Cyclic Cohomology At 40 Achievements And Future Prospects

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Cyclic Cohomology at 40: Achievements and Future Prospects

A. Connes,C. Consani,B. I. Dundas,M. Khalkhali,H. Moscovici
Cyclic Cohomology at 40 Achievements and Future Prospects Book PDF

Author : A. Connes,C. Consani,B. I. Dundas,M. Khalkhali,H. Moscovici
Publisher : American Mathematical Society
Page : 592 pages
File Size : 41,9 Mb
Release : 2023-02-23
Category : Mathematics
ISBN : 9781470469771

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Cyclic Cohomology at 40: Achievements and Future Prospects by A. Connes,C. Consani,B. I. Dundas,M. Khalkhali,H. Moscovici Pdf

This volume contains the proceedings of the virtual conference on Cyclic Cohomology at 40: Achievements and Future Prospects, held from September 27–October 1, 2021 and hosted by the Fields Institute for Research in Mathematical Sciences, Toronto, ON, Canada. Cyclic cohomology, since its discovery forty years ago in noncommutative differential geometry, has become a fundamental mathematical tool with applications in domains as diverse as analysis, algebraic K-theory, algebraic geometry, arithmetic geometry, solid state physics and quantum field theory. The reader will find survey articles providing a user-friendly introduction to applications of cyclic cohomology in such areas as higher categorical algebra, Hopf algebra symmetries, de Rham-Witt complex, quantum physics, etc., in which cyclic homology plays the role of a unifying theme. The researcher will find frontier research articles in which the cyclic theory provides a computational tool of great relevance. In particular, in analysis cyclic cohomology index formulas capture the higher invariants of manifolds, where the group symmetries are extended to Hopf algebra actions, and where Lie algebra cohomology is greatly extended to the cyclic cohomology of Hopf algebras which becomes the natural receptacle for characteristic classes. In algebraic topology the cyclotomic structure obtained using the cyclic subgroups of the circle action on topological Hochschild homology gives rise to remarkably significant arithmetic structures intimately related to crystalline cohomology through the de Rham-Witt complex, Fontaine's theory and the Fargues-Fontaine curve.

String-Math 2022

Ron Donagi,Adrian Langer,Piotr Sułkowski,Katrin Wendland
String Math 2022 Book PDF

Author : Ron Donagi,Adrian Langer,Piotr Sułkowski,Katrin Wendland
Publisher : American Mathematical Society
Page : 306 pages
File Size : 41,9 Mb
Release : 2024-04-18
Category : Mathematics
ISBN : 9781470472405

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String-Math 2022 by Ron Donagi,Adrian Langer,Piotr Sułkowski,Katrin Wendland Pdf

This is a proceedings volume from the String-Math conference which took place at the University of Warsaw in 2022. This 12th String-Math conference focused on several research areas actively developing these days. They included generalized (categorical) symmetries in quantum field theory and their relation to topological phases of matter; formal aspects of quantum field theory, in particular twisted holography; various developments in supersymmetric gauge theories, BPS counting and Donaldson–Thomas invariants. Other topics discussed at this conference included new advances in Gromov–Witten theory, curve counting, and Calabi–Yau manifolds. Another broad topic concerned algebraic aspects of conformal field theory, vertex operator algebras, and quantum groups. Furthermore, several other recent developments were presented during the conference, such as understanding the role of operator algebras in the presence of gravity, derivation of gauge-string duality, complexity of black holes, or mathematical aspects of the amplituhedron. This proceedings volume contains articles summarizing 14 conference lectures, devoted to the above topics.

Advances in Noncommutative Geometry

Ali Chamseddine,Caterina Consani,Nigel Higson,Masoud Khalkhali,Henri Moscovici,Guoliang Yu
Advances in Noncommutative Geometry Book PDF

Author : Ali Chamseddine,Caterina Consani,Nigel Higson,Masoud Khalkhali,Henri Moscovici,Guoliang Yu
Publisher : Springer Nature
Page : 753 pages
File Size : 52,7 Mb
Release : 2020-01-13
Category : Mathematics
ISBN : 9783030295974

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Advances in Noncommutative Geometry by Ali Chamseddine,Caterina Consani,Nigel Higson,Masoud Khalkhali,Henri Moscovici,Guoliang Yu Pdf

This authoritative volume in honor of Alain Connes, the foremost architect of Noncommutative Geometry, presents the state-of-the art in the subject. The book features an amalgam of invited survey and research papers that will no doubt be accessed, read, and referred to, for several decades to come. The pertinence and potency of new concepts and methods are concretely illustrated in each contribution. Much of the content is a direct outgrowth of the Noncommutative Geometry conference, held March 23–April 7, 2017, in Shanghai, China. The conference covered the latest research and future areas of potential exploration surrounding topology and physics, number theory, as well as index theory and its ramifications in geometry.

