[ { "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: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:8nc85-zp950", "collection": "authors", "collection_id": "8nc85-zp950", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20180529-110159594", "type": "book_section", "title": "Syntactic Phylogenetic Trees", "book_title": "Foundations of Mathematics and Physics One Century After Hilbert", "author": [ { "family_name": "Shu", "given_name": "Kevin", "clpid": "Shu-Kevin" }, { "family_name": "Aziz", "given_name": "Sharjeel", "clpid": "Aziz-S" }, { "family_name": "Huynh", "given_name": "Vy-Luan", "clpid": "Huynh-Vy-Luan" }, { "family_name": "Warrick", "given_name": "David", "clpid": "Warrick-D" }, { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" } ], "contributor": [ { "family_name": "Kouneiher", "given_name": "Joseph", "clpid": "Kouneiher-J" } ], "abstract": "In light of recent controversies surrounding the use of computational methods for the reconstruction of phylogenetic trees of language families (especially the Indo-European family), a possible approach based on syntactic information, complementing other linguistic methods, appeared as a promising possibility, largely developed in recent years in Longobardi's Parametric Comparison Method. In this paper we identify several serious problems that arise in the use of syntactic data from the SSWL database for the purpose of computational phylogenetic reconstruction. We show that the most naive approach fails to produce reliable linguistic phylogenetic trees. We identify some of the sources of the observed problems and we discuss how they may be, at least partly, corrected by using additional information, such as prior subdivision into language families and subfamilies, and a better use of the information about ancient languages. We also describe how the use of phylogenetic algebraic geometry can help in estimating to what extent the probability distribution at the leaves of the phylogenetic tree obtained from the SSWL data can be considered reliable, by testing it on phylogenetic trees established by other forms of linguistic analysis. In simple examples, we find that, after restricting to smaller language subfamilies and considering only those SSWL parameters that are fully mapped for the whole subfamily, the SSWL data match extremely well reliable phylogenetic trees, according to the evaluation of phylogenetic invariants. This is a promising sign for the use of SSWL data for linguistic phylogenetics. We also argue how dependencies and nontrivial geometry/topology in the space of syntactic parameters would have to be taken into consideration in phylogenetic reconstructions based on syntactic data. A more detailed analysis of syntactic phylogenetic trees and their algebro-geometric invariants will appear elsewhere [33].", "doi": "10.1007/978-3-319-64813-2_14", "isbn": "978-3-319-64812-5", "publisher": "Springer", "place_of_publication": "Cham, Switzerland", "publication_date": "2018-05-27", "pages": "417-441" }, { "id": "authors:g0301-tca57", "collection": "authors", "collection_id": "g0301-tca57", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20170712-110411817", "type": "book_section", "title": "Prevalence and recoverability of syntactic parameters in sparse distributed memories", "book_title": "Geometric Science of Information. GSI 2017", "author": [ { "family_name": "Park", "given_name": "Jeong Joon", "clpid": "Park-Jeong-Joon" }, { "family_name": "Boettcher", "given_name": "Ronnel", "clpid": "Boettcher-R" }, { "family_name": "Zhao", "given_name": "Andrew", "clpid": "Zhao-Andrew" }, { "family_name": "Mun", "given_name": "Alex", "clpid": "Mun-Alex" }, { "family_name": "Yuh", "given_name": "Kevin", "clpid": "Yuh-Kevin" }, { "family_name": "Kumar", "given_name": "Vibhor", "clpid": "Kumar-Vibhor" }, { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" } ], "contributor": [ { "family_name": "Nielsen", "given_name": "Frank", "clpid": "Nielsen-F" }, { "family_name": "Barbaresco", "given_name": "Fr\u00e9d\u00e9ric", "clpid": "Barbaresco-F" } ], "abstract": "We propose a new method, based on sparse distributed memory, for studying dependence relations between syntactic parameters in the Principles and Parameters model of Syntax. By storing data of syntactic structures of world languages in a Kanerva network and checking recoverability of corrupted data from the network, we identify two different effects: an overall underlying relation between the prevalence of parameters across languages and their degree of recoverability, and a finer effect that makes some parameters more easily recoverable beyond what their prevalence would indicate. The latter can be seen as an indication of the