[
{
"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
In this chapter, we sketch an approach to the problems of equivariant birational geometry developed by M. Kontsevich and Yu. Tschinkel [Invent. Math. (2019), 415–432], where Burnside invariants were introduced. We are making explicit the role of Nori constructions in this environment.
",
"doi": "10.4171/irma/34/31",
"isbn": "978-3-98547-024-2",
"publisher": "EMS Press",
"publication": "Essays in Geometry: Dedicated to Norbert A'Campo",
"publication_date": "2023-08-08",
"pages": "761-780"
},
{
"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:3z2pr-3dv08",
"collection": "authors",
"collection_id": "3z2pr-3dv08",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20210825-184621604",
"type": "monograph",
"title": "Moufang Patterns and Geometry of Information",
"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": "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. In 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 3-webs and Malcev algebras. We then present a new construction of (noncommutative) Moufang loops associated to almost-symplectic structures over finite fields, and use then to construct a new class of code loops with associated quantum error-correcting codes and networks of perfect tensors.",
"doi": "10.48550/arXiv.2107.07486",
"publisher": "arXiv",
"publication_date": "2021-07-15"
},
{
"id": "authors:c78xs-6qc80",
"collection": "authors",
"collection_id": "c78xs-6qc80",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20211222-457136400",
"type": "book_section",
"title": "Topological Model of Neural Information Networks",
"book_title": "Geometric Science of Information",
"author": [
{
"family_name": "Marcolli",
"given_name": "Matilde",
"orcid": "0000-0002-2045-2907",
"clpid": "Marcolli-M"
}
],
"contributor": [
{
"family_name": "Nielsen",
"given_name": "Frank",
"clpid": "Nielsen-Frank"
},
{
"family_name": "Barbaresco",
"given_name": "Fr\u00e9d\u00e9ric",
"clpid": "Barbaresco-Fr\u00e9d\u00e9ric"
}
],
"abstract": "This is a brief overview of an ongoing research project, involving topological models of neural information networks and the development of new versions of associated information measures that can be seen as possible alternatives to integrated information. Among the goals are a geometric modeling of a \"space of qualia\" and an associated mechanism that constructs and transforms representations from neural codes topologically. The more mathematical aspects of this project stem from the recent joint work of the author and Yuri Manin [18], while the neuroscience modeling aspects are part of an ongoing collaboration of the author with Doris Tsao.",
"doi": "10.1007/978-3-030-80209-7_67",
"isbn": "9783030802080",
"publisher": "Springer",
"place_of_publication": "Cham",
"publication_date": "2021-07-14",
"pages": "623-633"
},
{
"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:4713m-5zz85",
"collection": "authors",
"collection_id": "4713m-5zz85",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20210825-184618189",
"type": "monograph",
"title": "Syntactic structures and the general Markov models",
"author": [
{
"family_name": "Gakkhar",
"given_name": "Sitanshu",
"clpid": "Gakkhar-Sitanshu"
},
{
"family_name": "Marcolli",
"given_name": "Matilde",
"orcid": "0000-0002-2045-2907",
"clpid": "Marcolli-M"
}
],
"abstract": "We further the theme of studying syntactic structures data from Longobardi (2017b), Collins (2010), Ceolin et al. (2020) and Koopman (2011) using general Markov models initiated in Shu et al. (2017), exploring the question of how consistent the data is with the idea that general Markov models. The ideas explored in the present paper are more generally applicable than to the setting of syntactic structures, and can be used when analyzing consistency of data with general Markov models. Additionally, we give an interpretation of the methods of Ceolin et al. (2020) as an infinite sites evolutionary model and compare it to the Markov model and explore each in the context of evolutionary processes acting on human language syntax.",
"doi": "10.48550/arXiv.2104.08462",
"publisher": "arXiv",
"publication_date": "2021-04-17"
},
{
"id": "authors:yzhga-z9829",
"collection": "authors",
"collection_id": "yzhga-z9829",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20210825-184614782",
"type": "monograph",
"title": "Homotopy Spectra and Diophantine Equations",
