Noncommutative Geometry And Optimal Transport PDF Download
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| Author | : Pierre Martinetti |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 223 |
| Release | : 2016-10-26 |
| Genre | : Mathematical optimization |
| ISBN | : 1470422972 |
Download Noncommutative Geometry and Optimal Transport Book in PDF, ePub and Kindle
The distance formula in noncommutative geometry was introduced by Connes at the end of the 1980s. It is a generalization of Riemannian geodesic distance that makes sense in a noncommutative setting, and provides an original tool to study the geometry of the space of states on an algebra. It also has an intriguing echo in physics, for it yields a metric interpretation for the Higgs field. In the 1990s, Rieffel noticed that this distance is a noncommutative version of the Wasserstein distance of order 1 in the theory of optimal transport. More exactly, this is a noncommutative generalization of Kantorovich dual formula of the Wasserstein distance. Connes distance thus offers an unexpected connection between an ancient mathematical problem and the most recent discovery in high energy physics. The meaning of this connection is far from clear. Yet, Rieffel's observation suggests that Connes distance may provide an interesting starting point for a theory of optimal transport in noncommutative geometry. This volume contains several review papers that will give the reader an extensive introduction to the metric aspect of noncommutative geometry and its possible interpretation as a Wasserstein distance on a quantum space, as well as several topic papers.
| Author | : Pierre Martinetti |
| Publisher | : |
| Total Pages | : |
| Release | : 2016 |
| Genre | : Mathematical optimization |
| ISBN | : 9781470435608 |
Download Noncommutative Geometry and Optimal Transport Book in PDF, ePub and Kindle
| Author | : Ludovic Rifford |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 146 |
| Release | : 2014-04-03 |
| Genre | : Mathematics |
| ISBN | : 331904804X |
Download Sub-Riemannian Geometry and Optimal Transport Book in PDF, ePub and Kindle
The book provides an introduction to sub-Riemannian geometry and optimal transport and presents some of the recent progress in these two fields. The text is completely self-contained: the linear discussion, containing all the proofs of the stated results, leads the reader step by step from the notion of distribution at the very beginning to the existence of optimal transport maps for Lipschitz sub-Riemannian structure. The combination of geometry presented from an analytic point of view and of optimal transport, makes the book interesting for a very large community. This set of notes grew from a series of lectures given by the author during a CIMPA school in Beirut, Lebanon.
| Author | : Alain Connes |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 372 |
| Release | : 2003-12-08 |
| Genre | : Mathematics |
| ISBN | : 9783540203575 |
Download Noncommutative Geometry Book in PDF, ePub and Kindle
Noncommutative Geometry is one of the most deep and vital research subjects of present-day Mathematics. Its development, mainly due to Alain Connes, is providing an increasing number of applications and deeper insights for instance in Foliations, K-Theory, Index Theory, Number Theory but also in Quantum Physics of elementary particles. The purpose of the Summer School in Martina Franca was to offer a fresh invitation to the subject and closely related topics; the contributions in this volume include the four main lectures, cover advanced developments and are delivered by prominent specialists.
| Author | : Walter D. van Suijlekom |
| Publisher | : Springer |
| Total Pages | : 246 |
| Release | : 2014-07-21 |
| Genre | : Science |
| ISBN | : 9401791627 |
Download Noncommutative Geometry and Particle Physics Book in PDF, ePub and Kindle
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.
| Author | : Yann Ollivier |
| Publisher | : Cambridge University Press |
| Total Pages | : 317 |
| Release | : 2014-08-07 |
| Genre | : Mathematics |
| ISBN | : 1139993623 |
Download Optimal Transport Book in PDF, ePub and Kindle
The theory of optimal transportation has its origins in the eighteenth century when the problem of transporting resources at a minimal cost was first formalised. Through subsequent developments, particularly in recent decades, it has become a powerful modern theory. This book contains the proceedings of the summer school 'Optimal Transportation: Theory and Applications' held at the Fourier Institute in Grenoble. The event brought together mathematicians from pure and applied mathematics, astrophysics, economics and computer science. Part I of this book is devoted to introductory lecture notes accessible to graduate students, while Part II contains research papers. Together, they represent a valuable resource on both fundamental and advanced aspects of optimal transportation, its applications, and its interactions with analysis, geometry, PDE and probability, urban planning and economics. Topics covered include Ricci flow, the Euler equations, functional inequalities, curvature-dimension conditions, and traffic congestion.