existence of dependence relations, through which a given parameter can be determined using the remaining uncorrupted data.", "doi": "10.1007/978-3-319-68445-1_31", "isbn": "978-3-319-68444-4", "publisher": "Springer", "place_of_publication": "Cham, Switzerland", "publication_date": "2017-10-24", "pages": "265-272" }, { "id": "authors:vy2pb-qbm80", "collection": "authors", "collection_id": "vy2pb-qbm80", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20170712-091753130", "type": "book_section", "title": "Moduli Operad over F_1", "book_title": "Absolute arithmetic and F1-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" } ], "contributor": [ { "family_name": "Thas", "given_name": "Koen", "clpid": "Thas-K" } ], "abstract": "In this paper we answer a question raised in [25], Sec. 4, by showing that the genus zero moduli operad {M_(0,n+1)} can be endowed with natural descent data that allow it to be considered as the lift to Spec Z of an operad over F_1. The relevant descent data are based on a notion of constructible sets and constructible functions over F_1, which describes suitable differences of torifications with a positivity condition on the class in the Grothendieck ring. More generally, we do the same for the operads {T_(d,n+1)} (whose components were) introduced in [5]. Finally, we describe a blueprint structure on {M_(0,n)} and we discuss from this perspective the genus zero boundary modular operad {M^0_(g,n+1)}.", "doi": "10.4171/157-1/7", "isbn": "978-3-03719-157-6", "publisher": "European Mathematical Society", "place_of_publication": "Z\u00fcrich", "publication_date": "2016-07", "pages": "331-361" }, { "id": "authors:xqrv2-ffb73", "collection": "authors", "collection_id": "xqrv2-ffb73", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20160601-111532492", "type": "book_section", "title": "Information Algebras and Their Applications", "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-F" }, { "family_name": "Barbaresco", "given_name": "Fr\u00e9d\u00e9ric", "clpid": "Barbaresco-F" } ], "abstract": "In this lecture we will present joint work with Ryan Thorngren on thermodynamic semirings and entropy operads, with Nicolas Tedeschi on Birkhoff factorization in thermodynamic semirings, ongoing work with Marcus Bintz on tropicalization of Feynman graph hypersurfaces and Potts model hypersurfaces, and their thermodynamic deformations, and ongoing work by the author on applications of thermodynamic semirings to models of morphology and syntax in Computational Linguistics.", "doi": "10.1007/978-3-319-25040-3_30", "isbn": "978-3-319-25039-7", "publisher": "Springer", "place_of_publication": "Cham", "publication_date": "2016-04-03", "pages": "271-276" }, { "id": "authors:1zced-pkb20", "collection": "authors", "collection_id": "1zced-pkb20", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20170712-143931835", "type": "book_section", "title": "Noncommutative motives and their applications", "book_title": "Commutative algebra and noncommutative algebraic geometry", "author": [ { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" }, { "family_name": "Tabuada", "given_name": "Gon\u00e7alo", "clpid": "Tabuada-G" } ], "contributor": [ { "family_name": "Eisenbud", "given_name": "David", "clpid": "Eisenbud-D" }, { "family_name": "Iyengar", "given_name": "Srikanth B.", "clpid": "Iyengar-S-B" }, { "family_name": "Singh", "given_name": "Anurag K.", "clpid": "Singh-A-K" }, { "family_name": "Stafford", "given_name": "J. Joby", "clpid": "Stafford-J-J" }, { "family_name": "Van de Bergh", "given_name": "Michel", "clpid": "Van-de-Bergh-M" } ], "abstract": "This survey is based on lectures given by the authors during the program \"Noncommutative algebraic geometry and representation theory\" at the MSRI in the Spring 2013. It covers the recent work [44, 45, 46, 47, 48, 49, 50] on noncommutative motives and their applications, and is intended for a broad mathematical audience. In Section 1 we recall the main features of Grothendieck's theory of motives. In Sections 2 and 3 we introduce several categories of noncommutative motives and describe their relation with the classical commutative counterparts. In Section 4 we formulate the noncommutative analogues of Grothendieck's standard conjectures of type C and D, of Voevodsky's smash-nilpotence conjecture, and of Kimura-O'Sullivan finite-dimensionality conjecture. Section 5 is devoted to recollections of the (super-)Tannakian formalism. In Section 6 we introduce the noncommutative motivic Galois (super-)groups and their unconditional versions. In Section 7 we explain how the classical theory of (intermediate) Jacobians can be extended to the noncommutative world. Finally, in Section 8 we present some