "publication": "Journal of Singularities", "publication_date": "2016", "volume": "15", "pages": "80-117" }, { "id": "authors:8q30x-n0632", "collection": "authors", "collection_id": "8q30x-n0632", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20160301-093947103", "type": "article", "title": "Graph Grammars, Insertion Lie Algebras, and Quantum Field Theory", "author": [ { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" }, { "family_name": "Port", "given_name": "Alexander", "clpid": "Port-A" } ], "abstract": "Graph grammars extend the theory of formal languages in order to model distributed parallelism in theoretical computer science. We show here that to certain classes of context-free and context-sensitive graph grammars one can associate a Lie algebra, whose structure is reminiscent of the insertion Lie algebras of quantum field theory. We also show that the Feynman graphs of quantum field theories are graph languages generated by a theory dependent graph grammar.", "doi": "10.1007/s11786-015-0236-y", "issn": "1661-8270", "publisher": "Springer Verlag", "publication": "Mathematics in Computer Science", "publication_date": "2015-12", "series_number": "4", "volume": "9", "issue": "4", "pages": "391-408" }, { "id": "authors:fcebq-12z86", "collection": "authors", "collection_id": "fcebq-12z86", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20151016-150705985", "type": "article", "title": "Entropy algebras and Birkhoff factorization", "author": [ { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" }, { "family_name": "Tedeschi", "given_name": "Nicolas", "clpid": "Tedeschi-N" } ], "abstract": "We develop notions of Rota\u2013Baxter structures and associated Birkhoff factorizations, in the context of min-plus semirings and their thermodynamic deformations, including deformations arising from quantum information measures such as the von Neumann entropy. We consider examples related to Manin's renormalization and computation program, to Markov random fields and to counting functions and zeta functions of algebraic varieties.", "doi": "10.1016/j.geomphys.2015.07.018", "issn": "0393-0440", "publisher": "Elsevier", "publication": "Journal of Geometry and Physics", "publication_date": "2015-11", "volume": "97", "pages": "243-265" }, { "id": "authors:kz4x5-wm478", "collection": "authors", "collection_id": "kz4x5-wm478", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20151015-160216557", "type": "article", "title": "Potts models with magnetic field: Arithmetic, geometry, and computation", "author": [ { "family_name": "Dasu", "given_name": "Shival", "clpid": "Dasu-S" }, { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" } ], "abstract": "We give a sheaf theoretic interpretation of Potts models with external magnetic field, in terms of constructible sheaves and their Euler characteristics. We show that the polynomial countability question for the hypersurfaces defined by the vanishing of the partition function is affected by changes in the magnetic field: elementary examples suffice to see non-polynomially countable cases that become polynomially countable after a perturbation of the magnetic field. The same recursive formula for the Grothendieck classes, under edge-doubling operations, holds as in the case without magnetic field, but the closed formulae for specific examples like banana graphs differ in the presence of magnetic field. We give examples of computation of the Euler characteristic with compact support, for the set of real zeros, and find a similar exponential growth with the size of the graph. This can be viewed as a measure of topological and algorithmic complexity. We also consider the computational complexity question for evaluations of the polynomial, and show both tractable and NP-hard examples, using dynamic programming.", "doi": "10.1016/j.geomphys.2015.06.018", "issn": "0393-0440", "publisher": "Elsevier", "publication": "Journal of Geometry and Physics", "publication_date": "2015-11", "volume": "97", "pages": "14-24" }, { "id": "authors:1y3ee-5d087", "collection": "authors", "collection_id": "1y3ee-5d087", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20151005-140911430", "type": "article", "title": "Spectral Action for Bianchi Type-IX Cosmological Models", "author": [ { "family_name": "Fan", "given_name": "Wentao", "clpid": "Fan-Wentao" }, { "family_name": "Fathizadeh", "given_name": "Farzad", "orcid": "0000-0002-7863-4009", "clpid": "Fathizadeh-F" }, { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" } ], "abstract": "A rationality result previously proved for Robertson-Walker metrics is extended to a homogeneous anisotropic cosmological model, namely the Bianchi type-IX minisuperspace. It is shown that the Seeley-de Witt coefficients appearing in the expansion of the spectral action for the Bianchi type-IX geometry are expressed in terms of polynomials with rational coefficients in the cosmic evolution factors w_1(t), w_2(t), w)3(t), and their higher derivates with respect to time. We begin with the computation of the Dirac operator of this geometry and calculate the coefficients a_0 ,a_2 ,a_4 of the spectral action by using heat kernel methods and parametric pseudodifferential calculus. An efficient method is devised for computing the Seeley-de Witt coefficients of a geometry by making use of Wodzicki's noncommutative residue, and it is confirmed that the method checks out for the cosmological model studied in this article. The advantages of the new method are discussed, which combined with symmetries of the Bianchi type-IX metric, yield an elegant proof of the rationality result.", "doi": "10.1007/JHEP10(2015)085", "issn": "1126-6708", "publisher": "Springer", "publication": "Journal of High Energy Physics", "publication_date": "2015-10", "series_number": "10", "volume": "2015", "issue": "10", "pages": "Art. No. 085" }, { "id": "authors:648b1-q3x68", "collection": "authors", "collection_id": "648b1-q3x68", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20150616-152928658", "type": "article", "title": "Locality of Gravitational Systems from Entanglement of Conformal Field Theories", "author": [ { "family_name": "Lin", "given_name": "Jennifer", "clpid": "Lin-Jennifer" }, { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" }, { "family_name": "Ooguri", "given_name": "Hirosi", "orcid": "0000-0001-6021-3778", "clpid": "Ooguri-H" }, { "family_name": "Stoica", "given_name": "Bogdan", "clpid": "Stoica-B" } ], "abstract": "The Ryu-Takayanagi formula relates the entanglement entropy in a conformal field theory to the area of\na minimal surface in its holographic dual. We show that this relation can be inverted for any state in the\nconformal field theory to compute the bulk stress-energy tensor near the boundary of the bulk spacetime,\nreconstructing the local data in the bulk from the entanglement on the boundary. We also show that\npositivity, monotonicity, and convexity of the relative entropy for small spherical domains between the\nreduced density matrices of any state and of the ground state of the conformal field theory are guaranteed by\npositivity conditions on the bulk matter energy density. As positivity and monotonicity of the relative\nentropy are general properties of quantum systems, this can be interpreted as a derivation of bulk energy\nconditions in any holographic system for which the Ryu-Takayanagi prescription applies. We discuss an\ninformation theoretical interpretation of the convexity in terms of the Fisher metric.", "doi": "10.1103/PhysRevLett.114.221601", "issn": "0031-9007", "publisher": "American Physical Society", "publication": "Physical Review Letters", "publication_date": "2015-06-05", "series_number": "22", "volume": "114", "issue": "22", "pages": "Art. No. 221601" }, { "id": "authors:mvfjk-9gf09", "collection": "authors", "collection_id": "mvfjk-9gf09", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20150608-104133815", "type": "article", "title": "From exceptional collections to motivic decompositions via noncommutative motives", "author": [ { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" }, { "family_name": "Tabuada", "given_name": "Gon\u00e7alo", "clpid": "Tabuada-G" } ], "abstract": "Making use of noncommutative motives we relate exceptional collections (and more generally semi-orthogonal decompositions) to motivic decompositions. On one hand we prove that the Chow motive M(X)_Q of every smooth and proper Deligne\u2013Mumford stack X, whose bounded derived category D^b(X) of coherent schemes admits a full exceptional collection, decomposes into a direct sum of tensor powers of the Lefschetz motive. Examples include projective spaces, quadrics, toric varieties, homogeneous spaces, Fano threefolds, and moduli spaces. On the other hand we prove that if M(X)_Q decomposes into a direct sum of tensor powers of the Lefschetz motive and moreover D^b(X) admits a semiorthogonal decomposition, then the noncommutative motive of each one of the pieces of the\nsemi-orthogonal decomposition is a direct sum of \u2297-units. As an application we obtain a simplification of Dubrovin's conjecture.", "doi": "10.1515/crelle-2013-0027", "issn": "0075-4102", "publisher": "Walter de Gruyter", "publication": "Journal F\u00fcr Die Reine und Angewandte Mathematik", "publication_date": "2015-04", "series_number": "701", "volume": "2015", "issue": "701", "pages": "153-167" }, { "id": "authors:yn98w-t4t25", "collection": "authors", "collection_id": "yn98w-t4t25", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20150424-091752124", "type": "article", "title": "F_\u03b6-geometry, Tate motives, and the Habiro ring", "author": [ { "family_name": "Lo", "given_name": "Catharine Wing Kwan", "clpid": "Lo-Catharine-Wing-Kwan" }, { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" } ], "abstract": "In this paper, we propose different notions of F_zeta-geometry, for zeta a root of unity, generalizing notions of over finite fields, the Grothendieck class, and the notion of torification. We relate Fzeta-geometry to formal roots of Tate motives, and to functions in the Habiro ring, seen as counting functions of certain ind-varieties. We investigate the existence of Fzeta-structures in examples arising from general linear groups, matrix equations over finite fields, and some quantum modular forms.", "doi": "10.1142/S1793042115500189", "issn": "1793-0421", "publisher": "World Scientific Publishing", "publication": "International Journal of Number Theory", "publication_date": "2015-03-02", "series_number": "2", "volume": "11", "issue": "2", "pages": "311-339" }, { "id": "authors:gm33c-rsp82", "collection": "authors", "collection_id": "gm33c-rsp82", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20151209-134217011", "type": "article", "title": "Quantum Computation and Real Multiplication", "author": [ { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" }, { "family_name": "Napp", "given_name": "John", "clpid": "Napp-J" } ], "abstract": "We propose a construction of anyon systems associated to quantum tori with real multiplication and the embedding of quantum tori in AF algebras. These systems generalize the Fibonacci anyons, with weaker categorical properties, and are obtained from the basic modules and the real multiplication structure.", "doi": "10.1007/s11786-014-0179-8", "issn": "1661-8270", "publisher": "Springer", "publication": "Mathematics in Computer Science", "publication_date": "2015-03", "series_number": "1", "volume": "9", "issue": "1", "pages": "63-84" }, { "id": "authors:4n5fa-mq666", "collection": "authors", "collection_id": "4n5fa-mq666", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20140815-135716277", "type": "article", "title": "Big Bang, Blowup, and Modular Curves: Algebraic Geometry in Cosmology?", "author": [ { "family_name": "Manin", "given_name": "Yuri I.", "clpid": "Manin-Y-I" }, { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" } ], "abstract": "We introduce some algebraic geometric models in cosmology related to the ''boundaries'' of space-time: Big Bang, Mixmaster Universe, Penrose's crossovers between aeons. We suggest to model the kinematics of Big Bang using the algebraic geometric (or analytic) blow up of a point x. This creates a boundary which consists of the projective space of tangent directions to x and possibly of the light cone of x. We argue that time on the boundary undergoes the Wick rotation and becomes purely imaginary. The Mixmaster (Bianchi IX) model of