| Author | : Luigi Ambrosio |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 184 |
| Release | : 2003-06-12 |
| Genre | : Mathematics |
| ISBN | : 9783540401926 |
Download Optimal Transportation and Applications Book in PDF, ePub and Kindle
Leading researchers in the field of Optimal Transportation, with different views and perspectives, contribute to this Summer School volume: Monge-Ampère and Monge-Kantorovich theory, shape optimization and mass transportation are linked, among others, to applications in fluid mechanics granular material physics and statistical mechanics, emphasizing the attractiveness of the subject from both a theoretical and applied point of view. The volume is designed to become a guide to researchers willing to enter into this challenging and useful theory.
| Author | : Anna Beliakova |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 267 |
| Release | : 2017-02-21 |
| Genre | : Categories (Mathematics) |
| ISBN | : 1470428210 |
Download Categorification in Geometry, Topology, and Physics Book in PDF, ePub and Kindle
The emergent mathematical philosophy of categorification is reshaping our view of modern mathematics by uncovering a hidden layer of structure in mathematics, revealing richer and more robust structures capable of describing more complex phenomena. Categorification is a powerful tool for relating various branches of mathematics and exploiting the commonalities between fields. It provides a language emphasizing essential features and allowing precise relationships between vastly different fields. This volume focuses on the role categorification plays in geometry, topology, and physics. These articles illustrate many important trends for the field including geometric representation theory, homotopical methods in link homology, interactions between higher representation theory and gauge theory, and double affine Hecke algebra approaches to link homology. The companion volume (Contemporary Mathematics, Volume 683) is devoted to categorification and higher representation theory.
| Author | : Joseph C. Várilly |
| Publisher | : European Mathematical Society |
| Total Pages | : 134 |
| Release | : 2006 |
| Genre | : Mathematics |
| ISBN | : 9783037190241 |
Download An Introduction to Noncommutative Geometry Book in PDF, ePub and Kindle
Noncommutative geometry, inspired by quantum physics, describes singular spaces by their noncommutative coordinate algebras and metric structures by Dirac-like operators. Such metric geometries are described mathematically by Connes' theory of spectral triples. These lectures, delivered at an EMS Summer School on noncommutative geometry and its applications, provide an overview of spectral triples based on examples. This introduction is aimed at graduate students of both mathematics and theoretical physics. It deals with Dirac operators on spin manifolds, noncommutative tori, Moyal quantization and tangent groupoids, action functionals, and isospectral deformations. The structural framework is the concept of a noncommutative spin geometry; the conditions on spectral triples which determine this concept are developed in detail. The emphasis throughout is on gaining understanding by computing the details of specific examples. The book provides a middle ground between a comprehensive text and a narrowly focused research monograph. It is intended for self-study, enabling the reader to gain access to the essentials of noncommutative geometry. New features since the original course are an expanded bibliography and a survey of more recent examples and applications of spectral triples.
| Author | : Heather A. Harrington |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 277 |
| Release | : 2017-03-16 |
| Genre | : Commutative algebra -- Computational aspects and applications -- Applications of commutative algebra (e.g., to statistics, control theory, optimization, etc.) |
| ISBN | : 1470423219 |
Download Algebraic and Geometric Methods in Discrete Mathematics Book in PDF, ePub and Kindle
This volume contains the proceedings of the AMS Special Session on Algebraic and Geometric Methods in Applied Discrete Mathematics, held on January 11, 2015, in San Antonio, Texas. The papers present connections between techniques from “pure” mathematics and various applications amenable to the analysis of discrete models, encompassing applications of combinatorics, topology, algebra, geometry, optimization, and representation theory. Papers not only present novel results, but also survey the current state of knowledge of important topics in applied discrete mathematics. Particular highlights include: a new computational framework, based on geometric combinatorics, for structure prediction from RNA sequences; a new method for approximating the optimal solution of a sum of squares problem; a survey of recent Helly-type geometric theorems; applications of representation theory to voting theory and game theory; a study of fixed points of tensors; and exponential random graph models from the perspective of algebraic statistics with applications to networks. This volume was written for those trained in areas such as algebra, topology, geometry, and combinatorics who are interested in tackling problems in fields such as biology, the social sciences, data analysis, and optimization. It may be useful not only for experts, but also for students who wish to gain an applied or interdisciplinary perspective.