applications to motivic decompositions and to Dubrovin's conjecture.", "doi": "10.48550/arXiv.1311.2867", "isbn": "978-1-107-06562-8", "publisher": "Cambridge University Press", "place_of_publication": "Cambridge", "publication_date": "2015", "pages": "191-214" }, { "id": "authors:02am0-hba28", "collection": "authors", "collection_id": "02am0-hba28", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20151130-084148729", "type": "book_section", "title": "Dyson-Schwinger equations in the theory of computation", "book_title": "Feynman Amplitudes, Periods and Motives", "author": [ { "family_name": "Delaney", "given_name": "Colleen", "clpid": "Delaney-C-R" }, { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" } ], "contributor": [ { "family_name": "\u00c1lvarez-C\u00f3nsul", "given_name": "Luis", "clpid": "\u00c1lvarez-C\u00f3nsul-L" }, { "family_name": "Burgos-Gil", "given_name": "Jos\u00e9 Ignacio", "clpid": "Burgos-Gil-J-I" }, { "family_name": "Ebrahimi-Fard", "given_name": "Kurusch", "clpid": "Ebrahimi-Fard-K" } ], "abstract": "Following Manin's approach to renormalization in the theory of computation, we investigate Dyson-Schwinger equations on Hopf algebras, operads and properads of flow charts, as a way of encoding self-similarity structures in the theory of algorithms computing primitive and partial recursive functions and in the halting problem.", "doi": "10.48550/arXiv.1302.5040", "isbn": "978-1-4704-2247-9", "publisher": "American Mathematical Society", "place_of_publication": "Providence, RI", "publication_date": "2015", "pages": "79-107" }, { "id": "authors:gsya8-zeq05", "collection": "authors", "collection_id": "gsya8-zeq05", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20151119-101540033", "type": "book_section", "title": "Unconditional noncommutative motivic Galois groups", "book_title": "Hodge Theory and Classical Algebraic Geometry", "author": [ { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" }, { "family_name": "Tabuada", "given_name": "Gon\u00e7alo", "clpid": "Tabuada-G" } ], "contributor": [ { "family_name": "Kennedy", "given_name": "Gary", "clpid": "Kennedy-Gary" }, { "family_name": "Caib\u00e1r", "given_name": "Mirel", "clpid": "Caib\u00e1r-M" }, { "family_name": "Castravet", "given_name": "Ana-Maria", "clpid": "Castravet-A-M" }, { "family_name": "Macri", "given_name": "Emanuele", "clpid": "Macri-E" } ], "abstract": "In this short note we introduce the unconditional noncommutative motivic Galois groups and relate them with those of Andre-Kahn.", "doi": "10.48550/arXiv.1112.5422", "isbn": "978-1-4704-0990-6", "publisher": "American Mathematical Society", "place_of_publication": "Providence, RI", "publication_date": "2015", "pages": "109-115" }, { "id": "authors:b8fy6-qjj49", "collection": "authors", "collection_id": "b8fy6-qjj49", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20170615-124503245", "type": "book_section", "title": "The Wolf and the Street: Narrative Encounters with Mathematics", "book_title": "Imagine Math 3", "author": [ { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" } ], "contributor": [ { "family_name": "Emmer", "given_name": "Matilde", "clpid": "Emmer-M" } ], "abstract": "I was invited to this conference to present the results of an old experiment, a twenty year old exploration of narrative and mathematics, in the form of a collection of fairy tales. I began writing the stories of Racconti per il lupo (Tales for the Wolf) in 1986, when I was a 16 year old student of the Liceo Classico, more occupied with ancient Greek texts and philosophy than with mathematics. I finished the collection when I was about to graduate in Theoretical Physics, at the University of Milano, in 1993. Those few years spanned enormous transformations, at the personal level, going through the passage from late adolescence to adulthood, and from being a student of classical languages to a professional physicist, as well as on the larger scale of society and the world: those were the years that marked \"the end of the short century\", with all the upheaval, excitement and anxiety that came with it. In their minuscule cameo scale, the mathematical stories I was putting together, act as a small fragmented mirror of larger events. The stories, written in Italian, and illustrated by a series of collages I prepared in the style of Max Ernst, are now available online at the publisher Lulu.com, along with some of my more recent writings. In this presentation, I will try to describe the main ideas behind that old attempt at conveying in a narrative form some mathematical concepts, and I will contrast the spirit of that early encounter with mathematics, with the very different spirit in which I came to see the mathematical