the early history of the universe is neatly explained in this picture by postulating that the reverse Wick rotation follows a hyperbolic geodesic connecting imaginary time axis to the real one. Penrose's idea to see the Big Bang as a sign of crossover from \"the end of previous aeon'' of the expanding and cooling Universe to the ''beginning of the next aeon\" is interpreted as an identification of a natural boundary of Minkowski space at infinity with the Big Bang boundary.", "doi": "10.3842/SIGMA.2014.073", "issn": "1815-0659", "publisher": "Institute of Mathematics of National Academy of Sciences of Ukraine", "publication": "Symmetry, Integrability and Geometry, Methods and Applications (SIGMA)", "publication_date": "2014-07-09", "volume": "10", "pages": "Art. No. 73" }, { "id": "authors:2kn47-3pe52", "collection": "authors", "collection_id": "2kn47-3pe52", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20141111-073436103", "type": "article", "title": "Jacobians of noncommutative motives", "author": [ { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" }, { "family_name": "Tabuada", "given_name": "Gon\u00e7alo", "clpid": "Tabuada-G" } ], "abstract": "In this article one extends the classical theory of (intermediate) Jacobians to the \"noncommutative world\". Concretely, one constructs a \u211a-linear additive Jacobian functor N \u2192 J(N) from the category of noncommutative Chow motives to the category of abelian varieties up to isogeny, with the following properties: (i) the first de Rham cohomology group of J(N) agrees with the subspace of the odd periodic cyclic homology of N which is generated by algebraic curves; (ii) the abelian variety J(perf_(dg)(X)) (associated to the derived dg category perf_(dg)(X) of a smooth projective k-scheme X) identifies with the product of all the intermediate algebraic Jacobians of X. As an application, every semi-orthogonal decomposition of the derived category perf(X) gives rise to a decomposition of the intermediate algebraic Jacobians of X.", "doi": "10.48550/arXiv.1212.1118", "issn": "1609-3321", "publisher": "American Mathematical Society", "publication": "Moscow Mathematical Journal", "publication_date": "2014-07", "series_number": "3", "volume": "14", "issue": "3", "pages": "577-594" }, { "id": "authors:n1pmm-2jr02", "collection": "authors", "collection_id": "n1pmm-2jr02", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20140821-111916358", "type": "article", "title": "Kolmogorov complexity and the asymptotic bound for error-correcting codes", "author": [ { "family_name": "Manin", "given_name": "Yuri", "clpid": "Manin-Y-I" }, { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" } ], "abstract": "The set of all error-correcting block codes over a fixed alphabet with q letters determines a recursively enumerable set of rational points in the unit square with coordinates (R,\u03b4):= (relative transmission rate, relative minimal distance). Limit points of this set form a closed subset, defined by R\u2264\u03b1q(\u03b4), where \u03b1q(\u03b4) is a continuous decreasing function called the asymptotic bound. Its existence was proved by the first-named author in 1981, but no approaches to the computation of this function are known, and in it was even suggested that this function might be uncomputable in the sense of constructive analysis.\nIn this note we show that the asymptotic bound becomes computable with the assistance of an oracle producing codes in the order of their growing Kolmogorov complexity. Moreover, a natural partition function involving complexity allows us to interpret the asymptotic bound as a curve dividing two different thermodynamic phases of codes.", "doi": "10.48550/arXiv.1203.0653", "issn": "0022-040X", "publisher": "International Press", "publication": "Journal of Differential Geometry", "publication_date": "2014-05", "series_number": "1", "volume": "97", "issue": "1", "pages": "91-108" }, { "id": "authors:kx14s-qgy91", "collection": "authors", "collection_id": "kx14s-qgy91", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20140717-154554320", "type": "article", "title": "Multifractals, Mumford Curves and Eternal Inflation", "author": [ { "family_name": "Marcolli", "given_name": "M.", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" }, { "family_name": "Tedeschi", "given_name": "N.", "clpid": "Tedeschi-N" } ], "abstract": "We relate the Eternal Symmetree model of Harlow, Shenker, Stanford, and Susskind to constructions of stochastic processes related to quantum statistical mechanical systems on Cuntz-Krieger algebras. We extend the eternal inflation model from the Bruhat-Tits tree to quotients by p-adic Schottky groups, again using quantum statistical mechanics on graph algebras.", "doi": "10.1134/S2070046614020034", "issn": "2070-0474", "publisher": "Springer", "publication": "P-Adic Numbers, Ultrametric Analysis, and Applications", "publication_date": "2014-04", "series_number": "2", "volume": "6", "issue": "2", "pages": "135-154" }, { "id": "authors:pt9hm-jxk14", "collection": "authors", "collection_id": "pt9hm-jxk14", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20141212-091347163", "type": "article", "title": "Algebraic renormalization and Feynman integrals in configuration spaces", "author": [ { "family_name": "Ceyhan", "given_name": "\u00d6zg\u00fcr", "clpid": "Ceyhan-\u00d6" }, { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" } ], "abstract": "This paper continues our previous study of Feynman integrals in configuration spaces and their algebro-geometric and motivic aspects. We consider here both massless and massive Feynman amplitudes, from the point of view of potential theory. We consider a variant of the wonderful compactification of configuration spaces that works simultaneously for all graphs with a given number of vertices and that also accounts for the external structure of Feynman graph. As in our previous work, we consider two version of the Feynman amplitude in configuration space, which we refer to as the real and complex versions. In the real version, we show that we can extend to the massive case a method of evaluating Feynman integrals, based on expansion in Gegenbauer polynomials, that we investigated previously in the massless case. In the complex setting, we show that we can use algebro-geometric methods to renormalize the Feynman amplitudes, so that the renormalized values of the Feynman integrals are given by periods of a mixed Tate motive. The regularization and renormalization procedure is based on pulling back the form to the wonderful compactification and replace it with a cohomologous one with logarithmic poles. A complex of forms with logarithmic poles, endowed with an operator of pole subtraction, determine a Rota-Baxter algebra on the wonderful compactifications. We can then apply the renormalization procedure via Birkhoff factorization, after interpreting the regularization as an algebra homomorphism from the Connes-Kreimer Hopf algebra of Feynman graphs to the Rota-Baxter algebra. We obtain in this setting a description of the renormalization group.We also extend the period interpretation to the case of Dirac fermions and gauge bosons.", "doi": "10.4310/ATMP.2014.v18.n2.a5", "issn": "1095-0761", "publisher": "International Press", "publication": "Advances in Theoretical and Mathematical Physics", "publication_date": "2014-04", "series_number": "2", "volume": "18", "issue": "2", "pages": "469-511" }, { "id": "authors:mft2z-b4a35", "collection": "authors", "collection_id": "mft2z-b4a35", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20140718-073647086", "type": "article", "title": "Twisted Spectral Triples and Quantum Statistical Mechanical Systems", "author": [ { "family_name": "Greenfield", "given_name": "M.", "clpid": "Greenfield-M" }, { "family_name": "Marcolli", "given_name": "M.", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" }, { "family_name": "Teh", "given_name": "K.", "clpid": "Teh-Kevin" } ], "abstract": "Spectral triples and quantum statistical mechanical systems are two important constructions in noncommutative geometry. In particular, both lead to interesting reconstruction theorems for a broad range of geometric objects, including number fields, spin manifolds, graphs. There are similarities between the two structures, and we show that the notion of twisted spectral triple, introduced recently by Connes and Moscovici, provides a natural bridge between them. We investigate explicit examples, related to the Bost-Connes quantum statistical mechanical system and to Riemann surfaces and graphs.", "doi": "10.1134/S2070046614020010", "issn": "2070-0474", "publisher": "Springer", "publication": "P-Adic Numbers, Ultrametric Analysis, and Applications", "publication_date": "2014-04", "series_number": "2", "volume": "6", "issue": "2", "pages": "81-104" }, { "id": "authors:t5bwy-kv823", "collection": "authors", "collection_id": "t5bwy-kv823", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20140401-134330780", "type": "article", "title": "Endomotives of toric varieties", "author": [ { "family_name": "Jin", "given_name": "Zhaorong", "clpid": "Jin-Zhaorong" }, { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" } ], "abstract": "We construct endomotives associated to toric varieties, in terms of the decomposition of a toric variety into torus orbits and the action of a semigroup of toric morphisms. We show that the endomotives can be endowed with time evolutions and we discuss the resulting quantum statistical mechanical systems. We show that in particular, one can construct a time evolution related to the logarithmic height function. We discuss relations to F_1-geometry.", "doi": "10.1016/j.geomphys.2013.12.006", "issn": "0393-0440", "publisher": "Elsevier", "publication": "Journal of Geometry and Physics", "publication_date": "2014-03", "volume": "77", "pages": "48-71" }, { "id": "authors:qyc5h-rtg18", "collection": "authors", "collection_id": "qyc5h-rtg18", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20140324-104258394", "type": "article", "title": "Noncommutative motives, numerical equivalence, and semi-simplicity", "author": [ { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" }, { "family_name": "Tabuada", "given_name": "Gon\u00e7alo", "clpid": "Tabuada-G" } ], "abstract": "Making use of Hochschild homology, we introduce the correct category NNum(k)_F of noncommutative numerical motives (over a base ring k and with coefficients in a field F). We prove that NNum(k)_F is abelian semi-simple and that Grothendieck's category Num(k)_Q of numerical motives embeds into NNum(k)_Q after being factored out by the action of the Tate object. As an application we obtain an alternative proof of Jannsen's celebrate semi-simplicity result, which uses the noncommutative world instead of a classical Weil cohomology.", "doi": "10.1353/ajm.2014.0004", "issn": "0002-9327", "publisher": "Johns Hopkins University Press", "publication": "American Journal of Mathematics", "publication_date": "2014-02", "series_number": "1", "volume": "136", "issue": "1", "pages": "59-75" }, { "id": "authors:tnnkj-r6c80", "collection": "authors", "collection_id": "tnnkj-r6c80", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20140127-152321287", "type": "article", "title": "Noncommutative Artin motives", "author": [ { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" }, { "family_name": "Tabuada", "given_name": "Gon\u00e7alo", "clpid": "Tabuada-G" } ], "abstract": "In this article, we introduce the category of noncommutative Artin motives as well as the category of noncommutative mixed Artin motives. In the pure world, we start by proving that the classical category AM(k)_Q of Artin motives (over a base field k) can be characterized as the largest category inside Chow motives which fully embeds into noncommutative Chow motives. Making use of a refined bridge between pure motives and noncommutative pure motives, we then show that the image of this full embedding, which we call the category NAM(k)_Q of noncommutative Artin motives, is invariant under the different equivalence relations and modification of the symmetry isomorphism constraints. As an application, we recover the absolute Galois group Gal(k\u00af/k) from the Tannakian formalism applied to NAM(k)_Q . Then, we develop the base-change formalism in the world of noncommutative pure motives. As an application, we obtain new tools for the study of motivic decompositions and Schur/Kimura