"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": "Arguably, the first bridge between vast, ancient, but disjoint domains of mathematical knowledge, - topology and number theory, - was built only during the last fifty years. This bridge is the theory of spectra in stable homotopy theory. This connection poses the challenge: discover new information in number theory using the independently-developed machinery of homotopy theory. In this combined research/survey paper we suggest to apply homotopy spectra to the problem of distribution of rational points upon algebraic manifolds.",
"doi": "10.48550/arXiv.2101.00197",
"publisher": "arXiv",
"publication_date": "2021-01-01"
},
{
"id": "authors:fenqz-ggn88",
"collection": "authors",
"collection_id": "fenqz-ggn88",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20210825-184607934",
"type": "monograph",
"title": "Gluing Noncommutative 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-Schack complexes, for extending to quantized twistor spaces the Donaldson-Friedman 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 noncommutative geometry. We discuss specific aspects of the gluing construction for these different quantization procedures.",
"doi": "10.48550/arXiv.2012.02823",
"publisher": "arXiv",
"publication_date": "2020-12-04"
},
{
"id": "authors:ns6ke-29z07",
"collection": "authors",
"collection_id": "ns6ke-29z07",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20210825-184601111",
"type": "monograph",
"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",
"clpid": "Panangaden-Jane"
}
],
"abstract": "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 GL\u2082 and an analog for SL\u2082. The structure of KMS 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.",
"doi": "10.48550/arXiv.2006.16897",
"publisher": "arXiv",
"publication_date": "2020-06-30"
},
{
"id": "authors:099r5-twh05",
"collection": "authors",
"collection_id": "099r5-twh05",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20210825-184554285",
"type": "monograph",
"title": "Dessins for Modular Operad and Grothendieck-Teichm\u00fcller Group",
"author": [
{
"family_name": "Combe",
"given_name": "No\u00e9mie C.",
"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": "A part of Grothendieck's program for studying the Galois group G_\u211a of the field of all algebraic numbers \u211a emerged from his insight that one should lift its action upon \u211a to the action of G_\u211a upon the (appropriately defined) profinite completion of \u03c0\u2081(\u2119\u00b9\u2216{0,1,\u221e}). The latter admits a good combinatorial encoding via finite graphs \"dessins d'enfant\". \n\nThis part was actively developing during the last decades, starting with foundational works of A. Belyi, V. Drinfeld and Y. Ihara. \n\nOur brief note concerns another part of Grothendieck program, in which its geometric environment is extended to moduli spaces of algebraic curves, more specifically, stable curves of genus zero with marked/labelled points. Our main goal is to show that dual graphs of such curves may play the role of \"modular dessins\" in an appropriate operadic context.",
"doi": "10.48550/arXiv.2006.13663",
"publisher": "arXiv",
"publication_date": "2020-06-03"
},
{
"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:5390y-6yb83",
"collection": "authors",
"collection_id": "5390y-6yb83",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20200918-070725967",
"type": "book_section",
"title": "Homotopy types and geometries below Spec(\u2124)",
"book_title": "Dynamics: Topology and Numbers",
"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"
}
],
"contributor": [
{
"family_name": "Moree",
"given_name": "Pieter",
"clpid": "Moree-P"
},
{
"family_name": "Pohl",
"given_name": "Anke",
"clpid": "Pohl-A"
},
{
"family_name": "Snoha",
"given_name": "L'ubom\u00edr",
"clpid": "Snoha-L"
},
{
"family_name": "Ward",
"given_name": "Tom",
"clpid": "Ward-T"
}
],
"abstract": "After the first heuristic ideas about 'the field of one element' F\u2081 and 'geometry in characteristics 1' (J. Tits, C. Deninger, M. Kapranov, A. Smirnov et al.), there were developed several general approaches to the construction of 'geometries below Spec Z'. Homotopy theory and the 'the brave new algebra' were taking more and more important places in these developments, systematically explored by B. To\u00ebn and M. Vaqui\u00e9, among others.\nThis article contains a brief survey and some new results on counting problems in this context, including various approaches to zeta--functions and generalised scissors congruences.\nThe new version includes considerable extensions and revisions suggested by I. Zakharevich.",