Computational Homology

Tomasz Kaczynski,Konstantin Mischaikow,Marian Mrozek
Computational Homology Book PDF

Author : Tomasz Kaczynski,Konstantin Mischaikow,Marian Mrozek
Publisher : Springer Science & Business Media
Page : 488 pages
File Size : 55,5 Mb
Release : 2006-04-18
Category : Mathematics
ISBN : 9780387215976

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Computational Homology by Tomasz Kaczynski,Konstantin Mischaikow,Marian Mrozek Pdf

Homology is a powerful tool used by mathematicians to study the properties of spaces and maps that are insensitive to small perturbations. This book uses a computer to develop a combinatorial computational approach to the subject. The core of the book deals with homology theory and its computation. Following this is a section containing extensions to further developments in algebraic topology, applications to computational dynamics, and applications to image processing. Included are exercises and software that can be used to compute homology groups and maps. The book will appeal to researchers and graduate students in mathematics, computer science, engineering, and nonlinear dynamics.

Noncommutative Geometry and Particle Physics

Walter D. van Suijlekom
Noncommutative Geometry and Particle Physics Book PDF

Author : Walter D. van Suijlekom
Publisher : Springer
Page : 246 pages
File Size : 41,9 Mb
Release : 2014-07-21
Category : Science
ISBN : 9789401791625

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Noncommutative Geometry and Particle Physics by Walter D. van Suijlekom Pdf

This book provides an introduction to noncommutative geometry and presents a number of its recent applications to particle physics. It is intended for graduate students in mathematics/theoretical physics who are new to the field of noncommutative geometry, as well as for researchers in mathematics/theoretical physics with an interest in the physical applications of noncommutative geometry. In the first part, we introduce the main concepts and techniques by studying finite noncommutative spaces, providing a “light” approach to noncommutative geometry. We then proceed with the general framework by defining and analyzing noncommutative spin manifolds and deriving some main results on them, such as the local index formula. In the second part, we show how noncommutative spin manifolds naturally give rise to gauge theories, applying this principle to specific examples. We subsequently geometrically derive abelian and non-abelian Yang-Mills gauge theories, and eventually the full Standard Model of particle physics, and conclude by explaining how noncommutative geometry might indicate how to proceed beyond the Standard Model.

Variations on a Theme of Borel

Shmuel Weinberger
Variations on a Theme of Borel Book PDF

Author : Shmuel Weinberger
Publisher : Cambridge University Press
Page : 366 pages
File Size : 41,7 Mb
Release : 2022-12-08
Category : Mathematics
ISBN : 9781108916844

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Variations on a Theme of Borel by Shmuel Weinberger Pdf

In the middle of the last century, after hearing a talk of Mostow on one of his rigidity theorems, Borel conjectured in a letter to Serre a purely topological version of rigidity for aspherical manifolds (i.e. manifolds with contractible universal covers). The Borel conjecture is now one of the central problems of topology with many implications for manifolds that need not be aspherical. Since then, the theory of rigidity has vastly expanded in both precision and scope. This book rethinks the implications of accepting his heuristic as a source of ideas. Doing so leads to many variants of the original conjecture - some true, some false, and some that remain conjectural. The author explores this collection of ideas, following them where they lead whether into rigidity theory in its differential geometric and representation theoretic forms, or geometric group theory, metric geometry, global analysis, algebraic geometry, K-theory, or controlled topology.

String-Math 2011

Jonathan Block
String Math 2011 Book PDF

Author : Jonathan Block
Publisher : American Mathematical Soc.
Page : 506 pages
File Size : 54,9 Mb
Release : 2012
Category : Mathematics
ISBN : 9780821872956

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String-Math 2011 by Jonathan Block Pdf

The nature of interactions between mathematicians and physicists has been thoroughly transformed in recent years. String theory and quantum field theory have contributed a series of profound ideas that gave rise to entirely new mathematical fields and revitalized older ones. The influence flows in both directions, with mathematical techniques and ideas contributing crucially to major advances in string theory. A large and rapidly growing number of both mathematicians and physicists are working at the string-theoretic interface between the two academic fields. The String-Math conference series aims to bring together leading mathematicians and mathematically minded physicists working in this interface. This volume contains the proceedings of the inaugural conference in this series, String-Math 2011, which was held June 6-11, 2011, at the University of Pennsylvania.