profession nowadays, after twenty years of experience. The latter is best represented in some of my more recent writings, especially the lyric prose collection Street Science, composed in 2013, illustrated by mathematical street art graffiti, also available online from the same publisher.", "doi": "10.1007/978-3-319-01231-5_16", "isbn": "978-3-319-01230-8", "publisher": "Springer", "place_of_publication": "New York, NY", "publication_date": "2015", "pages": "225-234" }, { "id": "authors:29gfk-tr108", "collection": "authors", "collection_id": "29gfk-tr108", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20170628-141113183", "type": "book_section", "title": "Quantum Statistical Mechanics, L-Series and Anabelian Geometry I: Partition Functions", "book_title": "Trends in Contemporary Mathematics", "author": [ { "family_name": "Cornelissen", "given_name": "Gunther", "clpid": "Cornelissen-G" }, { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" } ], "contributor": [ { "family_name": "Ancona", "given_name": "Vincenzo", "clpid": "Ancona-V" }, { "family_name": "Strickland", "given_name": "Elisabetta", "clpid": "Strickland-E" } ], "abstract": "The zeta function of a number field can be interpreted as the partition function of an associated quantum statistical mechanical (QSM) system, built from abelian class field theory.\n\nWe introduce a general notion of isomorphism of QSM-systems and prove that it preserves (extremal) KMS equilibrium states.\n\nWe prove that two number fields with isomorphic quantum statistical mechanical systems are arithmetically equivalent, i.e., have the same zeta function. If one of the fields is normal over \u211a, this implies that the fields are isomorphic. Thus, in this case, isomorphism of QSM-systems is the same as isomorphism of number fields, and the noncommutative space built from the abelianized Galois group can replace the anabelian absolute Galois group from the theorem of Neukirch and Uchida.\n\nThis paper is an updated version of part of [9]. We have split the original preprint into various parts, depending on the methods that are used in them. In the current part, these belong mainly to mathematical physics.", "doi": "10.1007/978-3-319-05254-0_4", "isbn": "978-3-319-05253-3", "publisher": "Springer", "place_of_publication": "Cham, Switzerland", "publication_date": "2014-08-02", "pages": "47-57" }, { "id": "authors:9zq9z-rcg96", "collection": "authors", "collection_id": "9zq9z-rcg96", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20141125-125438021", "type": "book_section", "title": "Noncommutative geometry models for particle physics", "book_title": "Geometric, algebraic and topological methods for quantum field theory", "author": [ { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" } ], "contributor": [ { "family_name": "Cardona", "given_name": "Alexander", "clpid": "Cardona-A" } ], "abstract": "This is a writeup of part of a series of lectures delivered at the Villa de\nLeyva summer school \"Geometric, Algebraic and Topological Methods for\nQuantum Field Theory\".\nI am very grateful to Sylvi Paycha for the invitation and to the organizers and the participants for the very friendly\nand nice atmosphere and the very lively and useful discussions. This paper\nfollows the same format of the lectures and the same very informal style,\nbut we refer the reader, wherever needed, to more detailed references for\nadditional material and a more rigorous treatment. The lectures originally\ncovered both the particle physics applications and some recent applications\nto cosmology, along the lines of the work. This writeup\nonly deals with the particle physics part, while the cosmology part of the\nlectures will be written up elsewhere.", "doi": "10.1142/9789814460057_0005", "isbn": "9789814460057", "publisher": "World Scientific Publishing", "place_of_publication": "Hackensack, NJ", "publication_date": "2014-01", "pages": "189-223" }, { "id": "authors:dymh4-e3754", "collection": "authors", "collection_id": "dymh4-e3754", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130212-112112394", "type": "book_section", "title": "Motivic structures in quantum field theory", "book_title": "String-Math 2011", "author": [ { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" } ], "contributor": [ { "family_name": "Block", "given_name": "Jonathan", "clpid": "Block-J" }, { "family_name": "Distler", "given_name": "Jacques", "clpid": "Distler-J" }, { "family_name": "Donagi", "given_name": "Ron", "clpid": "Donagi-R" }, { "family_name": "Sharpe", "given_name": "Eric", "clpid": "Sharpe-E" } ], "abstract": "This is a writeup of the lecture given by the author at the String-Math\n2011 conference in Philadelphia. It gives an