"doi": "10.1090/conm/744/14978",
"isbn": "978-1-4704-5100-4",
"publisher": "American Mathematical Society",
"place_of_publication": "Providence, RI",
"publication_date": "2020",
"pages": "27-56"
},
{
"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
\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
\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
",
"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: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: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:mqm5j-zxy26",
"collection": "authors",
"collection_id": "mqm5j-zxy26",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20170713-091711317",
"type": "book_section",
"title": "Modular shadows and the Levy-Mellin \u221e-adic transform",
"book_title": "Modular Forms on Schiermonnikoog",
"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"
}
],
"contributor": [
{
"family_name": "Edixhoven",
"given_name": "Bas",
"clpid": "Edixhoven-B"
},
{
"family_name": "van der Geer",
"given_name": "Gerard",
"clpid": "van-der-Geer-G"
},
{
"family_name": "Moonen",
"given_name": "Ben",
"clpid": "Moonen-B"
}
],
"abstract": "This paper continues the study of the structures induced on the \"invisible boundary\" of the modular tower and extends some results of [MaMar1]. We start with a systematic formalism of pseudo\u2013measures generalizing the well\u2013known theory of modular symbols for SL(2). These pseudo\u2013measures, and the related integral formula which we call the L\u00e9vy\u2013Mellin transform, can be considered as an \"\u221e\u2013adic\" version of Mazur's p\u2013adic measures that have been introduced in the seventies in the theory of p\u2013adic interpolation of the Mellin transforms of cusp forms, cf. [Ma2]. A formalism of iterated L\u00e9vy\u2013Mellin transform in the style of [Ma3] is sketched. Finally, we discuss the invisible boundary from the perspective of non\u2013commutative geometry.",
"doi": "10.1017/CBO9780511543371.012",
"isbn": "9780521493543",
"publisher": "Cambridge University Press",
"place_of_publication": "Cambridge",
"publication_date": "2008",
"pages": "189-238"
},
{
"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:wzfhb-ngd47",
"collection": "authors",
"collection_id": "wzfhb-ngd47",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20170713-083009140",
"type": "monograph",
"title": "Noncommutative geometry on trees and buildings",
"author": [
{
"family_name": "Cornelissen",
"given_name": "Gunther",
"clpid": "Cornelissen-G"
},
{
"family_name": "Marcolli",
"given_name": "Matilde",
"orcid": "0000-0002-2045-2907",
"clpid": "Marcolli-M"
},
{
"family_name": "Reihani",
"given_name": "Kamran",
"clpid": "Reihani-K"
},
{
"family_name": "Vdovina",
"given_name": "Alina",
"clpid": "Vdovina-A"
}
],
"abstract": "We describe the construction of theta summable and finitely summable spectral triples associated to Mumford curves and some classes of higher dimensional buildings. The finitely summable case is constructed by considering the stabilization of the algebra of the dual graph of the special fiber of the Mumford curve and a variant of the Antonescu-Christensen spectral geometries for AF algebras. The information on the Schottky uniformization is encoded in the spectral geometry through the Patterson-Sullivan measure on the limit set. Some higher rank cases are obtained by adapting the construction for trees.",
"doi": "10.48550/arXiv.math/0604114",
"publication_date": "2006-04-05"
},
{
"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:jmgdc-tn948",
"collection": "authors",
"collection_id": "jmgdc-tn948",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20170713-081431578",
"type": "monograph",
"title": "A walk in the noncommutative garden",
"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 text is written for the volume of the school/conference \"Noncommutative Geometry 2005\" held at IPM Tehran. It gives a survey of methods and results in noncommutative geometry, based on a discussion of significant examples of noncommutative spaces in geometry, number theory, and physics. The paper also contains an outline (the \"Tehran program\") of ongoing joint work with Consani on the noncommutative geometry of the adeles class space and its relation to number theoretic questions.",
"doi": "10.48550/arXiv.0601054",
"publication_date": "2006-01-03"
},
{
"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: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:fmnf8-fne39",
"collection": "authors",