finiteness. Making use of this theory of base-change, we then construct a short exact sequence relating Gal(k\u00af/k) with the noncommutative motivic Galois groups of k and k\u00af . Finally, we describe a precise relationship between this short exact sequence and the one constructed by Deligne\u2013Milne. In the mixed world, we introduce the triangulated category NMAM(k)_Q of noncommutative mixed Artin motives and construct a faithful functor from the classical category MAM(k)_Q of mixed Artin motives to it. When k is a finite field, this functor is an equivalence. On the other hand, when k is of characteristic zero NMAM(k)_Q is much richer than MAM(k)_Q since its higher Ext-groups encode all the (rationalized) higher algebraic K -theory of finite \u00e9tale k-schemes. In the appendix, we establish a general result about short exact sequences of Galois groups which is of independent interest. As an application, we obtain a new proof of Deligne\u2013Milne's short exact sequence.", "doi": "10.1007/s00029-013-0131-9", "issn": "1022-1824", "publisher": "Springer", "publication": "Selecta Mathematica - New Series", "publication_date": "2014-01", "series_number": "1", "volume": "20", "issue": "1", "pages": "315-358" }, { "id": "authors:h2xnt-12531", "collection": "authors", "collection_id": "h2xnt-12531", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20140109-093610920", "type": "article", "title": "Gauge networks in noncommutative geometry", "author": [ { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" }, { "family_name": "van Suijlekom", "given_name": "Walter D.", "clpid": "van-Suijlekom-W-D" } ], "abstract": "We introduce gauge networks as generalizations of spin networks and lattice gauge fields to almost-commutative manifolds. The configuration space of quiver representations (modulo equivalence) in the category of finite spectral triples is studied; gauge networks appear as an orthonormal basis in a corresponding Hilbert space. We give many examples of gauge networks, also beyond the well-known spin network examples. We find a Hamiltonian operator on this Hilbert space, inducing a time evolution on the C^\u2217-algebra of gauge network correspondences.\n\nGiven a representation in the category of spectral triples of a quiver embedded in a spin manifold, we define a discretized Dirac operator on the quiver. We compute the spectral action of this Dirac operator on a four-dimensional lattice, and find that it reduces to the Wilson action for lattice gauge theories and a Higgs field lattice system. As such, in the continuum limit it reduces to the Yang\u2013Mills\u2013Higgs system. For the three-dimensional case, we relate the spectral action functional to the Kogut\u2013Susskind Hamiltonian.", "doi": "10.1016/j.geomphys.2013.09.002", "issn": "0393-0440", "publisher": "Elsevier", "publication": "Journal of Geometry and Physics", "publication_date": "2014-01", "volume": "75", "pages": "71-91" }, { "id": "authors:51fj0-sq365", "collection": "authors", "collection_id": "51fj0-sq365", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20141113-080729733", "type": "article", "title": "Thermodynamic semirings", "author": [ { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" }, { "family_name": "Thorngren", "given_name": "Ryan", "orcid": "0000-0001-9433-3399", "clpid": "Thorngren-R" } ], "abstract": "The Witt construction describes a functor from the category of Rings to the category\nof characteristic 0 rings. It is uniquely determined by a few associativity constraints which do\nnot depend on the types of the variables considered, in other words, by integer polynomials.\nThis universality allowed Alain Connes and Caterina Consani to devise an analogue of the Witt\nring for characteristic one, an attractive endeavour since we know very little about the arithmetic\nin this exotic characteristic and its corresponding field with one element. Interestingly, they\nfound that in characteristic one, the Witt construction depends critically on the Shannon entropy.\nIn the current work, we examine this surprising occurrence, defining a Witt operad for an\narbitrary information measure and a corresponding algebra we call a thermodynamic semiring.\nThis object exhibits algebraically many of the familiar properties of information measures,\nand we examine in particular the Tsallis and Renyi entropy functions and applications to nonextensive\nthermodynamics and multifractals. We find that the arithmetic of the thermodynamic\nsemiring is exactly that of a certain guessing game played using the given information measure.", "doi": "10.4171/JNCG/159", "issn": "1661-6952", "publisher": "European Mathematical Society", "publication": "Journal of Noncommutative Geometry", "publication_date": "2014", "series_number": "2", "volume": "8", "issue": "2", "pages": "337-392" }, { "id": "authors:b4b5w-07n42", "collection": "authors", "collection_id": "b4b5w-07n42", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20141114-110040989", "type": "article", "title": "Coupling of gravity to matter, spectral action and cosmic topology", "author": [ { "family_name": "\u0106a\u0107i\u0107", "given_name": "Branimir", "clpid": "\u0106a\u0107i\u0107-Branimir" }, { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" }, { "family_name": "Teh", "given_name": "Kevin", "clpid": "Teh-Kevin" } ], "abstract": "We consider a model of modified gravity based on the spectral action functional, for a cosmic topology given by a spherical space form, and the associated slow-roll inflation scenario. We consider then the coupling of gravity to matter determined by an almost-commutative geometry over the spherical space form. We show that this produces a multiplicative shift of the amplitude of the power spectra for the density fluctuations and the gravitational waves, by a multiplicative factor equal to the total number of fermions in the matter sector of the model. We obtain the result by an explicit nonperturbative computation, based on the Poisson summation formula and the spectra of twisted Dirac operators on spherical space forms, as well as, for more general spacetime manifolds, using a heat kernel computation.", "doi": "10.4171/JNCG/162", "issn": "1661-6952", "publisher": "European Mathematical Society", "publication": "Journal of Noncommutative Geometry", "publication_date": "2014", "series_number": "2", "volume": "8", "issue": "2", "pages": "473-504" }, { "id": "authors:9yds7-ya050", "collection": "authors", "collection_id": "9yds7-ya050", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130909-153558798", "type": "article", "title": "Graph reconstruction and quantum statistical mechanics", "author": [ { "family_name": "Cornelissen", "given_name": "Gunther", "clpid": "Cornelissen-G" }, { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" } ], "abstract": "We study how far it is possible to reconstruct a graph from various Banach algebras associated to its universal covering, and extensions thereof to quantum statistical mechanical systems. It turns out that most the boundary operator algebras reconstruct only topological information, but the statistical mechanical point of view allows for complete reconstruction of multigraphs with minimal degree three.", "doi": "10.1016/j.geomphys.2013.03.021", "issn": "0393-0440", "publisher": "Elsevier", "publication": "Journal of Geometry and Physics", "publication_date": "2013-10", "volume": "72", "pages": "110-117" }, { "id": "authors:1aqs2-grd16", "collection": "authors", "collection_id": "1aqs2-grd16", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20120911-145601334", "type": "article", "title": "Asymptotic safety, hypergeometric functions, and the Higgs mass in spectral action models", "author": [ { "family_name": "Estrada", "given_name": "Christopher", "clpid": "Estrada-C" }, { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" } ], "abstract": "We study the renormalization group flow for the Higgs self coupling in the presence of gravitational correction terms. We show that the resulting equation is equivalent to a singular linear ODE, which has explicit solutions in terms of hypergeometric functions. We discuss the implications of this model with gravitational corrections on the Higgs mass estimates in particle physics models based on the spectral action functional.", "doi": "10.1142/S0219887813500369", "issn": "0219-8878", "publisher": "World Scientific Publishing", "publication": "International Journal of Geometric Methods in Modern Physics", "publication_date": "2013-08", "series_number": "7", "volume": "10", "issue": "7", "pages": "Art. No. 1350036" }, { "id": "authors:cj1sj-bm485", "collection": "authors", "collection_id": "cj1sj-bm485", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130611-084746994", "type": "article", "title": "Quantum field theory over F_1", "author": [ { "family_name": "Bejleri", "given_name": "Dori", "clpid": "Bejleri-D" }, { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" } ], "abstract": "In this paper we discuss some questions about geometry over the field with one element, motivated by the properties of algebraic varieties that arise in perturbative quantum field theory. We follow the approach to F_1-geometry based on torified-schemes. We first discuss some simple necessary conditions in terms of the Euler characteristic and classes in the Grothendieck ring, then we give a blowup formula for torified varieties and we show that the wonderful compactifications of the graph configuration spaces, that arise in the computation of Feynman integrals in position space, admit an F_1-structure. By a similar argument we show that the moduli spaces of curves M_(0,n) admit an F_1-structure, thus answering a question of Manin. We also discuss conditions on hyperplane arrangements, a possible notion of embedded F_1-structure and its relation to Chern classes, and questions on Chern classes of varieties with regular torifications.", "doi": "10.1016/j.geomphys.2013.03.002", "issn": "0393-0440", "publisher": "Elsevier", "publication": "Journal of Geometry and Physics", "publication_date": "2013-07", "volume": "69", "pages": "40-59" }, { "id": "authors:bqndt-z3281", "collection": "authors", "collection_id": "bqndt-z3281", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130507-112308486", "type": "article", "title": "Arithmetic of Potts Model Hypersurfaces", "author": [ { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" }, { "family_name": "Su", "given_name": "Jessica", "clpid": "Su-Jessica" } ], "abstract": "We consider Potts model hypersurfaces defined by the multivariate Tutte polynomial of graphs (Potts model partition function). We focus on the behavior of the number of points over finite fields for these hypersurfaces, in comparison with the graph hypersurfaces of perturbative quantum field theory defined by the Kirchhoff graph polynomial. We give a very simple example of the failure of the \"fibration condition\" in the dependence of the Grothendieck class on the number of spin states and of the polynomial countability condition for these Potts model hypersurfaces. We then show that a period computation, formally similar to the parametric Feynman integrals of quantum field theory, arises by considering certain thermodynamic averages. One can show that these evaluate to combinations of multiple zeta values for Potts models on polygon polymer chains, while silicate tetrahedral chains provide a candidate for a possible occurrence of non-mixed Tate periods.", "doi": "10.1142/S0219887813500059", "issn": "0219-8878", "publisher": "World Scientific Publishing", "publication": "International Journal of Geometric Methods in Modern Physics", "publication_date": "2013-04", "series_number": "4", "volume": "10", "issue": "4", "pages": "Art. No. 1350005" }, { "id": "authors:hhdjh-a5t89", "collection": "authors", "collection_id": "hhdjh-a5t89", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130117-104305431", "type": "article", "title": "A motivic approach to phase transitions in Potts models", "author": [ { "family_name": "Aluffi", "given_name": "Paolo", "clpid": "Aluffi-P" }, { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" } ], "abstract": "We describe an approach to the study of phase transitions in Potts models based on an estimate of the complexity of the locus of real zeros of the partition function, computed in terms of the classes in the Grothendieck ring of the affine algebraic varieties defined by the vanishing