[ { "id": "authors:pr838-v2807", "collection": "authors", "collection_id": "pr838-v2807", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20180803-161235330", "type": "book_section", "title": "Introduction to motives", "book_title": "Noncommutative Geometry and Physics: Renormalisation, Motives, Index Theory", "author": [ { "family_name": "Ramdorai", "given_name": "Sujatha", "clpid": "Ramdorai-S" }, { "family_name": "Plazas", "given_name": "Jorge", "clpid": "Plazas-J" } ], "contributor": [ { "family_name": "Marcolli", "given_name": "Matilde", "clpid": "Marcolli-M" }, { "family_name": "Carey", "given_name": "Alan L.", "clpid": "Carey-A-L" } ], "abstract": "This article is based on the lectures of the same tittle given by the first author during the instructional workshop of the program \"number theory and physics\" at ESI Vienna during March 2009. An account of the topics treated during the lectures can be found in [50] where the categorical aspects of the theory are stressed. Although naturally overlapping, these two independent articles serve as complements to each other. In the present article we focus on the construction of the category of pure motives starting from the category of smooth projective varieties. The necessary preliminary material is discussed. Early accounts of the theory were given in Manin [42] and Kleiman [36], the material presented here reflects to some extent their treatment of the main aspects of the theory. We also survey the theory of endomotives developed in [11], this provides a link between the theory of motives and tools from quantum statistical mechanics which play an important role in results connecting number theory and noncommutative geometry. An extended appendix (by Matilde Marcolli) further elaborates these ideas and reviews the role of motives in noncommutative geometry.", "doi": "10.4171/008-1/2", "isbn": "978-3-03719-008-1", "publisher": "European Mathematical Society", "place_of_publication": "Z\u00fcrich", "publication_date": "2011-07", "pages": "41-87" }, { "id": "authors:esfyh-2pw10", "collection": "authors", "collection_id": "esfyh-2pw10", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20180808-084743368", "type": "book", "title": "Quantum Groups and Noncommutative Spaces: Perspectives on Quantum Geometry", "author": [ { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" }, { "family_name": "Parashar", "given_name": "Deepak" } ], "contributor": [ { "family_name": "Marcolli", "given_name": "Matilde", "clpid": "Marcolli-M" }, { "family_name": "Parashar", "given_name": "Deepak", "clpid": "Parashar-D" } ], "abstract": "This book is aimed at presenting different methods and perspectives in the theory of Quantum Groups, bridging between the algebraic, representation theoretic, analytic, and differential-geometric approaches. It also covers recent developments in Noncommutative Geometry, which have close relations to quantization and quantum group symmetries. The volume collects surveys by experts which originate from an acitvity at the Max-Planck-Institute for Mathematics in Bonn.", "doi": "10.1007/978-3-8348-9831-9", "isbn": "978-3-8348-1442-5", "publisher": "Springer", "place_of_publication": "Wiesbaden, Germany", "publication_date": "2011" }, { "id": "authors:mak25-kbx66", "collection": "authors", "collection_id": "mak25-kbx66", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20180808-091119205", "type": "book", "title": "Deformation Spaces: Perspectives on algebro-geometric moduli", "author": [ { "family_name": "Abbaspour", "given_name": "Hossein" }, { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" }, { "family_name": "Tradler", "given_name": "Thomas" } ], "contributor": [ { "family_name": "Abbaspour", "given_name": "Hossein", "clpid": "Abbaspour-H" }, { "family_name": "Marcolli", "given_name": "Matilde", "clpid": "Marcolli-M" }, { "family_name": "Tradler", "given_name": "Thomas", "clpid": "Tradler-T" } ], "abstract": "The first instances of deformation theory were given by Kodaira and Spencer for complex structures and by Gerstenhaber for associative algebras. Since then, deformation theory has been applied as a useful tool in the study of many other mathematical structures, and even today it plays an important role in many developments of modern mathematics.\nThis volume collects a few self-contained and peer-reviewed papers by experts which present up-to-date research topics in algebraic and motivic topology, quantum field theory, algebraic geometry, noncommutative geometry and the deformation theory of Poisson algebras. They originate from activities at the Max-Planck-Institute for Mathematics and the Hausdorff Center for Mathematics in Bonn.", "doi": "10.1007/978-3-8348-9680-3", "isbn": "978-3-8348-1271-1", "publisher": "Springer", "place_of_publication": "Wiesbaden, Germany", "publication_date": "2010" }, { "id": "authors:tpzen-8fd78", "collection": "authors", "collection_id": "tpzen-8fd78", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20180808-085720175", "type": "book", "title": "Arithmetic and Geometry Around Quantization", "author": [ { "family_name": "Ceyhan", "given_name": "\u00d6zg\u00fcr" }, { "family_name": "Manin", "given_name": "Yu. I." }, { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" } ], "contributor": [ { "family_name": "Ceyhan", "given_name": "\u00d6zg\u00fcr", "clpid": "Ceyhan-\u00d6" }, { "family_name": "Manin", "given_name": "Yu. I.", "clpid": "Manin-Yu-I" }, { "family_name": "Marcolli", "given_name": "Matilde", "clpid": "Marcolli-M" } ], "abstract": "In recent decades, quantization has led to interesting applications in various mathematical branches. This volume, comprised of research and survey articles, discusses key topics, including symplectic and algebraic geometry, representation theory, quantum groups, the geometric Langlands program, quantum ergodicity, and non-commutative geometry. A wide range of topics related to quantization are covered, giving a glimpse of the broad subject. The articles are written by distinguished mathematicians in the field and reflect subsequent developments following the Arithmetic and Geometry around Quantization conference held in Istanbul.", "doi": "10.1007/978-0-8176-4831-2", "isbn": "978-0-8176-4830-5", "publisher": "Springer", "place_of_publication": "Boston, MA", "publication_date": "2010" }, { "id": "authors:4w487-0sy76", "collection": "authors", "collection_id": "4w487-0sy76", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20180814-142452298", "type": "book", "title": "Quanta of Maths", "author": [ { "family_name": "Blanchard", "given_name": "Etienne" }, { "family_name": "Ellwood", "given_name": "David" }, { "family_name": "Khalkhali", "given_name": "Masoud" }, { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" }, { "family_name": "Moscovici", "given_name": "Henri" }, { "family_name": "Popa", "given_name": "Sorin" } ], "contributor": [ { "family_name": "Blanchard", "given_name": "Etienne", "clpid": "Blanchard-E" }, { "family_name": "Ellwood", "given_name": "David", "clpid": "Ellwood-D" }, { "family_name": "Khalkhali", "given_name": "Masoud", "clpid": "Khalkhali-M" }, { "family_name": "Marcolli", "given_name": "Matilde", "clpid": "Marcolli-M" }, { "family_name": "Moscovici", "given_name": "Henri", "clpid": "Moscovici-H" }, { "family_name": "Popa", "given_name": "Sorin", "clpid": "Popa-S" } ], "abstract": "The work of Alain Connes has cut a wide swath across several areas of math- ematics and physics. Reflecting its broad spectrum and profound impact on the contemporary mathematical landscape, this collection of articles covers a wealth of topics at the forefront of research in operator algebras, analysis, noncommutative geometry, topology, number theory and physics.", "isbn": "9780821852033", "publisher": "American Mathematical Society", "place_of_publication": "Providence, RI", "publication_date": "2010" }, { "id": "authors:wdpyf-71h20", "collection": "authors", "collection_id": "wdpyf-71h20", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20180808-075818895", "type": "book", "title": "Feynman Motives", "author": [ { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" } ], "contributor": [ { "family_name": "Marcolli", "given_name": "Matilde", "clpid": "Marcolli-M" } ], "abstract": "This book presents recent and ongoing research work aimed at understanding the mysterious relation between the computations of Feynman integrals in perturbative quantum field theory and the theory of motives of algebraic varieties and their periods. One of the main questions in the field is understanding when the residues of Feynman integrals in perturbative quantum field theory evaluate to periods of mixed Tate motives. The question originates from the occurrence of multiple zeta values in Feynman integrals calculations observed by Broadhurst and Kreimer. \n\nTwo different approaches to the subject are described. The first, a \"bottom-up\" approach, constructs explicit algebraic varieties and periods from Feynman graphs and parametric Feynman integrals. This approach, which grew out of work of Bloch\u2013Esnault\u2013Kreimer and was more recently developed in joint work of Paolo Aluffi and the author, leads to algebro-geometric and motivic versions of the Feynman rules of quantum field theory and concentrates on explicit constructions of motives and classes in the Grothendieck ring of varieties associated to Feynman integrals. While the varieties obtained in this way can be arbitrarily complicated as motives, the part of the cohomology that is involved in the Feynman integral computation might still be of the special mixed Tate kind. A second, \"top-down\" approach to the problem, developed in the work of Alain Connes and the author, consists of comparing a Tannakian category constructed out of the data of renormalization of perturbative scalar field theories, obtained in the form of a Riemann\u2013Hilbert correspondence, with Tannakian categories of mixed Tate motives. The book draws connections between these two approaches and gives an overview of other ongoing directions of research in the field, outlining the many connections of perturbative quantum field theory and renormalization to motives, singularity theory, Hodge structures, arithmetic geometry, supermanifolds, algebraic and non-commutative geometry. \n\nThe text is aimed at researchers in mathematical physics, high energy physics, number theory and algebraic geometry. Partly based on lecture notes for a graduate course given by the author at Caltech in the fall of 2008, it can also be used by graduate students interested in working in this area.", "doi": "10.1142/7245", "isbn": "978-981-4271-20-2", "publisher": "World Scientific", "place_of_publication": "Singapore", "publication_date": "2009-12" } ]