"collection_id": "fmnf8-fne39",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20170712-154917444",
"type": "monograph",
"title": "Towards the fractional quantum Hall effect: a noncommutative geometry perspective",
"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": "In this paper we give a survey of some models of the integer and fractional quantum Hall effect based on noncommutative geometry. We begin by recalling some classical geometry of electrons in solids and the passage to noncommutative geometry produced by the presence of a magnetic field. We recall how one can obtain this way a single electron model of the integer quantum Hall effect. While in the case of the integer quantum Hall effect the underlying geometry is Euclidean, we then discuss a model of the fractional quantum Hall effect, which is based on hyperbolic geometry simulating the multi-electron interactions. We derive the fractional values of the Hall conductance as integer multiples of orbifold Euler characteristics. We compare the results with experimental data.",
"doi": "10.48550/arXiv.0502356",
"publication_date": "2005-02-15"
},
{
"id": "authors:7557s-69t47",
"collection": "authors",
"collection_id": "7557s-69t47",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20170713-080604764",
"type": "monograph",
"title": "From Physics to Number Theory via Noncommutative Geometry, Part II: Renormalization, the Riemann-Hilbert correspondence, 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 give here a comprehensive treatment of the mathematical theory of perturbative renormalization (in the minimal subtraction scheme with dimensional regularization), in the framework of the Riemann\u2013Hilbert correspondence and\nmotivic Galois theory. We give a detailed overview of the work of Connes\u2013Kreimer [31], [32]. We also cover some background material on affine group schemes, Tannakian categories, the Riemann\u2013Hilbert problem in the regular singular and irregular case, and a brief introduction to motives and motivic Galois theory. We then give a complete account of our results on renormalization and motivic Galois theory announced in [35].",
"doi": "10.48550/arXiv.0411114",
"publication_date": "2004-11-11"
},
{
"id": "authors:04m4t-1vr74",
"collection": "authors",
"collection_id": "04m4t-1vr74",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20170713-083048947",
"type": "monograph",
"title": "Lectures on Arithmetic Noncommutative Geometry",
"author": [
{
"family_name": "Marcolli",
"given_name": "Matilde",
"orcid": "0000-0002-2045-2907",
"clpid": "Marcolli-M"
}
],
"abstract": "This is the text of a series of five lectures given by the author at the \"Second Annual Spring Institute on Noncommutative Geometry and Operator Algebras\" held at Vanderbilt University in May 2004. It is meant as an overview of recent results illustrating the interplay between noncommutative geometry and arithmetic geometry/number theory.",
"doi": "10.48550/arXiv.math/0409520",
"publication_date": "2004-09-27"
},
{
"id": "authors:dejrb-xjx76",
"collection": "authors",
"collection_id": "dejrb-xjx76",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20170713-085828455",
"type": "monograph",
"title": "Archimedean cohomology revisited",
"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": "Archimedean cohomology provides a cohomological interpretation for the calculation of the local L-factors at archimedean places as zeta regularized determinant of a log of Frobenius. In this paper we investigate further the properties of the Lefschetz and log of monodromy operators on this cohomology. We use the Connes-Kreimer formalism of renormalization to obtain a fuchsian connection whose residue is the log of the monodromy. We also present a dictionary of analogies between the geometry of a tubular neighborhood of the \"fiber at arithmetic infinity\" of an arithmetic variety and the complex of nearby cycles in the geometry of a degeneration over a disk, and we recall Deninger's approach to the archimedean cohomology through an interpretation as global sections of a analytic Rees sheaf. We show that action of the Lefschetz, the log of monodromy and the log of Frobenius on the archimedean cohomology combine to determine a spectral triple in the sense of Connes. The archimedean part of the Hasse-Weil L-function appears as a zeta function of this spectral triple. We also outline some formal analogies between this cohomological theory at arithmetic infinity and Givental's homological geometry on loop spaces.",
"doi": "10.48550/arXiv.0407480",
