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Functor Categories, Model Theory, Algebraic Analysis and Constructive Methods
This volume comprises selected contributions by the participants of the second "Functor Categories, Model Theory, Algebraic Analysis and Constructive Methods" conference, which took place at the University of Almería, Spain, in July 2022. The conference was devoted to several seemingly unrelated fields: functor categories, model theory of modules, algebraic analysis (including linear control systems), and constructive category theory, to mention just a few. The fact that these fields are actually related is a very recent realization. The connections between these disciplines are changing in real time, and the goal of this volume is to provide an initial reference point for this emerging interdisciplinary field. Besides research articles, the volume includes two extended lectures: one on constructive methods in algebraic analysis and the other on the functorial approach to algebraic systems theory. Hence, in addition to its interest for researchers, the volume will also be an invaluable resource for newcomers.
Representations of Algebras, Geometry and Physics
This volume contains selected expository lectures delivered at the 2018 Maurice Auslander Distinguished Lectures and International Conference, held April 25–30, 2018, at the Woods Hole Oceanographic Institute, Woods Hole, MA. Reflecting recent developments in modern representation theory of algebras, the selected topics include an introduction to a new class of quiver algebras on surfaces, called “geodesic ghor algebras”, a detailed presentation of Feynman categories from a representation-theoretic viewpoint, connections between representations of quivers and the structure theory of Coxeter groups, powerful new applications of approximable triangulated categories, new results on the heart of a t t-structure, and an introduction to methods of constructive category theory.
Representation Theory and Algebraic Geometry
For any researcher working in representation theory, algebraic or arithmetic geometry.
Surveys in Representation Theory of Algebras
This volume contains selected expository lectures delivered at the annual Maurice Auslander Distinguished Lectures and International Conference over the last several years. Reflecting the diverse landscape of modern representation theory of algebras, the selected articles include: a quick introduction to silting modules; a survey on the first decade of co-t-structures in triangulated categories; a functorial approach to the notion of module; a representation-theoretic approach to recollements in abelian categories; new examples of applications of relative homological algebra; connections between Coxeter groups and quiver representations; and recent progress on limits of approximation theory.
Expository Lectures on Representation Theory
This volume contains the proceedings of the Maurice Auslander Distinguished Lectures and International Conference, held April 25-30, 2012, in Falmouth, MA. The representation theory of finite dimensional algebras and related topics, especially cluster combinatorics, is a very active topic of research. This volume contains papers covering both the history and the latest developments in this topic. In particular, Otto Kerner gives a review of basic theorems and latest results about wild hereditary algebras, Yuri Berest develops the theory of derived representation schemes, and Markus Schmidmeier presents new applications of arc diagrams.
Recent Developments in Representation Theory
This volume contains selected expository lectures delivered at the Maurice Auslander Distinguished Lectures and International Conference, held May 1–6, 2014, at the Woods Hole Oceanographic Institute, Woods Hole, MA. Several significant developments of the last decade in representation theory of finite-dimensional algebras are related to combinatorics. Three of the five lectures in this volume deal, respectively, with the Catalan combinatorics, the combinatorics of Gelfand-Zetlin polytopes, and the combinatorics of tilting modules. The remaining papers present history and recent advances in the study of left orders in left Artinian rings and a survey on invariant theory of Artin-Schelter regular algebras.
Infinite Length Modules
This book is concerned with the role played by modules of infinite length when dealing with problems in the representation theory of groups and algebras, but also in topology and geometry; it shows the intriguing interplay between finite and infinite length modules.
Groups, Generators, Syzygies, and Orbits in Invariant Theory
The history of invariant theory spans nearly a century and a half, with roots in certain problems from number theory, algebra, and geometry appearing in the work of Gauss, Jacobi, Eisenstein, and Hermite. Although the connection between invariants and orbits was essentially discovered in the work of Aronhold and Boole, a clear understanding of this connection had not been achieved until recently, when invariant theory was in fact subsumed by a general theory of algebraic groups. Written by one of the major leaders in the field, this book provides an excellent, comprehensive exposition of invariant theory. Its point of view is unique in that it combines both modern and classical approaches to the subject. The introductory chapter sets the historical stage for the subject, helping to make the book accessible to nonspecialists.
Number-Theoretic Algorithms in Cryptography
Algorithmic number theory is a rapidly developing branch of number theory, which, in addition to its mathematical importance, has substantial applications in computer science and cryptography. Among the algorithms used in cryptography, the following are especially important: algorithms for primality testing; factorization algorithms for integers and for polynomials in one variable; applications of the theory of elliptic curves; algorithms for computation of discrete logarithms; algorithms for solving linear equations over finite fields; and, algorithms for performing arithmetic operations on large integers. The book describes the current state of these and some other algorithms. It also contains extensive bibliography. For this English translation, additional references were prepared and commented on by the author.