Fields Medallists' Lectures

Michael Atiyah,Daniel Iagolnitzer
Fields Medallists Lectures Book PDF

Author : Michael Atiyah,Daniel Iagolnitzer
Publisher : World Scientific
Page : 644 pages
File Size : 42,7 Mb
Release : 1997-10-13
Category : Mathematics
ISBN : 9789814497510

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Fields Medallists' Lectures by Michael Atiyah,Daniel Iagolnitzer Pdf

Although the Fields Medal does not have the same public recognition as the Nobel Prizes, they share a similar intellectual standing. It is restricted to one field - that of mathematics - and an age limit of 40 has become an accepted tradition. Mathematics has in the main been interpreted as pure mathematics, and this is not so unreasonable since major contributions in some applied areas can be (and have been) recognized with Nobel Prizes. The restriction to 40 years is of marginal significance, since most mathematicians have made their mark long before this age.A list of Fields Medallists and their contributions provides a bird's eye view of mathematics over the past 60 years. It highlights the areas in which, at various times, greatest progress has been made. This volume does not pretend to be comprehensive, nor is it a historical document. On the other hand, it presents contributions from 22 Fields Medallists and so provides a highly interesting and varied picture.The contributions themselves represent the choice of the individual Medallists. In some cases the articles relate directly to the work for which the Fields Medals were awarded. In other cases new articles have been produced which relate to more current interests of the Medallists. This indicates that while Fields Medallists must be under 40 at the time of the award, their mathematical development goes well past this age. In fact the age limit of 40 was chosen so that young mathematicians would be encouraged in their future work.The Fields Medallists' Lectures is now available on CD-ROM. Sections can be accessed at the touch of a button, and similar topics grouped together using advanced keyword searches.

The Brauer-Hasse-Noether Theorem in Historical Perspective

Peter Roquette
The Brauer Hasse Noether Theorem in Historical Perspective Book PDF

Author : Peter Roquette
Publisher : Springer Science & Business Media
Page : 92 pages
File Size : 43,8 Mb
Release : 2006-03-30
Category : Mathematics
ISBN : 9783540269687

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The Brauer-Hasse-Noether Theorem in Historical Perspective by Peter Roquette Pdf

The unpublished writings of Helmut Hasse, consisting of letters, manuscripts and other papers, are kept at the Handschriftenabteilung of the University Library at Göttingen. Hasse had an extensive correspondence; he liked to exchange mathematical ideas, results and methods freely with his colleagues. There are more than 8000 documents preserved. Although not all of them are of equal mathematical interest, searching through this treasure can help us to assess the development of Number Theory through the 1920s and 1930s. The present volume is largely based on the letters and other documents its author has found concerning the Brauer-Hasse-Noether Theorem in the theory of algebras; this covers the years around 1931. In addition to the documents from the literary estates of Hasse and Brauer in Göttingen, the author also makes use of some letters from Emmy Noether to Richard Brauer that are preserved at the Bryn Mawr College Library (Pennsylvania, USA).

Calabi-Yau Varieties and Mirror Symmetry

Noriko Yui,James Dominic Lewis
Calabi Yau Varieties and Mirror Symmetry Book PDF

Author : Noriko Yui,James Dominic Lewis
Publisher : American Mathematical Soc.
Page : 388 pages
File Size : 55,8 Mb
Release : 2024-07-02
Category : Mathematics
ISBN : 0821871439

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Calabi-Yau Varieties and Mirror Symmetry by Noriko Yui,James Dominic Lewis Pdf

The idea of mirror symmetry originated in physics, but in recent years, the field of mirror symmetry has exploded onto the mathematical scene. It has inspired many new developments in algebraic and arithmetic geometry, toric geometry, the theory of Riemann surfaces, and infinite-dimensional Lie algebras among others. The developments in physics stimulated the interest of mathematicians in Calabi-Yau varieties. This led to the realization that the time is ripe for mathematicians, armed with many concrete examples and alerted by the mirror symmetry phenomenon, to focus on Calabi-Yau varieties and to test for these special varieties some of the great outstanding conjectures, e.g., the modularity conjecture for Calabi-Yau threefolds defined over the rationals, the Bloch-Beilinson conjectures, regulator maps of higher algebraic cycles, Picard-Fuchs differential equations, GKZ hypergeometric systems, and others. The articles in this volume report on current developments. The papers are divided roughly into two categories: geometric methods and arithmetic methods. One of the significant outcomes of the workshop is that we are finally beginning to understand the mirror symmetry phenomenon from the arithmetic point of view, namely, in terms of zeta-functions and L-series of mirror pairs of Calabi-Yau threefolds. The book is suitable for researchers interested in mirror symmetry and string theory.