"publication_date": "2004-07-28"
},
{
"id": "authors:t8va9-8hj75",
"collection": "authors",
"collection_id": "t8va9-8hj75",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20170713-081805785",
"type": "monograph",
"title": "From Physics to Number Theory via Noncommutative Geometry. Part I: Quantum Statistical Mechanics of Q-lattices",
"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": "Several recent results reveal a surprising connection between modular forms and noncommutative geometry. The first occurrence came from the classification of noncommutative three spheres, [C\u2013DuboisViolette-I] [C\u2013DuboisViolette-II]. Hard computations with the noncommutative analog of the Jacobian involving the ninth power of the Dedekind eta function were necessary in order to analyze the relation between such spheres and noncommutative nilmanifolds. Another occurrence can be seen in the computation of the explicit cyclic cohomology\nChern character of a spectral triple on SU_q(2) [C\u201302]. Another surprise came recently from a remarkable action of the Hopf algebra of transverse geometry of foliations of codimension one on the space of lattices modulo Hecke correspondences, described in the framework of noncommutative geometry, using a modular Hecke algebra obtained as the cross product of modular forms by the\naction of Hecke correspondences [C\u2013Moscovici-I] [C\u2013Moscovici-II]. This action determines a differentiable structure on this noncommutative space, related to the Rankin\u2013Cohen brackets of modular forms, and shows their compatibility with Hecke operators. Another instance where properties of modular forms can be recast in the context of noncommutative geometry can be found in the theory\nof modular symbols and Mellin transforms of cusp forms of weight two, which can be recovered from the geometry of the moduli space of Morita equivalence classes of noncommutative tori viewed as boundary of the modular curve [Manin\u2013M].",
"doi": "10.48550/arXiv.0404128",
"publication_date": "2004-04-06"
},
{
"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:qcvad-fpq10",
"collection": "authors",
"collection_id": "qcvad-fpq10",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20170713-082602443",
"type": "monograph",
"title": "Modular curves, C^* algebras, and chaotic cosmology",
"author": [
{
"family_name": "Marcolli",
"given_name": "Matilde",
"orcid": "0000-0002-2045-2907",
"clpid": "Marcolli-M"
}
],
"abstract": "We make some brief remarks on the relation of the mixmaster universe model of chaotic cosmology to the geometry of modular curves and to noncommutative geometry. We show that the full dynamics of the mixmaster universe\nis equivalent to the geodesic flow on the modular curve X_(\u03930(2)). We then consider a special class of solutions, with bounded number of cycles in each Kasner era, and describe their dynamical properties (invariant density, Lyapunov exponent, topological pressure). We relate these properties to the noncommutative geometry of a moduli space of such solutions, which is given by a Cuntz\u2013Krieger C^\u2217-algebra.",
"doi": "10.48550/arXiv.0312035",
"publication_date": "2003-12-11"
},
{
"id": "authors:4mq62-k2841",
"collection": "authors",
"collection_id": "4mq62-k2841",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20170713-075105845",
"type": "monograph",
"title": "Non-commutative geometry, dynamics, and infinity-adic 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 Arakelov theory a completion of an arithmetic surface is achieved by enlarging the group of divisors by formal linear combinations of the \"closed fibers at infinity\". Manin described the dual graph of any such closed fiber in terms of an infinite tangle of bounded geodesics in a hyperbolic\nhandlebody endowed with a Schottky uniformization. In this paper we consider arithmetic surfaces over the ring of integers in a number field, with fibers of genus g \u2265 2. We use Connes' theory of spectral triples to relate the hyperbolic geometry of the handlebody to Deninger's Archimedean cohomology and the cohomology of the cone of the local monodromy N at arithmetic infinity as introduced by the first author of this paper. First, we consider derived (cohomological) spectral data (A,H\u00b7(X^\u2217),\u0424), where the algebra is obtained from the SL(2,R) action on the cohomology of the cone, induced by the presence of a polarized Lefschetz module structure, and its restriction to the group ring of a Fuchsian Schottky group. In this setting we