of the multivariate Tutte polynomial. We give completely explicit calculations for the examples of the chains of linked polygons and of the graphs obtained by replacing the polygons with their dual graphs. These are based on a deletion\u2013contraction formula for the Grothendieck classes and on generating functions for splitting and doubling edges.", "doi": "10.1016/j.geomphys.2012.09.003", "issn": "0393-0440", "publisher": "Elsevier", "publication": "Journal of Geometry and Physics", "publication_date": "2013-01", "volume": "63", "pages": "6-31" }, { "id": "authors:ph1ec-73643", "collection": "authors", "collection_id": "ph1ec-73643", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130514-095557537", "type": "article", "title": "Noncommutative Mixmaster Cosmologies", "author": [ { "family_name": "Estrada", "given_name": "Christopher", "clpid": "Estrada-C" }, { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" } ], "abstract": "In this paper we investigate a variant of the classical mixmaster universe model of anisotropic cosmology, where the spatial sections are noncommutative 3-tori. We consider ways in which the discrete dynamical system describing the mixmaster dynamics can be extended to act on the noncommutative torus moduli, and how the resulting dynamics differs from the classical one, for example, in the appearance of exotic smooth structures. We discuss properties of the spectral action, focussing on how the slow-roll inflation potential determined by the spectral action affects the mixmaster dynamics. We relate the model to other recent results on spectral action computation and we identify other physical contexts in which this model may be relevant.", "doi": "10.1142/S0219887812500867", "issn": "0219-8878", "publisher": "World Scientific Publishing", "publication": "International Journal of Geometric Methods in Modern Physics", "publication_date": "2013-01", "series_number": "1", "volume": "10", "issue": "1", "pages": "Art. No. 1250086" }, { "id": "authors:ddbyt-wk891", "collection": "authors", "collection_id": "ddbyt-wk891", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130102-141740985", "type": "article", "title": "Kontsevich's noncommutative numerical motives", "author": [ { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" }, { "family_name": "Tabuada", "given_name": "Gon\u00e7alo", "clpid": "Tabuada-G" } ], "abstract": "In this article we prove that Kontsevich's category NC_(num)(k)_F of noncommutative numerical motives is equivalent to the one constructed by the authors in [Marcolli and Tabuada, Noncommutative motives, numerical equivalence, and semisimplicity, Amer. J. Math., to appear, available at arXiv:1105.2950]. As a consequence, we conclude that NC_(num)(k)_F is abelian semi-simple as conjectured by Kontsevich.", "doi": "10.1112/S0010437X12000383", "issn": "0010-437X", "publisher": "Cambridge University Press", "publication": "Compositio Mathematica", "publication_date": "2012-11", "series_number": "6", "volume": "148", "issue": "6", "pages": "1811-1820" }, { "id": "authors:r5mq8-5n617", "collection": "authors", "collection_id": "r5mq8-5n617", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130328-095244455", "type": "article", "title": "Codes as Fractals and Noncommutative Spaces", "author": [ { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" }, { "family_name": "Perez", "given_name": "Christopher", "clpid": "Perez-C" } ], "abstract": "We consider the CSS algorithm relating self-orthogonal classical linear codes to q-ary quantum stabilizer codes and we show that to such a pair of a classical and a quantum code one can associate geometric spaces constructed using methods from noncommutative geometry, arising from rational noncommutative tori and finite abelian group actions on Cuntz algebras and fractals associated to the classical codes.", "doi": "10.1007/s11786-012-0114-9", "issn": "1661-8270", "publisher": "Springer Verlag", "publication": "Mathematics in Computer Science", "publication_date": "2012-09-01", "series_number": "3", "volume": "6", "issue": "3", "pages": "199-215" }, { "id": "authors:6q7np-qtf73", "collection": "authors", "collection_id": "6q7np-qtf73", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20120820-083919381", "type": "article", "title": "The Ricci Flow on Noncommutative Two-Tori", "author": [ { "family_name": "Bhuyain", "given_name": "Tanvir Ahamed", "clpid": "Bhuyain-T-A" }, { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" } ], "abstract": "In this paper we construct a version of Ricci flow for noncommutative two-tori, based on a spectral formulation in terms of the eigenvalues and eigenfunction of the Laplacian and recent results on the Gauss\u2013Bonnet theorem for noncommutative tori.", "doi": "10.1007/s11005-012-0550-0", "issn": "0377-9017", "publisher": "Springer", "publication": "Letters in Mathematical Physics", "publication_date": "2012-08", "series_number": "2", "volume": "101", "issue": "2", "pages": "173-194" }, { "id": "authors:yreyx-gys95", "collection": "authors", "collection_id": "yreyx-gys95", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20120710-100433105", "type": "article", "title": "Feynman Integrals and Motives of Configuration Spaces", "author": [ { "family_name": "Ceyhan", "given_name": "\u00d6zg\u00fcr", "clpid": "Ceyhan-\u00d6" }, { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" } ], "abstract": "We formulate the problem of renormalization of Feynman integrals and its relation to periods of motives in configuration space instead of momentum space. The algebro-geometric setting is provided by the wonderful compactifications Conf \u0393(X) of arrangements of subvarieties associated to the subgraphs of a Feynman graph \u0393, with X a (quasi)projective variety. The motive and the class in the Grothendieck ring are computed explicitly for these wonderful compactifications, in terms of the motive of X and the combinatorics of the Feynman graph, using recent results of Li Li. The pullback to the wonderful compactification of the form defined by the unrenormalized Feynman amplitude has singularities along a hypersurface, whose real locus is contained in the exceptional divisors of the iterated blowup that gives the wonderful compactification. A regularization of the Feynman integrals can be obtained by modifying the cycle of integration, by replacing the divergent locus with a Leray coboundary. The ambiguities are then defined by Poincar\u00e9 residues. While these residues give periods associated to the cohomology of the exceptional divisors and their intersections, the regularized integrals give rise to periods of the hypersurface complement in the wonderful compactification.", "doi": "10.1007/s00220-012-1484-1", "issn": "0010-3616", "publisher": "Springer", "publication": "Communications in Mathematical Physics", "publication_date": "2012-07", "series_number": "1", "volume": "313", "issue": "1", "pages": "35-70" }, { "id": "authors:xargs-prw72", "collection": "authors", "collection_id": "xargs-prw72", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20120203-142242831", "type": "article", "title": "The Coupling of Topology and Inflation in Noncommutative Cosmology", "author": [ { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" }, { "family_name": "Pierpaoli", "given_name": "Elena", "orcid": "0000-0002-7957-8993", "clpid": "Pierpaoli-Elena" }, { "family_name": "Teh", "given_name": "Kevin", "clpid": "Teh-Kevin" } ], "abstract": "We show that, in a model of modified gravity based on the spectral action functional, there is a nontrivial coupling between cosmic topology and inflation, in the sense that the shape of the possible slow-roll inflation potentials obtained in the model from the nonperturbative form of the spectral action is sensitive not only to the geometry (flat or positively curved) of the universe, but also to the different possible non-simply connected topologies. We show this by explicitly computing the nonperturbative spectral action for some candidate flat cosmic topologies given by Bieberbach manifolds and showing that the resulting inflation potential differs from that of the flat torus by a multiplicative factor, similarly to what happens in the case of the spectral action of the spherical forms in relation to the case of the 3-sphere. We then show that, while the slow-roll parameters differ between the spherical and flat manifolds but do not distinguish different topologies within each class, the power spectra detect the different scalings of the slow-roll potential and therefore distinguish between the various topologies, both in the spherical and in the flat case.", "doi": "10.1007/s00220-011-1352-4", "issn": "0010-3616", "publisher": "Springer", "publication": "Communications in Mathematical Physics", "publication_date": "2012-01", "series_number": "2", "volume": "309", "issue": "2", "pages": "341-369" }, { "id": "authors:9gbkd-x2c34", "collection": "authors", "collection_id": "9gbkd-x2c34", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20111207-093921796", "type": "article", "title": "Building Cosmological Models via Noncommutative Geometry", "author": [ { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" } ], "abstract": "This is an overview of new and ongoing research developments aimed at constructing cosmological\nmodels based on noncommutative geometry, via the spectral action functional,\nthought of as a modified gravity action, which includes the coupling with matter when\ncomputed on an almost commutative geometry. This survey is mostly based on recent\nresults obtained in collaboration with Elena Pierpaoli and Kevin Teh. We describe various\naspects of cosmological models of the very early universe, developed by the author\nand Pierpaoli, based on the asymptotic expansion of the spectral action functional and\non renormalization group analysis of the associated particle physics model (an extension\nof the standard model with right-handed neutrinos and Majorana mass terms previously\ndeveloped in collaboration with Chamseddine and Connes). We also describe nonperturbative\nresults, more recently obtained by Pierpaoli, Teh, and the author, which extend\nto the more modern universe, which show that, for different candidate cosmic topologies,\nthe form of the slow-roll inflation potentials obtained from the nonperturbative calculation\nof the spectral action are strongly coupled to the underlying geometry and topology.\nWe discuss some ongoing directions of research and open questions in this new field of\n\"noncommutative cosmology\". The paper is based on the talk given by the author at\nthe conference \"Geometry and Quantum Field Theory\" at the MPI, in honor of Alan\nCarey.", "doi": "10.1142/S0219887811005592", "issn": "0219-8878", "publisher": "World Scientific Publishing", "publication": "International Journal of Geometric Methods in Modern Physics", "publication_date": "2011-08", "series_number": "5", "volume": "8", "issue": "5", "pages": "1131-1168" }, { "id": "authors:dxkb0-r7s18", "collection": "authors", "collection_id": "dxkb0-r7s18", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20170712-080808371", "type": "article", "title": "Error-Correcting Codes and Phase Transitions", "author": [ { "family_name": "Manin", "given_name": "Yuri I.", "clpid": "Manin-Y-I" }, { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" } ], "abstract": "The theory of error-correcting codes is concerned with constructing codes that optimize simultaneously transmission rate and relative minimum distance. These conflicting requirements determine an asymptotic bound, which is a continuous curve in the space of parameters. The main goal of this paper is to relate the asymptotic bound to phase diagrams of quantum statistical mechanical systems. We first identify the code parameters with Hausdorff and von Neumann dimensions, by considering fractals consisting of infinite sequences of code words. We then construct operator algebras associated to individual codes. These are Toeplitz algebras with a time evolution for which the KMS state at critical temperature gives the Hausdorff measure on the corresponding fractal. We extend this construction to algebras associated to limit points of codes, with non-uniform multi-fractal measures, and to tensor products over varying parameters.", "doi": "10.1007/s11786-010-0031-8", "issn": "1661-8270", "publisher": "Springer Verlag", "publication": "Mathematics in Computer