Cyclic Cohomology and Noncommutative Geometry

Joachim J. R. Cuntz,Masoud Khalkhali
Cyclic Cohomology and Noncommutative Geometry Book PDF

Author : Joachim J. R. Cuntz,Masoud Khalkhali
Publisher : American Mathematical Soc.
Page : 202 pages
File Size : 53,8 Mb
Release : 1997-01-01
Category : Mathematics
ISBN : 0821871242

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Cyclic Cohomology and Noncommutative Geometry by Joachim J. R. Cuntz,Masoud Khalkhali Pdf

Noncommutative geometry is a new field that is among the great challenges of present-day mathematics. Its methods allow one to treat noncommutative algebras - such as algebras of pseudodifferential operators, group algebras, or algebras arising from quantum field theory - on the same footing as commutative algebras, that is, as spaces. Applications range over many fields of mathematics and mathematical physics. This volume contains the proceedings of the workshop on "Cyclic Cohomology and Noncommutative Geometry" held at The Fields Institute (Waterloo, ON) in June 1995. The workshop was part of the program for the special year on operator algebras and its applications.

Love and Math

Edward Frenkel
Love and Math Book PDF

Author : Edward Frenkel
Publisher : Basic Books
Page : 314 pages
File Size : 44,6 Mb
Release : 2013-10-01
Category : Mathematics
ISBN : 9780465069958

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Love and Math by Edward Frenkel Pdf

An awesome, globe-spanning, and New York Times bestselling journey through the beauty and power of mathematics What if you had to take an art class in which you were only taught how to paint a fence? What if you were never shown the paintings of van Gogh and Picasso, weren't even told they existed? Alas, this is how math is taught, and so for most of us it becomes the intellectual equivalent of watching paint dry. In Love and Math, renowned mathematician Edward Frenkel reveals a side of math we've never seen, suffused with all the beauty and elegance of a work of art. In this heartfelt and passionate book, Frenkel shows that mathematics, far from occupying a specialist niche, goes to the heart of all matter, uniting us across cultures, time, and space. Love and Math tells two intertwined stories: of the wonders of mathematics and of one young man's journey learning and living it. Having braved a discriminatory educational system to become one of the twenty-first century's leading mathematicians, Frenkel now works on one of the biggest ideas to come out of math in the last 50 years: the Langlands Program. Considered by many to be a Grand Unified Theory of mathematics, the Langlands Program enables researchers to translate findings from one field to another so that they can solve problems, such as Fermat's last theorem, that had seemed intractable before. At its core, Love and Math is a story about accessing a new way of thinking, which can enrich our lives and empower us to better understand the world and our place in it. It is an invitation to discover the magic hidden universe of mathematics.

Foundations of Mathematics and Physics One Century After Hilbert

Joseph Kouneiher
Foundations of Mathematics and Physics One Century After Hilbert Book PDF

Author : Joseph Kouneiher
Publisher : Springer
Page : 441 pages
File Size : 48,9 Mb
Release : 2018-05-26
Category : Science
ISBN : 9783319648132

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Foundations of Mathematics and Physics One Century After Hilbert by Joseph Kouneiher Pdf

This book explores the rich and deep interplay between mathematics and physics one century after David Hilbert’s works from 1891 to 1933, published by Springer in six volumes. The most prominent scientists in various domains of these disciplines contribute to this volume providing insight to their works, and analyzing the impact of the breakthrough and the perspectives of their own contributions. The result is a broad journey through the most recent developments in mathematical physics, such as string theory, quantum gravity, noncommutative geometry, twistor theory, Gauge and Quantum fields theories, just to mention a few. The reader, accompanied on this journey by some of the fathers of these theories, explores some far reaching interfaces where mathematics and theoretical physics interact profoundly and gets a broad and deep understanding of subjects which are at the core of recent developments in mathematical physics. The journey is not confined to the present state of the art, but sheds light on future developments of the field, highlighting a list of open problems. Graduate students and researchers working in physics, mathematics and mathematical physics will find this journey extremely fascinating. All those who want to benefit from a comprehensive description of all the latest advances in mathematics and mathematical physics, will find this book very useful too.