recover the alternating product of the Archimedean factors from a zeta function of a spectral triple. Then, we introduce a different\nconstruction, which is related to Manin's description of the dual graph of the fiber at infinity. We provide a geometric model for the dual graph as the mapping torus of a dynamical system T on a Cantor set. We consider a noncommutative space which describes the action of the Schottky group on its limit set and parameterizes the \"components of the closed fiber at infinity\". This can be identified with a Cuntz\u2013Krieger algebra O_A associated to a subshift of finite type. We\nconstruct a spectral triple for this noncommutative space, via a representation on the cochains of a \"dynamical cohomology\", defined in terms of the tangle of bounded geodesics in the handlebody. In both constructions presented in the paper, the Dirac operator agrees with the grading operator \u0424, that represents the \"logarithm of a Frobenius\u2013type operator\" on the Archimedean cohomology. In fact, the Archimedean cohomology embeds in the dynamical cohomology, compatibly with the action of a real Frobenius F_\u221e, so that the local factor can again be recovered from these data. The\nduality isomorphism on the cohomology of the cone of N corresponds to the pairing of dynamical homology and cohomology. This suggests the existence of a duality between the monodromy N and the dynamical map 1\u2212T. Moreover, the \"reduction mod infinity\" is described in terms of the\nhomotopy quotient associated to the noncommutative space OA and the \u03bc-map of Baum\u2013Connes. The geometric model of the dual graph can also be described as a homotopy quotient.",
"doi": "10.48550/arXiv.0205306",
"publication_date": "2003-10-27"
},
{
"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:k6wbf-5k411",
"collection": "authors",
"collection_id": "k6wbf-5k411",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20170713-080540298",
"type": "monograph",
"title": "Variants of 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": "For a rational homology 3-sphere Y with a Spin^c structure s, we show that simple algebraic manipulations of our construction of equivariant Seiberg-Witten Floer homology in [5] lead to a collection of variants HF^(SW,\u2212)_(*,U(1))(Y,s), HF^(SW,\u221e)_(*,U(1)(Y,s) HF^(SW,+)_(*,U(1))(Y,s), HF^(SW)_*(Y,s) and HF^(SW)_(red,*)(Y,s) which are topological invariants. We establish a long exact sequence relating HF^(SW,\u00b1)_(*,U(1))(Y,s) and HF^(SW,\u221e) _(*,U(1))(Y,s). We show they satisfy a duality under orientation reversal, and we explain their relation to the equivariant Seiberg-Witten Floer (co)homologies introduced in [5]. We conjecture the equivalence of these versions of equivariant Seiberg-Witten Floer homology with the Heegaard Floer invariants introduced by Ozsv\u00e1th and Szab\u00f3.",
"doi": "10.48550/arXiv.0211238",
"publication_date": "2002-11-15"
},
{
"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:3cw7z-prv14",
"collection": "authors",
"collection_id": "3cw7z-prv14",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20170713-075847963",
"type": "monograph",
"title": "Spectral triples from Mumford curves",
"author": [
{
"family_name": "Consani",
"given_name": "C.",
"clpid": "Consani-C"
},
{
"family_name": "Marcolli",
"given_name": "M.",
"orcid": "0000-0002-2045-2907",
"clpid": "Marcolli-M"
}
],
"abstract": "We construct spectral triples associated to Schottky--Mumford curves, in such a way that the local Euler factor can be recovered from the zeta functions of such spectral triples. We propose a way of extending this construction to the case where the curve is not k-split degenerate.",
"doi": "10.48550/arXiv.math/0210435",
"publication_date": "2002-10-28"
},
{
"id": "authors:p2n5v-9e676",
"collection": "authors",
"collection_id": "p2n5v-9e676",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20170713-075158422",
"type": "monograph",
"title": "New perspectives 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 survey, written for the proceedings of the VII meeting of the CNTA held in May 2002 in Montreal, we describe how Connes' theory of spectral triples provides a unified view, via noncommutative geometry, of the archimedean and the totally split degenerate fibers of an arithmetic surface.",
"doi": "10.48550/arXiv.math/0210357",
"publication_date": "2002-10-23"
},
{
"id": "authors:0h2cb-9e174",
"collection": "authors",
"collection_id": "0h2cb-9e174",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20170713-071539391",