Science", "publication_date": "2011-06", "series_number": "2", "volume": "5", "issue": "2", "pages": "133-170" }, { "id": "authors:acv60-f4w82", "collection": "authors", "collection_id": "acv60-f4w82", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20110509-092722190", "type": "article", "title": "The Spectral Action and Cosmic Topology", "author": [ { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" }, { "family_name": "Pierpaoli", "given_name": "Elena", "orcid": "0000-0002-7957-8993", "clpid": "Pierpaoli-Elena" }, { "family_name": "Teh", "given_name": "Kevin", "clpid": "Teh-Kevin" } ], "abstract": "The spectral action functional, considered as a model of gravity coupled to\nmatter, provides, in its non-perturbative form, a slow-roll potential for inflation, whose\nform and corresponding slow-roll parameters can be sensitive to the underlying cosmic\ntopology. We explicitly compute the non-perturbative spectral action for some of the\nmain candidates for cosmic topologies, namely the quaternionic space, the Poincar\u00e9\ndodecahedral space, and the flat tori. We compute the corresponding slow-roll parameters\nand we check that the resulting inflation model behaves in the same way as for\na simply-connected spherical topology in the case of the quaternionic space and the\nPoincar\u00e9 homology sphere, while it behaves differently in the case of the flat tori. We\nadd an appendix with a discussion of the case of lens spaces.", "doi": "10.1007/s00220-011-1211-3", "issn": "0010-3616", "publisher": "Springer", "publication": "Communications in Mathematical Physics", "publication_date": "2011-05", "series_number": "1", "volume": "304", "issue": "1", "pages": "125-174" }, { "id": "authors:wd916-xk848", "collection": "authors", "collection_id": "wd916-xk848", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20110322-095729744", "type": "article", "title": "Cuntz\u2013Krieger Algebras and Wavelets on Fractals", "author": [ { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" }, { "family_name": "Paolucci", "given_name": "Anna Maria", "clpid": "Paolucci-A-M" } ], "abstract": "We consider representations of Cuntz\u2013Krieger algebras on the Hilbert space of square integrable functions on the limit set, identified with a Cantor set in the unit interval. We use these representations and the associated Perron\u2013Frobenius and Ruelle operators to construct families of wavelets on these Cantor sets.", "doi": "10.1007/s11785-009-0044-y", "issn": "1661-8254", "publisher": "Springer", "publication": "Complex Analysis and Operator Theory", "publication_date": "2011-03", "series_number": "1", "volume": "5", "issue": "1", "pages": "41-81" }, { "id": "authors:csyfa-1vk23", "collection": "authors", "collection_id": "csyfa-1vk23", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20110318-142126748", "type": "article", "title": "Graph Hypersurfaces and a Dichotomy in the Grothendieck Ring", "author": [ { "family_name": "Aluffi", "given_name": "Paolo", "clpid": "Aluffi-P" }, { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" } ], "abstract": "The subring of the Grothendieck ring of varieties generated by the graph\nhypersurfaces of quantum field theory maps to the monoid ring of stable birational equivalence\nclasses of varieties. We show that the image of this map is the copy of Z generated\nby the class of a point. This clarifies the extent to which the graph hypersurfaces 'generate\nthe Grothendieck ring of varieties': while it is known that graph hypersurfaces generate\nthe Grothendieck ring over a localization of Z[L] in which L becomes invertible, the span\nof the graph hypersurfaces in the Grothendieck ring itself is nearly killed by setting the\nLefschetz motive L to zero. In particular, this shows that the graph hypersurfaces do not\ngenerate the Grothendieck ring prior to localization. The same result yields some information\non the mixed Hodge structures of graph hypersurfaces, in the form of a constraint on\nthe terms in their Deligne\u2013Hodge polynomials. These observations are certainly not surprising\nfor the expert reader, but are somewhat hidden in the literature. The treatment in\nthis note is straightforward and self-contained.", "doi": "10.1007/s11005-011-0461-5", "issn": "0377-9017", "publisher": "Springer", "publication": "Letters in Mathematical Physics", "publication_date": "2011-03", "series_number": "3", "volume": "95", "issue": "3", "pages": "223-232" }, { "id": "authors:phjsx-gsv45", "collection": "authors", "collection_id": "phjsx-gsv45", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20111205-113131774", "type": "article", "title": "Algebro-Geometric Feynman Rules", "author": [ { "family_name": "Aluffi", "given_name": "Paolo", "clpid": "Aluffi-P" }, { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" } ], "abstract": "We give a general procedure to construct \"algebro-geometric Feynman rules\", that is, characters of the Connes\u2013Kreimer Hopf algebra of Feynman graphs that factor through a Grothendieck ring of immersed conical varieties, via the class of the complement of the affine graph hypersurface. In particular, this maps to the usual Grothendieck ring of varieties, defining \"motivic Feynman rules\". We also construct an algebro-geometric Feynman rule with values in a polynomial ring, which does not factor through the usual Grothendieck ring, and which is defined in terms of characteristic classes of singular varieties. This invariant recovers, as a special value, the Euler characteristic of the projective graph hypersurface complement. The main result underlying the construction of this invariant is a formula for the characteristic classes of the join of two projective varieties. We discuss the BPHZ renormalization procedure in this algebro-geometric context and some motivic zeta functions arising from the partition functions associated to motivic Feynman rules.", "doi": "10.1142/S0219887811005099", "issn": "0219-8878", "publisher": "World Scientific Publishing", "publication": "International Journal of Geometric Methods in Modern Physics", "publication_date": "2011-02", "series_number": "1", "volume": "8", "issue": "1", "pages": "203-237" }, { "id": "authors:2wgfd-k8h37", "collection": "authors", "collection_id": "2wgfd-k8h37", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20101012-141717687", "type": "article", "title": "Spin foams and noncommutative geometry", "author": [ { "family_name": "Denicola", "given_name": "Domenic", "clpid": "Denicola-D" }, { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" }, { "family_name": "Zainy al-Yasry", "given_name": "Ahmad", "clpid": "Zainy-al-Yasry-A" } ], "abstract": "We extend the formalism of embedded spin networks and spin foams to include topological data that encode the underlying three-manifold or four-manifold as a branched cover. These data are expressed as monodromies, in a way similar to the encoding of the gravitational field via holonomies. We then describe convolution algebras of spin networks and spin foams, based on the different ways in which the same topology can be realized as a branched covering via covering moves, and on possible composition operations on spin foams. We illustrate the case of the groupoid algebra of the equivalence relation determined by covering moves and a 2-semigroupoid algebra arising from a 2-category of spin foams with composition operations corresponding to a fibered product of the branched coverings and the gluing of cobordisms. The spin foam amplitudes then give rise to dynamical flows on these algebras, and the existence of low temperature equilibrium states of the Gibbs form is related to questions on the existence of topological invariants of embedded graphs and embedded two-complexes with given properties. We end by sketching a possible approach to combining the spin network and spin foam formalism with matter within the framework of spectral triples in noncommutative geometry.", "doi": "10.1088/0264-9381/27/20/205025", "issn": "0264-9381", "publisher": "IOP", "publication": "Classical and Quantum Gravity", "publication_date": "2010-10-21", "series_number": "20", "volume": "27", "issue": "20", "pages": "Art. No. 205025" }, { "id": "authors:e1ja5-m0f90", "collection": "authors", "collection_id": "e1ja5-m0f90", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20111111-141226446", "type": "article", "title": "Early universe models from noncommutative geometry", "author": [ { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" }, { "family_name": "Pierpaoli", "given_name": "Elena", "orcid": "0000-0002-7957-8993", "clpid": "Pierpaoli-Elena" } ], "abstract": "We investigate cosmological predictions on the early universe based on the noncommutative geometry (NCG) models of gravity coupled to matter. Using the renormalization group analysis for the standard model with right-handed neutrinos and Majorana mass terms, which is the particle\nphysics content of the most recent NCG models, we analyze the behavior of the coefficients of the gravitational and cosmological terms in the Lagrangian derived from the asymptotic expansion of the spectral action functional of NCG. We find emergent Hoyle\u2013Narlikar and conformal\ngravity at the see-saw scales and a running effective gravitational constant, which affects the propagation of gravitational waves and the evaporation law of primordial black holes and provides Linde models of negative gravity in the early universe. The same renormalization group\nanalysis also governs the running of the effective cosmological constant of the model. The model also provides a Higgs-based slow-roll inflationary mechanism, for which one can explicitly compute the slow-roll parameters.\nThe particle physics content allows for dark matter models based on sterile neutrinos with Majorana mass terms.", "doi": "10.4310/ATMP.2010.v14.n5.a2", "issn": "1095-0761", "publisher": "International Press", "publication": "Advances in Theoretical and Mathematical Physics", "publication_date": "2010-10", "series_number": "5", "volume": "14", "issue": "5", "pages": "1373-1432" }, { "id": "authors:dt8kw-njc19", "collection": "authors", "collection_id": "dt8kw-njc19", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20101103-111640771", "type": "article", "title": "Boundary conditions of the RGE flow in the noncommutative geometry approach to particle physics and cosmology", "author": [ { "family_name": "Kolodrubetz", "given_name": "Daniel", "clpid": "Kolodrubetz-D" }, { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" } ], "abstract": "We investigate the effect of varying boundary conditions on the renormalization group flow in a recently developed noncommutative geometry model of particle physics and cosmology. We first show that there is a sensitive dependence on the initial conditions at unification, so that, varying a parameter even slightly can be shown to have drastic effects on the running of the model parameters. We compare the running in the case of the default and the maximal mixing conditions at unification. We then exhibit explicitly a particular choice of initial conditions at the unification scale, in the form of modified maximal mixing conditions, which have the property that they satisfy all the geometric constraints imposed by the noncommutative geometry of the model at unification, and at the same time, after running them down to lower energies with the renormalization group flow, they still agree in order of magnitude with the predictions at the electroweak scale.", "doi": "10.1016/j.physletb.2010.08.018", "issn": "0370-2693", "publisher": "Elsevier", "publication": "Physics Letters B", "publication_date": "2010-09-27", "series_number": "2", "volume": "693", "issue": "2", "pages": "166-174" }, { "id": "authors:5hb0k-44d56", "collection": "authors", "collection_id": "5hb0k-44d56", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20100920-094734398", "type": "article", "title": "Open string theory and planar algebras", "author": [ { "family_name": "Ceyhan", "given_name": "\u00d6zg\u00fcr", "clpid": "Ceyhan-\u00d6" }, { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" } ], "abstract": "In this paper, we show that abstract planar algebras are algebras over the topological operad of moduli spaces of stable maps with Lagrangian boundary conditions, which in the case of the projective line are described in terms of real rational functions. These moduli spaces appear naturally in the formulation of open string theory