overview of recent work of\nthe author, in collaboration with Aluffi and with Ceyhan, on some aspects of\nthe occurrence of motivic structures in perturbative quantum field theory.", "isbn": "9780821872956", "publisher": "American Mathematical Society", "place_of_publication": "Providence, RI", "publication_date": "2012", "pages": "173-190" }, { "id": "authors:1j6aw-wxs82", "collection": "authors", "collection_id": "1j6aw-wxs82", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20110613-094812678", "type": "book_section", "title": "Feynman motives and deletion-contraction relations", "book_title": "Topology of Algebraic Varieties and Singularities", "author": [ { "family_name": "Aluffi", "given_name": "Paolo", "clpid": "Aluffi-P" }, { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" } ], "contributor": [ { "family_name": "Libgober", "given_name": "A.", "clpid": "Libgober-A" }, { "family_name": "Cogolludo-Agust\u00edn", "given_name": "Jos\u00e9 Ignacio", "clpid": "Cogolludo-Agust\u00edn-J-I" }, { "family_name": "Hironaka", "given_name": "Eriko", "clpid": "Hironaka-E" } ], "abstract": "We prove a deletion-contraction formula for motivic Feynman rules given by the classes of the affine graph hypersurface complement in the Grothendieck ring of varieties. We derive explicit recursions and generating series for these motivic Feynman rules under the operation of multiplying edges in a graph and we compare it with similar formulae for the Tutte polynomial of graphs, both being specializations of the same universal recursive relation. We obtain similar recursions for outerplanar graphs (given in full for chains of polygons) and for graphs obtained by replacing an edge by a chain of triangles. We show that the deletion-contraction relation can be lifted to the level of the category of mixed motives in the form of a distinguished triangle, similarly to what happens in categorfications of graph invariants.", "doi": "10.48550/arXiv.0907.3225", "isbn": "978-0-8218-4890-6", "publisher": "American Mathematical Society", "place_of_publication": "Providence, RI", "publication_date": "2011", "pages": "21-64" }, { "id": "authors:n2xft-jzr11", "collection": "authors", "collection_id": "n2xft-jzr11", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20120712-101547699", "type": "book_section", "title": "Modular Index Invariants of Mumford Curves", "book_title": "Noncommutative geometry, arithmetic, and related topics: proceedings of the Twenty-first Meeting of the Japan-U.S. Mathematics Institute", "author": [ { "family_name": "Carey", "given_name": "Alan", "clpid": "Carey-A" }, { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" }, { "family_name": "Rennie", "given_name": "Adam", "clpid": "Rennie-A" } ], "contributor": [ { "family_name": "Consani", "given_name": "Caterina", "clpid": "Consani-C" }, { "family_name": "Connes", "given_name": "Alain", "clpid": "Connes-A" } ], "abstract": "We continue an investigation initiated by Consani-Marcolli of the relation between the algebraic geometry of p-adic Mumford curves and the noncommutative geometry of graph C*-algebras associated to the action of the uniformizing p-adic Schottky group on the Bruhat-Tits tree. We reconstruct invariants of Mumford curves related to valuations of generators of the associated Schottky group, by developing a graphical theory for KMS weights on the associated graph C*-algebra, and using modular index theory for KMS weights. We give explicit examples of the construction of graph weights for low genus Mumford curves. We then show that the theta functions of Mumford curves, and the induced currents on the Bruhat-Tits tree, define functions that generalize the graph weights. We show that such inhomogeneous graph weights can be constructed from spectral flows, and that one can reconstruct theta functions from such graphical data.", "doi": "10.48550/arXiv.0905.3157", "isbn": "9781421403526", "publisher": "Johns Hopkins University Press", "place_of_publication": "Baltimore", "publication_date": "2011", "pages": "31-73" }, { "id": "authors:xf64a-qg632", "collection": "authors", "collection_id": "xf64a-qg632", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20170713-091114280", "type": "book_section", "title": "The Weil Proof and the Geometry of the Adel\u00e8s Class Space", "book_title": "Algebra, Arithmetic, and Geometry. Volume I: In Honor of Yu. I. Manin", "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" } ], "contributor": [ { "family_name": "Tschinkel", "given_name": "Yuri", "clpid": "Tschinkel-Y" }, { "family_name": "Zarhin", "given_name": "Yuri", "clpid": "Zarhin-Y" } ], "abstract": "This paper explores analogies between the Weil proof of the Riemann hypothesis for function