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XVIIth International Congress on Mathematical Physics
This is an in-depth study of not just about Tan Kah-kee, but also the making of a legend through his deeds, self-sacrifices, fortitude and foresight. This revised edition sheds new light on his political agonies in Mao's China over campaigns against capitalists and intellectuals.
Approaching the Kannan-Lovász-Simonovits and Variance Conjectures
Focusing on two central conjectures of Asymptotic Geometric Analysis, the Kannan-Lovász-Simonovits spectral gap conjecture and the variance conjecture, these Lecture Notes present the theory in an accessible way, so that interested readers, even those who are not experts in the field, will be able to appreciate the treated topics. Offering a presentation suitable for professionals with little background in analysis, geometry or probability, the work goes directly to the connection between isoperimetric-type inequalities and functional inequalities, giving the interested reader rapid access to the core of these conjectures. In addition, four recent and important results in this theory are presented in a compelling way. The first two are theorems due to Eldan-Klartag and Ball-Nguyen, relating the variance and the KLS conjectures, respectively, to the hyperplane conjecture. Next, the main ideas needed prove the best known estimate for the thin-shell width given by Guédon-Milman and an approach to Eldan's work on the connection between the thin-shell width and the KLS conjecture are detailed.
Linear Differential Equations in the Complex Domain
This book provides a detailed introduction to recent developments in the theory of linear differential systems and integrable total differential systems. Starting from the basic theory of linear ordinary differential equations and integrable systems, it proceeds to describe Katz theory and its applications, extending it to the case of several variables. In addition, connection problems, deformation theory, and the theory of integral representations are comprehensively covered. Complete proofs are given, offering the reader a precise account of the classical and modern theory of linear differential equations in the complex domain, including an exposition of Pfaffian systems and their monodromy problems. The prerequisites are a course in complex analysis and the basics of differential equations, topology and differential geometry. This book will be useful for graduate students, specialists in differential equations, and for non-specialists who want to use differential equations.
Berkovich Spaces and Applications
We present an introduction to Berkovich’s theory of non-archimedean analytic spaces that emphasizes its applications in various fields. The first part contains surveys of a foundational nature, including an introduction to Berkovich analytic spaces by M. Temkin, and to étale cohomology by A. Ducros, as well as a short note by C. Favre on the topology of some Berkovich spaces. The second part focuses on applications to geometry. A second text by A. Ducros contains a new proof of the fact that the higher direct images of a coherent sheaf under a proper map are coherent, and B. Rémy, A. Thuillier and A. Werner provide an overview of their work on the compactification of Bruhat-Tits building...
Random Matrices
Random matrix theory has many roots and many branches in mathematics, statistics, physics, computer science, data science, numerical analysis, biology, ecology, engineering, and operations research. This book provides a snippet of this vast domain of study, with a particular focus on the notations of universality and integrability. Universality shows that many systems behave the same way in their large scale limit, while integrability provides a route to describe the nature of those universal limits. Many of the ten contributed chapters address these themes, while others touch on applications of tools and results from random matrix theory. This book is appropriate for graduate students and researchers interested in learning techniques and results in random matrix theory from different perspectives and viewpoints. It also captures a moment in the evolution of the theory, when the previous decade brought major break-throughs, prompting exciting new directions of research.
Decomposition of Jacobians by Prym Varieties
This monograph studies decompositions of the Jacobian of a smooth projective curve, induced by the action of a finite group, into a product of abelian subvarieties. The authors give a general theorem on how to decompose the Jacobian which works in many cases and apply it for several groups, as for groups of small order and some series of groups. In many cases, these components are given by Prym varieties of pairs of subcovers. As a consequence, new proofs are obtained for the classical bigonal and trigonal constructions which have the advantage to generalize to more general situations. Several isogenies between Prym varieties also result.
A Modern View of the Riemann Integral
This monograph uncovers the full capabilities of the Riemann integral. Setting aside all notions from Lebesgue’s theory, the author embarks on an exploration rooted in Riemann’s original viewpoint. On this journey, we encounter new results, numerous historical vignettes, and discover a particular handiness for computations and applications. This approach rests on three basic observations. First, a Riemann integrability criterion in terms of oscillations, which is a quantitative formulation of the fact that Riemann integrable functions are continuous a.e. with respect to the Lebesgue measure. Second, the introduction of the concepts of admissible families of partitions and modified Rieman...
In Memoriam Marc Yor - Séminaire de Probabilités XLVII
This volume is dedicated to the memory of Marc Yor, who passed away in 2014. The invited contributions by his collaborators and former students bear testament to the value and diversity of his work and of his research focus, which covered broad areas of probability theory. The volume also provides personal recollections about him, and an article on his essential role concerning the Doeblin documents. With contributions by P. Salminen, J-Y. Yen & M. Yor; J. Warren; T. Funaki; J. Pitman& W. Tang; J-F. Le Gall; L. Alili, P. Graczyk & T. Zak; K. Yano & Y. Yano; D. Bakry & O. Zribi; A. Aksamit, T. Choulli & M. Jeanblanc; J. Pitman; J. Obloj, P. Spoida & N. Touzi; P. Biane; J. Najnudel; P. Fitzsimmons, Y. Le Jan & J. Rosen; L.C.G. Rogers & M. Duembgen; E. Azmoodeh, G. Peccati & G. Poly, timP-L Méliot, A. Nikeghbali; P. Baldi; N. Demni, A. Rouault & M. Zani; N. O'Connell; N. Ikeda & H. Matsumoto; A. Comtet & Y. Tourigny; P. Bougerol; L. Chaumont; L. Devroye & G. Letac; D. Stroock and M. Emery.
Supergeometry, Super Riemann Surfaces and the Superconformal Action Functional
This book treats the two-dimensional non-linear supersymmetric sigma model or spinning string from the perspective of supergeometry. The objective is to understand its symmetries as geometric properties of super Riemann surfaces, which are particular complex super manifolds of dimension 1|1. The first part gives an introduction to the super differential geometry of families of super manifolds. Appropriate generalizations of principal bundles, smooth families of complex manifolds and integration theory are developed. The second part studies uniformization, U(1)-structures and connections on Super Riemann surfaces and shows how the latter can be viewed as extensions of Riemann surfaces by a gr...
Mean Field Games
This volume provides an introduction to the theory of Mean Field Games, suggested by J.-M. Lasry and P.-L. Lions in 2006 as a mean-field model for Nash equilibria in the strategic interaction of a large number of agents. Besides giving an accessible presentation of the main features of mean-field game theory, the volume offers an overview of recent developments which explore several important directions: from partial differential equations to stochastic analysis, from the calculus of variations to modeling and aspects related to numerical methods. Arising from the CIME Summer School "Mean Field Games" held in Cetraro in 2019, this book collects together lecture notes prepared by Y. Achdou (with M. Laurière), P. Cardaliaguet, F. Delarue, A. Porretta and F. Santambrogio. These notes will be valuable for researchers and advanced graduate students who wish to approach this theory and explore its connections with several different fields in mathematics.