on the projective line. We also show two geometric ways to obtain planar algebras from real algebraic geometry, one based on string topology and one on Gromov\u2013Witten theory. In particular, through the well-known relation between planar algebras and subfactors, these results establish a connection between open string theory, real algebraic geometry and subfactors of von Neumann algebras.", "doi": "10.1088/1751-8113/43/38/385401", "issn": "0305-4470", "publisher": "IOP", "publication": "Journal of Physics A: Mathematical and General", "publication_date": "2010-09-24", "series_number": "38", "volume": "43", "issue": "38", "pages": "Art. No. 385401" }, { "id": "authors:av0g1-g3461", "collection": "authors", "collection_id": "av0g1-g3461", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20180806-102243722", "type": "article", "title": "Noncommutative Geometry and Loop Quantum Gravity: Loops, Algebras and Spectral Triples", "author": [ { "family_name": "Fleischhack", "given_name": "Christian", "clpid": "Fleischhack-C" }, { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" }, { "family_name": "Nest", "given_name": "Ryszard", "clpid": "Nest-R" } ], "abstract": "Spectral triples have recently turned out to be relevant for different approaches that aim at quantizing gravity and the other fundamental forces of nature in a mathematically rigorous way. The purpose of this workshop was to bring together researchers mainly from noncommutative geometry and loop quantum gravity \u2013two major fields that have used spectraltriples independently so far\u2013 in order to share their results and open issues.", "doi": "10.4171/OWR/2010/09", "issn": "1660-8933", "publisher": "European Mathematical Society", "publication": "Oberwolfach Reports", "publication_date": "2010-09-01", "series_number": "1", "volume": "7", "issue": "1", "pages": "373-413" }, { "id": "authors:ch1t4-h8378", "collection": "authors", "collection_id": "ch1t4-h8378", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20110725-070951650", "type": "article", "title": "Parametric Feynman integrals and determinant hypersurfaces", "author": [ { "family_name": "Aluffi", "given_name": "Paolo", "clpid": "Aluffi-P" }, { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" } ], "abstract": "The purpose of this paper is to show that, under certain combinatorial conditions on the graph, parametric Feynman integrals can be realized as periods on the complement of the determinant hypersurface in an affine space depending on the number of loops of the Feynman graph. The question of whether the Feynman integrals are periods of mixed Tate motives can then be reformulated (modulo divergences) as a question on a relative cohomology being a realization of a mixed Tate motive. This is the cohomology of the pair of the determinant hypersurface complement and a normal crossings divisor depending only on the number of loops and the genus of the graph. We show explicitly that this relative cohomology is a realization of a mixed Tate motive in the case of three loops and we give alternative formulations of the main question in the general case, by describing the locus of intersection of the divisor with the determinant hypersurface complement in terms of intersections of unions of Schubert cells in flag varieties. We also discuss different methods of regularization aimed at removing the divergences of the Feynman integral.", "doi": "10.48550/arXiv.0901.2107", "issn": "1095-0761", "publisher": "International Press", "publication": "Advances in Theoretical and Mathematical Physics", "publication_date": "2010-06", "series_number": "3", "volume": "14", "issue": "3", "pages": "911-963" }, { "id": "authors:wkawb-8zd24", "collection": "authors", "collection_id": "wkawb-8zd24", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20100414-093452256", "type": "article", "title": "Cyclotomy and Endomotives", "author": [ { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" } ], "abstract": "We compare two different models of noncommutative geometry of the cyclotomic tower, both based on an arithmetic algebra of functions of roots of unity and an action by endomorphisms, the first based on the Bost-Connes (BC) quantum statistical mechanical system and the second on the Habiro ring, where the Habiro functions have, in addition to evaluations at roots of unity, also full Taylor expansions. Both have compatible endomorphisms actions of the multiplicative semigroup of positive integers. As a higher dimensional generalization, we consider a crossed product ring obtained using Manin's multivariable generalizations of the Habiro functions and an action by endomorphisms of the semigroup of integer matrices with positive determinant. We then construct a corresponding class of multivariable BC endomotives, which are obtained geometrically from self maps of higher dimensional algebraic tori, and we discuss some of their quantum statistical mechanical properties. These multivariable BC endomotives are universal for (torsion free) \u039b-rings, compatibly with the Frobenius action. Finally, we discuss briefly how Habiro's universal Witten-Reshetikhin-Turaev invariant of integral homology 3-spheres may relate invariants of 3-manifolds to gadgets over F_1 and semigroup actions on homology 3-spheres to endomotives.", "doi": "10.1134/S2070046609030042", "issn": "2070-0474", "publisher": "Springer", "publication": "P-Adic Numbers, Ultrametric Analysis, and Applications", "publication_date": "2009-09", "series_number": "3", "volume": "1", "issue": "3", "pages": "217-263" }, { "id": "authors:yzr3e-d7h13", "collection": "authors", "collection_id": "yzr3e-d7h13", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20090714-095640002", "type": "article", "title": "Fun with F_1", "author": [ { "family_name": "Connes", "given_name": "Alain", "clpid": "Connes-A" }, { "family_name": "Consani", "given_name": "Caterina", "clpid": "Consani-C" }, { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" } ], "abstract": "We show that the algebra and the endomotive of the quantum statistical mechanical system of Bost\u2013Connes naturally arises by extension of scalars from the \"field with one element\" to rational numbers. The inductive structure of the abelian part of the endomotive corresponds to the tower of finite extensions of that \"field,\" while the endomorphisms reflect the Frobenius correspondences. This gives in particular an explicit model over the integers for this endomotive, which is related to the original Hecke algebra description. We study the reduction at a prime of the endomotive and of the corresponding noncommutative crossed product algebra.\n\nVideo. For a video summary of this paper, please visit http://\nwww.youtube.com/watch?v=az_0pxm1jrI.", "doi": "10.1016/j.jnt.2008.08.007", "issn": "0022-314X", "publisher": "Elsevier", "publication": "Journal of Number Theory", "publication_date": "2009-06", "series_number": "6", "volume": "129", "issue": "6", "pages": "1532-1561" }, { "id": "authors:vpznr-hfn23", "collection": "authors", "collection_id": "vpznr-hfn23", "cite_using_url": "https://authors.library.caltech.edu/records/vpznr-hfn23", "type": "article", "title": "Feynman motives of banana graphs", "author": [ { "family_name": "Aluffi", "given_name": "Paolo", "clpid": "Aluffi-P" }, { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" } ], "abstract": "
\n
\n

We consider the infinite family of Feynman graphs known as the “banana graphs” and compute explicitly the classes of the corresponding graph hypersurfaces in the Grothendieck ring of varieties as well as their Chern–Schwartz–MacPherson classes, using the classical Cremona transformation and the dual graph, and a blowup formula for characteristic classes. We outline the interesting similarities between these operations and we give formulae for cones obtained by simple operations on graphs. We formulate a positivity conjecture for characteristic classes of graph hypersurfaces and discuss briefly the effect of passing to noncommutative spacetime.

\n
\n
\n
 
", "doi": "10.4310/CNTP.2009.v3.n1.a1", "issn": "1931-4523", "publisher": "International Press", "publication": "Communications in Number Theory and Physics", "publication_date": "2009-03", "series_number": "1", "volume": "3", "issue": "1", "pages": "1-57" }, { "id": "authors:5t5qc-r1s05", "collection": "authors", "collection_id": "5t5qc-r1s05", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20170712-075152597", "type": "article", "title": "Coverings, correspondences, and noncommutative geometry", "author": [ { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" }, { "family_name": "Zainy al-Yasry", "given_name": "Ahmad", "clpid": "Zainy-al-Yasry-A" } ], "abstract": "We construct an additive category where objects are embedded graphs in the 3-sphere and morphisms are geometric correspondences given by 3-manifolds realized in different ways as branched covers of the 3-sphere, up to branched cover cobordisms. We consider dynamical systems obtained from associated convolution algebras endowed with time evolutions defined in terms of the underlying geometries. We describe the relevance of our construction to the problem of spectral correspondences in noncommutative geometry.", "doi": "10.1016/j.geomphys.2008.07.007", "issn": "0393-0440", "publisher": "Elsevier", "publication": "Journal of Geometry and Physics", "publication_date": "2008-12", "series_number": "12", "volume": "58", "issue": "12", "pages": "1639-1661" }, { "id": "authors:d67wj-s0e37", "collection": "authors", "collection_id": "d67wj-s0e37", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:MARjpa08", "type": "article", "title": "Supermanifolds from Feynman graphs", "author": [ { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" }, { "family_name": "Rej", "given_name": "Abhijnan", "clpid": "Rej-A" } ], "abstract": "We generalize the computation of Feynman integrals of log divergent graphs in terms of the Kirchhoff polynomial to the case of graphs with both fermionic and bosonic edges, to which we assign a set of ordinary and Grassmann variables. This procedure gives a computation of the Feynman integrals in terms of a period on a supermanifold, for graphs admitting a basis of the first homology satisfying a condition generalizing the log divergence in this context. The analog in this setting of the graph hypersurfaces is a graph supermanifold given by the divisor of zeros and poles of the Berezinian of a matrix associated with the graph, inside a superprojective space. We introduce a Grothendieck group for supermanifolds and identify the subgroup generated by the graph supermanifolds. This can be seen as a general procedure for constructing interesting classes of supermanifolds with associated periods.", "doi": "10.1088/1751-8113/41/31/315402", "issn": "1751-8113", "publisher": "IOP", "publication": "Journal of Physics A: Mathematical and Theoretical", "publication_date": "2008-08-08", "series_number": "31", "volume": "41", "issue": "31", "pages": "315402" }, { "id": "authors:pzkne-vh292", "collection": "authors", "collection_id": "pzkne-vh292", "cite_using_url": "https://authors.library.caltech.edu/records/pzkne-vh292", "type": "article", "title": "Solvmanifolds and noncommutative tori with real multiplication", "author": [ { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" } ], "abstract": "

We prove that the Shimizu L -function of a real quadratic fieldis obtained from a (Lorentzian) spectral triple on anoncommutative torus with real multiplication, as an adiabaticlimit of the Dirac operator on a 3-dimensional solvmanifold. TheDirac operator on this 3-dimensional geometry gives, via theConnes–Landi isospectral deformations, a spectral triple forthe noncommutative tori obtained by deforming the fiber tori tononcommutative spaces. The 3-dimensional solvmanifold is thehomotopy quotient in the sense of Baum–Connes of thenoncommutative space obtained as the crossed product of thenoncommutative torus by the action of the units of the realquadratic field. This noncommutative space is identified withthe twisted group C∗ -algebra of the fundamental group of the3-manifold. The twisting can be interpreted as the cocyclearising from a magnetic field, as in the theory of the quantumHall effect. We prove a twisted index theorem that computes therange of the trace on the K -theory of this noncommutativespace and gives an estimate on the gaps in the spectrum of theassociated Harper operator.

