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Ergodic Theory and Topological Dynamics of Group Actions on Homogeneous Spaces
This book, first published in 2000, focuses on developments in the study of geodesic flows on homogenous spaces.
Unitary Representations of Groups, Duals, and Characters
Unitary representations of groups play an important role in many subjects, including number theory, geometry, probability theory, partial differential equations, and quantum mechanics. This monograph focuses on dual spaces associated to a group, which are spaces of building blocks of general unitary representations. Special attention is paid to discrete groups for which the unitary dual, the most common dual space, has proven to be not useful in general and for which other duals spaces have to be considered, such as the primitive dual, the normal quasi-dual, or spaces of characters. The book offers a detailed exposition of these alternative dual spaces and covers the basic facts about unitar...
Higher Structures in Topology, Geometry, and Physics
This volume contains the proceedings of the AMS Special Session on Higher Structures in Topology, Geometry, and Physics, held virtually on March 26–27, 2022. The articles give a snapshot survey of the current topics surrounding the mathematical formulation of field theories. There is an intricate interplay between geometry, topology, and algebra which captures these theories. The hallmark are higher structures, which one can consider as the secondary algebraic or geometric background on which the theories are formulated. The higher structures considered in the volume are generalizations of operads, models for conformal field theories, string topology, open/closed field theories, BF/BV formalism, actions on Hochschild complexes and related complexes, and their geometric and topological aspects.
Sampling in Combinatorial and Geometric Set Systems
Understanding the behavior of basic sampling techniques and intrinsic geometric attributes of data is an invaluable skill that is in high demand for both graduate students and researchers in mathematics, machine learning, and theoretical computer science. The last ten years have seen significant progress in this area, with many open problems having been resolved during this time. These include optimal lower bounds for epsilon-nets for many geometric set systems, the use of shallow-cell complexity to unify proofs, simpler and more efficient algorithms, and the use of epsilon-approximations for construction of coresets, to name a few. This book presents a thorough treatment of these probabilistic, combinatorial, and geometric methods, as well as their combinatorial and algorithmic applications. It also revisits classical results, but with new and more elegant proofs. While mathematical maturity will certainly help in appreciating the ideas presented here, only a basic familiarity with discrete mathematics, probability, and combinatorics is required to understand the material.
Recovery Methodologies: Regularization and Sampling
The goal of this book is to introduce the reader to methodologies in recovery problems for objects, such as functions and signals, from partial or indirect information. The recovery of objects from a set of data demands key solvers of inverse and sampling problems. Until recently, connections between the mathematical areas of inverse problems and sampling were rather tenuous. However, advances in several areas of mathematical research have revealed deep common threads between them, which proves that there is a serious need for a unifying description of the underlying mathematical ideas and concepts. Freeden and Nashed present an integrated approach to resolution methodologies from the perspe...
Maximal Cohen–Macaulay Modules and Tate Cohomology
This book is a lightly edited version of the unpublished manuscript Maximal Cohen–Macaulay modules and Tate cohomology over Gorenstein rings by Ragnar-Olaf Buchweitz. The central objects of study are maximal Cohen–Macaulay modules over (not necessarily commutative) Gorenstein rings. The main result is that the stable category of maximal Cohen–Macaulay modules over a Gorenstein ring is equivalent to the stable derived category and also to the homotopy category of acyclic complexes of projective modules. This assimilates and significantly extends earlier work of Eisenbud on hypersurface singularities. There is also an extensive discussion of duality phenomena in stable derived categories, extending Tate duality on cohomology of finite groups. Another noteworthy aspect is an extension of the classical BGG correspondence to super-algebras. There are numerous examples that illustrate these ideas. The text includes a survey of developments subsequent to, and connected with, Buchweitz's manuscript.
Iwasawa Theory and Its Perspective, Volume 2
Iwasawa theory began in the late 1950s with a series of papers by Kenkichi Iwasawa on ideal class groups in the cyclotomic tower of number fields and their relation to $p$-adic $L$-functions. The theory was later generalized by putting it in the context of elliptic curves and modular forms. The main motivation for writing this book was the need for a total perspective of Iwasawa theory that includes the new trends of generalized Iwasawa theory. Another motivation is to update the classical theory for class groups, taking into account the changed point of view on Iwasawa theory. The goal of this second part of the three-part publication is to explain various aspects of the cyclotomic Iwasawa theory of $p$-adic Galois representations.
Completion Problems on Operator Matrices
Completion problems for operator matrices are concerned with the question of whether a partially specified operator matrix can be completed to form an operator of a desired type. The research devoted to this topic provides an excellent means to investigate the structure of operators. This book provides an overview of completion problems dealing with completions to different types of operators and can be considered as a natural extension of classical results concerned with matrix completions. The book assumes some basic familiarity with functional analysis and operator theory. It will be useful for graduate students and researchers interested in operator theory and the problem of matrix completions.
Translation Surfaces
This textbook offers an accessible introduction to translation surfaces. Building on modest prerequisites, the authors focus on the fundamentals behind big ideas in the field: ergodic properties of translation flows, counting problems for saddle connections, and associated renormalization techniques. Proofs that go beyond the introductory nature of the book are deftly omitted, allowing readers to develop essential tools and motivation before delving into the literature. Beginning with the fundamental example of the flat torus, the book goes on to establish the three equivalent definitions of translation surface. An introduction to the moduli space of translation surfaces follows, leading int...
Tool Kit for Groupoid C∗ -Algebras
The construction of a C∗-algebra from a locally compact groupoid is an important generalization of the group C∗-algebra construction and of the transformation group C∗-algebra construction. Since their introduction in 1980, groupoid C∗-algebras have been intensively studied with diverse applications, including graph algebras, classification theory, variations on the Baum-Connes conjecture, and noncommutative geometry. This book provides a detailed introduction to this vast subject and is suitable for graduate students or any researcher who wants to use groupoid C∗-algebras in their work. The main focus is to equip the reader with modern versions of the basic technical tools used in...