fields and the geometry of the ad\u00e8les class space, which is the noncommutative space underlying Connes' spectral realization of the zeros of the Riemann zeta function. We consider the cyclic homology of the cokernel (in the abelian category of cyclic modules) of the \"restriction map\" defined by the inclusion of the id\u00e8les class group of a global field in the noncommutative ad\u00e8les class space. Weil's explicit formula can then be formulated as a Lefschetz trace formula for the induced action of the id\u00e8les class group on this cohomology. In this formulation the Riemann hypothesis becomes equivalent to the positivity of the relevant trace pairing. This result suggests a possible dictionary between the steps in the Weil proof and corresponding notions involving the noncommutative geometry of the ad\u00e8les class space, with good working notions of correspondences, degree, and codegree etc. In particular, we construct an analog for number fields of the algebraic points of the curve for function fields, realized here as classical points (low temperature KMS states) of quantum statistical mechanical systems naturally associated to the periodic orbits of the action of the id\u00e8les class group, that is, to the noncommutative spaces on which the geometric side of the trace formula is supported.", "doi": "10.1007/978-0-8176-4745-2_8", "isbn": "978-0-8176-4744-5", "publisher": "Springer", "place_of_publication": "Boston, MA", "publication_date": "2010-11-21", "pages": "339-405" }, { "id": "authors:ny2y1-hs438", "collection": "authors", "collection_id": "ny2y1-hs438", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20100921-095023951", "type": "book_section", "title": "Feynman integrals and motives", "author": [ { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" } ], "abstract": "This article gives an overview of recent results on the relation between quantum\nfield theory and motives, with an emphasis on two different approaches: a \"bottom-up\"\napproach based on the algebraic geometry of varieties associated to Feynman graphs, and\na \"top-down\" approach based on the comparison of the properties of associated categorical\nstructures. This survey is mostly based on joint work of the author with Paolo Aluffi, along\nthe lines of the first approach, and on previous work of the author with Alain Connes on\nthe second approach.", "doi": "10.48550/arXiv.0907.0321", "publisher": "European Mathematical Society", "publication_date": "2010" }, { "id": "authors:0nnbr-rds46", "collection": "authors", "collection_id": "0nnbr-rds46", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20110324-084219567", "type": "book_section", "title": "Motivic renormalization and singularities", "book_title": "Quanta of Maths: Conference in Honor of Alain Connes", "author": [ { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" } ], "contributor": [ { "family_name": "Connes", "given_name": "Alain", "clpid": "Connes-A" }, { "family_name": "Blanchard", "given_name": "Etienne", "clpid": "Blanchard-E" } ], "abstract": "We consider parametric Feynman integrals and their dimensional regularization from the point of view of differential forms on hypersurface complements and the approach to mixed Hodge structures via oscillatory integrals. We consider restrictions to linear subspaces that slice the singular locus, to handle the presence\nof non-isolated singularities. In order to account for all possible choices of slicing, we encode this extra datum as an enrichment of the Hopf algebra of Feynman graphs.\nWe introduce a new regularization method for parametric Feynman integrals, which is based on Leray coboundaries and, like dimensional regularization, replaces a divergent\nintegral with a Laurent series in a complex parameter. The Connes\u2013Kreimer formulation of renormalization can be applied to this regularization method. We relate the\ndimensional regularization of the Feynman integral to the Mellin transforms of certain Gelfand\u2013Leray forms and we show that, upon varying the external momenta, the\nFeynman integrals for a given graph span a family of subspaces in the cohomological Milnor fibration. We show how to pass from regular singular Picard\u2013Fuchs equations\nto irregular singular flat equisingular connections. In the last section, which is more speculative in nature, we propose a geometric model for dimensional regularization in\nterms of logarithmic motives and motivic sheaves.", "doi": "10.48550/arXiv.0804.4824v3", "isbn": "978-0-8218-5203-3", "publisher": "American Mathematical Society", "place_of_publication": "Providence, RI", "publication_date": "2010", "pages": "409-458" }, { "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" } ]