", "doi": "10.4310/CNTP.2008.v2.n2.a4", "issn": "1931-4523", "publisher": "International Press", "publication": "Communications in Number Theory and Physics", "publication_date": "2008-06", "series_number": "2", "volume": "2", "issue": "2", "pages": "421-476" }, { "id": "authors:4j82w-zbv21", "collection": "authors", "collection_id": "4j82w-zbv21", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:CORjgp08", "type": "article", "title": "Zeta functions that hear the shape of a Riemann surface", "author": [ { "family_name": "Cornelissen", "given_name": "Gunther", "clpid": "Cornelissen-G" }, { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" } ], "abstract": "To a compact hyperbolic Riemann surface, we associate a finitely summable spectral triple whose underlying topological space is the limit set of a corresponding Schottky group, and whose \"Riemannian\" aspect (Hilbert space and Dirac operator) encode the boundary action through its Patterson-Sullivan measured. We prove that the ergodic rigidity theorem for this boundary action implies that the zeta functions of the spectral triple suffice to characterize the (anti-)complex isomorphism class of the corresponding Riemann surface. Thus, you can hear the complex analytic shape of a Riemann surface, by listening to a suitable spectral triple.", "doi": "10.1016/j.geomphys.2007.12.011", "issn": "0393-0440", "publisher": "Elsevier", "publication": "Journal of Geometry and Physics", "publication_date": "2008-05", "series_number": "5", "volume": "58", "issue": "5", "pages": "619-632" }, { "id": "authors:f9z4h-qck67", "collection": "authors", "collection_id": "f9z4h-qck67", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:CORaam08", "type": "article", "title": "On the K-Theory of Graph C^\u2217-Algebras", "author": [ { "family_name": "Cornelissen", "given_name": "Gunther", "clpid": "Cornelissen-G" }, { "family_name": "Lorscheid", "given_name": "Oliver", "clpid": "Lorscheid-O" }, { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" } ], "abstract": "We classify graph C^*-algebras, namely, Cuntz-Krieger algebras associated to the Bass-Hashimoto edge incidence operator of a finite graph, up to strict isomorphism. This is done by a purely graph theoretical calculation of the K-theory of the C^*-algebras and the method also provides an independent proof of the classification up to Morita equivalence and stable equivalence of such algebras, without using the boundary operator algebra. A direct relation is given between the K_1-group of the algebra and the cycle space of the graph.", "doi": "10.1007/s10440-008-9208-4", "issn": "0167-8019", "publisher": "Springer", "publication": "Acta Applicandae Mathematicae", "publication_date": "2008-05", "series_number": "1", "volume": "102", "issue": "1", "pages": "57-69" }, { "id": "authors:zrw1x-b5m79", "collection": "authors", "collection_id": "zrw1x-b5m79", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:CONam07", "type": "article", "title": "Noncommutative geometry and motives: the thermodynamics of endomotives", "author": [ { "family_name": "Connes", "given_name": "Alain", "clpid": "Connes-A" }, { "family_name": "Consani", "given_name": "Caterina", "clpid": "Consani-C" }, { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" } ], "abstract": "We combine aspects of the theory of motives in algebraic geometry with noncommutative geometry and the classification of factors to obtain a cohomological interpretation of the spectral realization of zeros of L-functions. The analogue in characteristic zero of the action of the Frobenius on \u2113-adic cohomology is the action of the scaling group on the cyclic homology of the cokernel (in a suitable category of motives) of a restriction map of noncommutative spaces. The latter is obtained through the thermodynamics of the quantum statistical system associated to an endomotive (a noncommutative generalization of Artin motives). Semigroups of endomorphisms of algebraic varieties give rise canonically to such endomotives, with an action of the absolute Galois group. The semigroup of endomorphisms of the multiplicative group yields the Bost\u2013Connes system, from which one obtains, through the above procedure, the desired cohomological interpretation of the zeros of the Riemann zeta function. In the last section we also give a Lefschetz formula for the archimedean local L-factors of arithmetic varieties.", "doi": "10.1016/j.aim.2007.03.006", "issn": "0001-8708", "publisher": "Elsevier", "publication": "Advances in Mathematics", "publication_date": "2007-10-01", "series_number": "2", "volume": "214", "issue": "2", "pages": "761-831" }, { "id": "authors:6kna4-ffz46", "collection": "authors", "collection_id": "6kna4-ffz46", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:CONjnt07", "type": "article", "title": "Quantum statistical mechanics over function fields", "author": [ { "family_name": "Consani", "given_name": "Caterina", "clpid": "Consani-C" }, { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" } ], "abstract": "In this paper we construct a noncommutative space of \"pointed Drinfeld modules\" that generalizes to the case of function fields the noncommutative spaces of commensurability classes of Q-lattices. It extends the usual moduli spaces of Drinfeld modules to possibly degenerate level structures. In the second part of the paper we develop some notions of quantum statistical mechanics in positive characteristic and we show that, in the case of Drinfeld modules of rank one, there is a natural time evolution on the associated noncommutative space, which is closely related to the positive characteristic L-functions introduced by Goss. The points of the usual moduli space of Drinfeld modules define KMS functionals for this time evolution. We also show that the scaling action on the dual system is induced by a Frobenius action, up to a Wick rotation to imaginary time.", "doi": "10.1016/j.jnt.2006.12.002", "issn": "0022-314X", "publisher": "Elsevier", "publication": "Journal of Number Theory", "publication_date": "2007-04", "series_number": "2", "volume": "123", "issue": "2", "pages": "487-528" }, { "id": "authors:kb2xz-vwr68", "collection": "authors", "collection_id": "kb2xz-vwr68", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:CHAatmp07", "type": "article", "title": "Gravity and the standard model with neutrino mixing", "author": [ { "family_name": "Chamseddine", "given_name": "Ali H.", "clpid": "Chamseddine-A-H" }, { "family_name": "Connes", "given_name": "Alain", "clpid": "Connes-A" }, { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" } ], "abstract": "We present an effective unified theory based on noncommutative geometry for the standard model with neutrino mixing, minimally coupled to gravity. The unification is based on the symplectic unitary group in Hilbert space and on the spectral action. It yields all the detailed structure of the standard model with several predictions at unification scale. Besides the familiar predictions for the gauge couplings as for GUT theories, it predicts the Higgs scattering parameter and the sum of the squares of Yukawa couplings. From these relations, one can extract predictions at low energy, giving in particular a Higgs mass around 170 GeV and a top mass compatible with present experimental value. The geometric picture that emerges is that space-time is the product of an ordinary spin manifold (for which the theory would deliver Einstein gravity) by a finite noncommutative geometry F. The discrete space F is of KO-dimension 6 modulo 8 and of metric dimension 0, and accounts for all the intricacies of the standard model with its spontaneous symmetry breaking Higgs sector.", "doi": "10.4310/ATMP.2007.v11.n6.a3", "issn": "1095-0761", "publisher": "International Press", "publication": "Advances in Theoretical and Mathematical Physics", "publication_date": "2007", "series_number": "6", "volume": "11", "issue": "6", "pages": "991-1089" }, { "id": "authors:6ap12-1wh22", "collection": "authors", "collection_id": "6ap12-1wh22", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20110808-140919868", "type": "article", "title": "KMS states and complex multiplication", "author": [ { "family_name": "Connes", "given_name": "Alain", "clpid": "Connes-A" }, { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" }, { "family_name": "Ramachandran", "given_name": "Niranjan", "clpid": "Ramachandran-N" } ], "abstract": "We construct a quantum statistical mechanical system which generalizes the Bost\u2013Connes system to imaginary quadratic fields K of arbitrary class number and fully incorporates the explicit class field theory for such fields. This system admits the Dedekind zeta function as partition function and the id\u00e8le class group as group of symmetries. The extremal KMS states at zero temperature intertwine this symmetry with the Galois action on the values of the states on the arithmetic subalgebra. The geometric notion underlying the construction is that of commensurability of K-lattices.", "doi": "10.1007/s00029-005-0013-x", "issn": "1022-1824", "publisher": "Springer", "publication": "Selecta Mathematica - New Series", "publication_date": "2006-04", "series_number": "3-4", "volume": "11", "issue": "3-4", "pages": "325-347" }, { "id": "authors:63zzg-arw72", "collection": "authors", "collection_id": "63zzg-arw72", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20110224-082841614", "type": "article", "title": "Q-lattices: Quantum statistical mechanics and Galois theory", "author": [ { "family_name": "Connes", "given_name": "Alain", "clpid": "Connes-A" }, { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" } ], "abstract": "We review our recent results on the noncommutative geometry of Q-lattices modulo commensurability. We discuss the cases of 1-dimensional and 2-dimensional Q-lattices. In the first case, we show that, by considering commensurability classes of 1-dimensional Q-lattices up to scaling, one recovers the Bost-Connes quantum statistical mechanical system, whose zero temperature KMS states intertwine the symmetries of the system with the Galois action of Gal(Q/Q). In the 2-dimensional case, commensurability classes of Q-lattices up to scaling give rise to another quantum statistical mechanical system, whose symmetries are the automorphisms of the modular field, and whose (generic) zero temperature KMS states intertwine the action of these symmetries with the Galois action on an embedding in C of the modular field. Following our joint work with Ramachandran, we then show how the noncommutative spaces associated to commensurability classes of Q-lattices up to scale have a natural geometric interpretation as noncommutative versions of the Shimura varieties Sh(GL_1,{\u00b11}) in the Bost-Connes case and Sh(GL_2, H^\u00b1) in the case of the GL_2 system. We also show how this leads naturally to the construction of a system generalizing the Bost-Connes system that fully recovers the explicit class field theory of imaginary quadratic fields.", "doi": "10.1016/j.geomphys.2005.04.010", "issn": "0393-0440", "publisher": "Elsevier", "publication": "Journal of Geometry and Physics", "publication_date": "2006-01", "series_number": "1", "volume": "56", "issue": "1", "pages": "2-23" }, { "id": "authors:aw0e1-bab18", "collection": "authors", "collection_id": "aw0e1-bab18", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20110224-082841783", "type": "article", "title": "Quantum fields and motives", "author": [ { "family_name": "Connes", "given_name": "Alain", "clpid": "Connes-A" }, { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" } ], "abstract": "This is a survey of our results on the relation between perturbative renormalization and motivic Galois theory. The main result is that all quantum field theories share a common universal symmetry realized as a motivic Galois group, whose action is dictated by the divergences and generalizes that of the renormalization group. The existence of such a group was conjectured by P. Cartier based on number theoretic evidence and on the Connes-Kreimer theory of perturbative renormalization. The group provides a universal formula for counterterms and is obtained via a Riemann-Hilbert correspondence classifying equivalence classes of flat equisingular bundles, where the equisingularity condition corresponds to the independence of the counterterms on the mass scale.", "doi": "10.1016/j.geomphys.2005.04.004", "issn": "0393-0440", "publisher": "Elsevier", "publication": "Journal of Geometry and Physics", "publication_date": "2006-01", "series_number": "1", "volume": "56", "issue": "1", "pages": "55-85" }, { "id": "authors:fd2v8-vwd20", "collection": "authors", "collection_id": "fd2v8-vwd20", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20110817-140437615", "type": "article", "title": "Renormalization and motivic galois theory", "author": [ { "family_name": "Connes", "given_name": "Alain", "clpid": "Connes-A" }, { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" } ], "abstract": "We investigate the nature of divergences in quantum field theory, showing that they are organized in the structure of a certain \"motivic Galois group\" U*, which is uniquely determined and universal with respect to the set of physical theories. The renormalization group can be identified canonically with a one-parameter subgroup of U*. The group U* arises through a Riemann-Hilbert correspondence. Its representations classify equisingular flat vector bundles, where the equisingularity condition is a geometric formulation of the fact that in quantum field theory the counterterms are independent of the choice of a unit of mass. As an algebraic group scheme, U* is a semidirect product by the multiplicative group G_m of a prounipotent group scheme whose Lie algebra is freely generated by one generator in each positive integer degree. There is a universal singular frame in which all divergences disappear. When computed as iterated integrals, its coefficients are certain rational numbers that appear in the local index formula of Connes-Moscovici. When working with formal Laurent series over \u211a, the data of equisingular flat vector bundles define a Tannakian category whose properties are reminiscent of a category of mixed Tate motives.", "doi": "10.1155/S1073792804143122", "issn": "1073-7928", "publisher": "Oxford University Press", "publication": "International Mathematics Research Notices", "publication_date": "2004", "volume": "76", "pages": "4073-4091" }, { "id": "authors:ay5x0-y5c29", "collection": "authors", "collection_id": "ay5x0-y5c29", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20111013-134118921", "type": "article", "title": "Limiting modular symbols and the Lyapunov spectrum", "author": [ { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" } ], "abstract": "This paper consists of variations upon a theme, that of limiting modular symbols introduced in [17]. We show that, on level sets of the Lyapunov exponent for the shift map T of the continued fraction expansion, the limiting modular symbol can be computed as a Birkhoff average. We show that the limiting modular symbols vanish almost everywhere on T-invariant subsets for which a corresponding transfer operator has a good spectral theory, thus improving the weak convergence result proved in [17]. We also show that, even when the limiting modular symbol vanishes, it is possible to construct interesting non-trivial homology classes on modular curves that are associated to non-closed geodesics. These classes are related to \"automorphic series\", defined in terms of successive denominators of continued fraction expansion, and their integral averages are related to certain Mellin transforms of modular forms of weight two considered in [17]. We discuss some variants of the Selberg zeta function that sum over certain classes of closed geodesics, and their relation to Fredholm determinants of transfer operators. Finally, we argue that one can use T-invariant subsets to enrich the picture of non-commutative geometry at the boundary of modular curves presented in [17].", "doi": "10.1016/S0022-314X(02)00044-6", "issn": "0022-314X", "publisher": "Elsevier", "publication": "Journal of Number Theory", "publication_date": "2003-02", "series_number": "2", "volume": "98", "issue": "2", "pages": "348-376" }, { "id": "authors:6q22f-nrs08", "collection": "authors", "collection_id": "6q22f-nrs08", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20111019-090624789", "type": "article", "title": "Spectral triples in Arakelov geometry", "author": [ { "family_name": "Consani", "given_name": "Caterina", "clpid": "Consani-C" }, { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" } ], "abstract": "In this Note, we use Connes' theory of spectral triples to provide a connection between Manin's model of the dual graph of the fiber at infinity of an Arakelov surface and the cohomology of the mapping cone of the local monodromy.", "doi": "10.1016/S1631-073X(02)02569-4", "issn": "1631-073X", "publisher": "Elsevier", "publication": "Comptes Rendus Mathematique", "publication_date": "2002-11-15", "series_number": "10", "volume": "335", "issue": "10", "pages": "779-784" }, { "id": "authors:fjrf4-taq62", "collection": "authors", "collection_id": "fjrf4-taq62", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20111019-085917147", "type": "article", "title": "Weak UCP and Perturbed Monopole Equations", "author": [ { "family_name": "Booss-Bavnbek", "given_name": "Bernhelm", "clpid": "Booss-Bavnbek-B" }, { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" }, { "family_name": "Wang", "given_name": "Bai-Ling", "clpid": "Wang-B-L" } ], "abstract": "We give a simple proof of weak Unique Continuation Property for perturbed Dirac operators, using the Carleman inequality. We apply the result to a class of perturbations of the Seiberg\u2013Witten monopole equations that arise in Floer theory.", "doi": "10.1142/S0129167X02001551", "issn": "0129-167X", "publisher": "World Scientific Publishing", "publication": "International Journal of Mathematics", "publication_date": "2002-11", "series_number": "9", "volume": "13", "issue": "9", "pages": "987-1008" }, { "id": "authors:rf9k8-w4643", "collection": "authors", "collection_id": "rf9k8-w4643", "cite_using_url": "https://authors.library.caltech.edu/records/rf9k8-w4643", "type": "article", "title": "Continued fractions, modular symbols, and noncommutative geometry", "author": [ { "family_name": "Manin", "given_name": "Yuri I.", "clpid": "Manin-Y-I" }, { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" } ], "abstract": "
\n
\n

Using techniques introduced by D. Mayer, we prove an extension of the classical Gauss-Kuzmin theorem about the distribution of continued fractions, which in particular allows one to take into account some congruence properties of successive convergents. This result has an application to the Mixmaster Universe model in general relativity. We then study some averages involving modular symbols and show that Dirichlet series related to modular forms of weight 2 can be obtained by integrating certain functions on real axis defined in terms of continued fractions. We argue that the quotient PGL(2, Z)\\P1(R) should be considered as noncommutative modular curve, and show that the modular complex can be seen as a sequence of K 0-groups of the related crossed-product C *-algebras.