"type": "monograph",
"title": "Triplets spectraux en g\u00e9om\u00e9trie d'Arakelov",
"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": "10.48550/arXiv.math/0209182",
"publication_date": "2002-09-15"
},
{
"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
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:p9bdm-qw833",
"collection": "authors",
"collection_id": "p9bdm-qw833",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20170713-073152604",
"type": "monograph",
"title": "Exact triangles in monopole homology and the Casson-Walker invariant",
"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": "The purpose of this paper is to give a general outline of the problem of the exact triangles in Seiberg\u2013Witten\u2013Floer theory. We present here the most general case, where the problem consists of producing a surgery formula relating the monopole homology of a compact oriented 3\u2013manifold Y with an embedded knot K, and the monopole homologies of some 3\u2013manifolds obtained by Dehn surgery on K.",
"doi": "10.48550/arXiv.math/0101127",
"publication_date": "2001-01-16"
},
{
"id": "authors:zhvr5-v9796",
"collection": "authors",
"collection_id": "zhvr5-v9796",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20170712-151923729",
"type": "monograph",
"title": "Exact triangles in Seiberg-Witten Floer theory. Part IV: Z-graded monopole 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-B-L"
}
],
"abstract": "Some aspects of the construction of SW Floer homology for manifolds with non-trivial rational homology are analyzed. In particular, the case of manifolds that are obtained as zero-surgery on a knot in a homology sphere, and for torsion spinc structures. We discuss relative invariants in the case of torsion spinc structures.",
"doi": "10.48550/arXiv.math/0009159",
"publication_date": "2000-09-15"
},
{
"id": "authors:s0sjj-kfm53",
"collection": "authors",
"collection_id": "s0sjj-kfm53",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20170712-152627045",
"type": "monograph",
"title": "Exact triangles in Seiberg-Witten Floer theory. Part III: proof of exactness",
"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": "This is the third part of the work on the exact triangles. We construct chain homomorphisms and show exactness of the resulting sequence.",
"doi": "10.48550/arXiv.math/0009157",
"publication_date": "2000-09-15"
},
{
"id": "authors:8cb8c-89p40",
"collection": "authors",
"collection_id": "8cb8c-89p40",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20170713-094456115",
"type": "monograph",
"title": "Exact triangles in Seiberg-Witten Floer theory. Part II: geometric limits of flow lines",
"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": "This is the second part of the proof of the exact traiangles in Seiberg-Witten Floer theory. We analyse the splitting and gluing of flow lines of the Chern-Simons-Dirac functional when the underlying three-manifold splits along a torus. (two corrections added)",
"doi": "10.48550/arXiv.math/9907080",
"publication_date": "2000-09-15"
},
{
"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:023hn-a7782",
"collection": "authors",
"collection_id": "023hn-a7782",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20170713-093723319",
"type": "monograph",
"title": "Twisted higher index theory on good orbifolds and 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": "In this paper, we 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, and we apply these results to obtain qualitative results, related\nto generalizations of the Bethe-Sommerfeld conjecture, on the spectrum of self adjoint elliptic operators which are invariant under a projective action of the orbifold fundamental group. We also compute the range of the higher traces on K-theory, which we then apply to compute the\nrange of values of the Hall conductance in the quantum Hall effect on the hyperbolic plane. The new phenomenon that we observe in this case is that the Hall conductance again has plateaus at all energy levels belonging to any gap in the spectrum of the Hamiltonian, where it is now shown to be equal to an integral multiple of a fractional valued invariant. Moreover the set of possible denominators is finite and has been explicitly determined. It is plausible that this might shed light on the mathematical mechanism responsible for fractional quantum numbers.",
"doi": "10.48550/arXiv.math/9803051",
"publication_date": "1998-03-12"
},
{
"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"
}
]