Graphs on Surfaces and Their Applications

Sergei K. Lando,Alexander K. Zvonkin
Graphs on Surfaces and Their Applications Book PDF

Author : Sergei K. Lando,Alexander K. Zvonkin
Publisher : Springer Science & Business Media
Page : 455 pages
File Size : 45,6 Mb
Release : 2013-04-17
Category : Mathematics
ISBN : 9783540383611

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Graphs on Surfaces and Their Applications by Sergei K. Lando,Alexander K. Zvonkin Pdf

Graphs drawn on two-dimensional surfaces have always attracted researchers by their beauty and by the variety of difficult questions to which they give rise. The theory of such embedded graphs, which long seemed rather isolated, has witnessed the appearance of entirely unexpected new applications in recent decades, ranging from Galois theory to quantum gravity models, and has become a kind of a focus of a vast field of research. The book provides an accessible introduction to this new domain, including such topics as coverings of Riemann surfaces, the Galois group action on embedded graphs (Grothendieck's theory of "dessins d'enfants"), the matrix integral method, moduli spaces of curves, the topology of meromorphic functions, and combinatorial aspects of Vassiliev's knot invariants and, in an appendix by Don Zagier, the use of finite group representation theory. The presentation is concrete throughout, with numerous figures, examples (including computer calculations) and exercises, and should appeal to both graduate students and researchers.

Renewing U.S. Mathematics

National Research Council,Division on Engineering and Physical Sciences,Commission on Physical Sciences, Mathematics, and Applications,Board on Mathematical Sciences,Committee on the Mathematical Sciences: Status and Future Directions
Renewing U S Mathematics Book PDF

Author : National Research Council,Division on Engineering and Physical Sciences,Commission on Physical Sciences, Mathematics, and Applications,Board on Mathematical Sciences,Committee on the Mathematical Sciences: Status and Future Directions
Publisher : National Academies Press
Page : 148 pages
File Size : 53,6 Mb
Release : 1990-02-01
Category : Mathematics
ISBN : 9780309042284

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Renewing U.S. Mathematics by National Research Council,Division on Engineering and Physical Sciences,Commission on Physical Sciences, Mathematics, and Applications,Board on Mathematical Sciences,Committee on the Mathematical Sciences: Status and Future Directions Pdf

As requested by the National Science Foundation (NSF) and the Interagency Committee for Extramural Mathematics Programs (ICEMAP), this report updates the 1984 Report known as the "David Report." Specifically, the charge directed the committee to (1) update that report, describing the infrastructure and support for U.S. mathematical sciences research; (2) assess trends and progress over the intervening five years against the recommendations of the 1984 Report; (3) briefly assess the field scientifically and identify significant opportunities for research, including cross-disciplinary collaboration; and (4) make appropriate recommendations designed to ensure that U.S. mathematical sciences research will meet national needs in coming years. Of the several components of the mathematical sciences community requiring action, its wellspring--university research departments--is the primary focus of this report. The progress and promise of research--described in the 1984 Report relative to theoretical development, new applications, and the refining and deepening of old applications--have if anything increased since 1984, making mathematics research ever more valuable to other sciences and technology. Although some progress has been made since 1984 in the support for mathematical sciences research, the goals set in the 1984 Report have not been achieved. Practically all of the increase in funding has gone into building the infractructure, which had deteriorated badly by 1984. While graduate and postdoctoral research, computer facilities, and new institutes have benefited from increased resources, some of these areas are still undersupported by the standards of other sciences. And in the area of research support for individual investigators, almost no progress has been made. A critical storage of qualified mathematical sciences researchers still looms, held at bay for the moment by a large influx of foreign researchers, an uncertain solution in the longer term. While government has responded substantially to the 1984 Report's recommendations, particularly in the support of infrastructure, the universities generally have not, so that the academic foundations of the mathematical sciences research enterprise are as shaky now as in 1984. The greatet progress has been made in the mathematics sciences community, whose members have shown a growing awareness of the problems confronting their discipline and increased interest in dealing with the problems, particularly in regard to communication with the public and government agencies and involvement in education. (AA)