\n
\n
\n\n
\n
\n\n\n
\n\n
\n
", "doi": "10.1007/s00029-002-8113-3", "issn": "1022-1824", "publisher": "Birkh\u00e4user", "publication": "Selecta Mathematica (New Series)", "publication_date": "2002-09", "volume": "8", "pages": "475-521" }, { "id": "authors:772f4-g1n44", "collection": "authors", "collection_id": "772f4-g1n44", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20111026-110403593", "type": "article", "title": "Seiberg\u2013Witten and Casson\u2013Walker Invariants for Rational Homology 3-Spheres", "author": [ { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" }, { "family_name": "Wang", "given_name": "Bai-Ling", "clpid": "Wang-B-L" } ], "abstract": "We consider a modified version of the Seiberg\u2013Witten invariants for rational homology\n3-spheres, obtained by adding to the original invariants a correction term which is a combination\nof \u03b7-invariants. We show that these modified invariants are topological invariants.\nWe prove that an averaged version of these modified invariants equals the Casson\u2013Walker\ninvariant. In particular, this result proves an averaged version of a conjecture of Ozsv\u00e1th\nand Szab\u00f3 on the equivalence between their \u03b8 invariant and the Seiberg\u2013Witten invariant of\nrational homology 3-spheres.", "doi": "10.1023/A:1016299716922", "issn": "0046-5755", "publisher": "Springer", "publication": "Geometriae Dedicata", "publication_date": "2002-04", "series_number": "1", "volume": "91", "issue": "1", "pages": "45-58" }, { "id": "authors:nttvq-zkb10", "collection": "authors", "collection_id": "nttvq-zkb10", "cite_using_url": "https://authors.library.caltech.edu/records/nttvq-zkb10", "type": "article", "title": "Holography principle and arithmetic of algebraic curves", "author": [ { "family_name": "Manin", "given_name": "Yuri I.", "clpid": "Manin-Y-I" }, { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" } ], "abstract": "

According to the holography principle (due to G. 't Hooft, L. Susskind, J. Maldacena, et al.), quantum gravity and string theory on certain manifolds with boundary can be studied in terms of a conformal field theory on the boundary. Only a few mathematically exact results corroborating this exciting program are known. In this paper we interpret from this perspective several constructions which arose initially in the arithmetic geometry of algebraic curves. We show that the relation between hyperbolic geometry and Arakelov geometry at arithmetic infinity involves exactly the same geometric data as the Euclidean AdS3 holography of black holes. Moreover, in the case of Euclidean AdS2 holography, we present some results on bulk/boundary correspondence where the boundary is a non-commutative space.

", "doi": "10.4310/ATMP.2001.v5.n3.a6", "issn": "1095-0753", "publisher": "International Press", "publication": "Advances in Theoretical and Mathematical Physics", "publication_date": "2002", "series_number": "3", "volume": "5", "issue": "3", "pages": "617-650" }, { "id": "authors:a3e9r-6bh46", "collection": "authors", "collection_id": "a3e9r-6bh46", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20111104-094350111", "type": "article", "title": "Equivariant Seiberg-Witten Floer homology", "author": [ { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" }, { "family_name": "Wang", "given_name": "Bai-Ling", "clpid": "Wang-Bai-Ling" } ], "abstract": "In this paper we construct, for all compact oriented three- manifolds, a U(1)-equivariant version of Seiberg-Witten Floer homology, which is invariant under the choice of metric and perturbation. We give a detailed analysis of the boundary structure of the monopole moduli spaces, compactified to smooth manifolds with corners. The proof of the independence of metric and perturbation is then obtained via an analysis of all the relevant obstruction bundles and sections, and the corresponding gluing theorems. The paper also contains a discussion of the chamber structure for the Seiberg-Witten invariant for rational homology 3-spheres, and proofs of the wall crossing formula, obtained by studying the exact sequences relating the equivariant and the non-equivariant Floer homologies and by a local model at the reducible monopole.", "doi": "10.4310/CAG.2001.v9.n3.a1", "issn": "1019-8385", "publisher": "International Press", "publication": "Communications in Analysis and Geometry", "publication_date": "2001-07", "series_number": "3", "volume": "9", "issue": "3", "pages": "451-639" }, { "id": "authors:4fxym-f9f97", "collection": "authors", "collection_id": "4fxym-f9f97", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20111104-093842954", "type": "article", "title": "Twisted Index Theory on Good Orbifolds, II: Fractional Quantum Numbers", "author": [ { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" }, { "family_name": "Mathai", "given_name": "Varghese", "clpid": "Mathai-V" } ], "abstract": "This paper uses techniques in noncommutative geometry as developed by Alain Connes [Co2], in order to study the twisted higher index theory of elliptic operators on orbifold covering spaces of compact good orbifolds, which are invariant under a projective action of the orbifold fundamental group, continuing our earlier work [MM]. We also compute the range of the higher cyclic traces on K-theory for cocompact Fuchsian groups, which is then applied to determine the range of values of the Connes\u2013Kubo Hall conductance in the discrete model of the quantum Hall effect on the hyperbolic plane, generalizing earlier results in [Bel+E+S], [CHMM]. The new phenomenon that we observe in our case is that the Connes\u2013Kubo Hall conductance has plateaux at integral multiples of a fractional valued topological invariant, namely the orbifold Euler characteristic. Moreover the set of possible fractions has been determined, and is compared with recently available experimental data. It is plausible that this might shed some light on the mathematical mechanism responsible for fractional quantum numbers.", "doi": "10.1007/s002200000351", "issn": "0010-3616", "publisher": "Springer", "publication": "Communications in Mathematical Physics", "publication_date": "2001-02", "series_number": "1", "volume": "217", "issue": "1", "pages": "55-87" }, { "id": "authors:bscs2-sjw02", "collection": "authors", "collection_id": "bscs2-sjw02", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20170713-095029363", "type": "article", "title": "Twisted index theory on good orbifolds, I: noncommutative Bloch theory", "author": [ { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" }, { "family_name": "Mathai", "given_name": "Varghese", "clpid": "Mathai-V" } ], "abstract": "We study the twisted index theory of elliptic operators on orbifold covering spaces of compact good orbifolds, which are invariant under a projective action of the orbifold fundamental group. We apply these results to obtain qualitative results on real and complex hyperbolic spaces in two and four dimensions, related to generalizations of the Bethe\u2013Sommerfeld conjecture and the Ten Martini Problem, on the spectrum of self adjoint elliptic operators which are invariant under a projective action of a discrete cocompact group.", "doi": "10.1142/S0219199799000213", "issn": "0219-1997", "publisher": "World Scientific Publishing", "publication": "Communications in Contemporary Mathematics", "publication_date": "1999-11", "series_number": "4", "volume": "1", "issue": "4", "pages": "553-587" }, { "id": "authors:6vswr-vqc95", "collection": "authors", "collection_id": "6vswr-vqc95", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20170713-092558409", "type": "article", "title": "Seiberg-Witten Floer Homology and Heegaard Splittings", "author": [ { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" } ], "abstract": "The dimensional reduction of Seiberg-Witten theory defines a gauge theory of compact connected three-manifolds. Solutions of the equations modulo gauge symmetries on a three-manifold Y can be interpreted as the critical points of a functional defined on an infinite dimensional configuration space of U(1)-connections and spinors. The original Seiberg-Witten equations on the infinite cylinder Y \u00d7 IR are the downward gradient flow of the functional. Thus, it is possible to construct an infinite dimensional\nMorse theory. The associated Morse homology is the analogue in the context of Seiberg-Witten theory of Floer's instanton homology constructed using Yang-Mills gauge theory. The construction and the properties of\nthis Seiberg-Witten Floer homology are essentially different according to whether the three-manifold Y is a homology sphere or has non-trivial rational homology. In this work we construct the Seiberg-Witten Floer homology for three-manifolds with b^1(Y ) > 0. We define an associated Casson-like invariant and we prove that it satisfies the expected intersection formula under a Heegaard splitting of the three-manifold.", "doi": "10.1142/S0129167X96000359", "issn": "0129-167X", "publisher": "World Scientific Publishing", "publication": "International Journal of Mathematics", "publication_date": "1996-10", "series_number": "5", "volume": "7", "issue": "5", "pages": "671-696" } ]