Pavel Etingof – författare

The text is based on an established graduate course given at MIT that provides an introduction to the theory of the dynamical Yang-Baxter equation and its applications, which is an important area in representation theory and quantum groups. The book, which contains many detailed proofs and explicit calculations, will be accessible to graduate students of mathematics, who are familiar with the basics of representation theory of semisimple Lie algebras.

(Very preliminary)A tribute to the vision and legacy of Israel Gelfand, the invited papers in this volume reflect the unity of mathematics as a whole, with particular emphasis on the many connections among the fields of geometry, physics, and representation theory. Written by leading mathematicians, the text is broadly divided into two sections: the first is devoted to developments at the intersection of geometry and physics, and the second to representation theory and algebraic geometry. Topics include conformal field theory, K-theory, noncommutative geometry, gauge theory, representations of infinite-dimensional Lie algebras, and various aspects of the Langlands program. Graduate students and researchers will benefit from and find inspiration in this broad and unique work, which brings together fundamental results in a number of disciplines and highlights the rewards of an interdisciplinary approach to mathematics and physics.Contributors: M. Atiyah, A. Beilinson, J. Bernstein, A. Connes, P. Deligne, R. Dijkgraaf, D. Gaitsgory, M. Gromov, F. Hirzebruch, M. Hopkins, D. Kazhdan, F. Kirwan, M. Kontsevich, B. Kostant, G. Lusztig, D. McDuff, H. Nakajima, S. Novikov, P. Sarnak, A.Schwartz, I. Singer, R. Taylor, C. Vafa, A. Vershik, E. Witten.

Is there a vector space whose dimension is the golden ratio? Of course not-the golden ratio is not an integer! But this can happen for generalizations of vector spaces-objects of a tensor category. The theory of tensor categories is a relatively new field of mathematics that generalizes the theory of group representations. It has deep connections with many other fields, including representation theory, Hopf algebras, operator algebras, low-dimensional topology (in particular, knot theory), homotopy theory, quantum mechanics and field theory, quantum computation, theory of motives, etc. This book gives a systematic introduction to this theory and a review of its applications. While giving a detailed overview of general tensor categories, it focuses especially on the theory of finite tensor categories and fusion categories (in particular, braided and modular ones), and discusses the main results about them with proofs. In particular, it shows how the main properties of finite-dimensional Hopf algebras may be derived from the theory of tensor categories. Many important results are presented as a sequence of exercises, which makes the book valuable for students and suitable for graduate courses. Many applications, connections to other areas, additional results, and references are discussed at the end of each chapter.

The development of quantum field theory and string theory in the last half-century has taken interactions between mathematics and physics to an unprecedented level, incorporating into this area of physics (often called ""high-energy physics"") such traditionally ""pure"" areas of mathematics as algebraic topology, category theory, differential and algebraic geometry, representation theory, combinatorics, and even number theory. This interaction has been highly fruitful in both directions, leading to the necessity for high-energy physicists to know the basics of modern mathematics and for mathematicians to know the basics of modern physics. However, in their attempts to learn these basics, many mathematicians are deterred by lack of rigor in physical texts and by a different manner of presentation. In particular, even the basic setting of quantum of field theory-necessary for understanding its more advanced (and more mathematically exciting) parts-is already largely unfamiliar to mathematicians. Nevertheless, many of the basic ideas of quantum field theory can in fact be presented rigorously, and even in cases when a rigorous exposition is impractical, difficult, or impossible, one can still attempt to explain the material in a mathematically natural way. Doing so is the main goal of the present book. It attempts to give a mathematical introduction to the basic ideas of quantum field theory, providing precise definitions and statements, motivation, examples, and rigorous proofs when possible. In other words, it provides a description of the basic settings of quantum field theory which is palatable to a mathematically minded audience.