Mathematical Surveys and Monographs, Volume: 222
2017; Hardcover
Print ISBN: 978-1-4704-3623-0
The aim of this book is to introduce and develop an arithmetic analogue of classical differential geometry. In this new geometry the ring of integers plays the role of a ring of functions on an infinite dimensional manifold. The role of coordinate functions on this manifold is played by the prime numbers. The role of partial derivatives of functions with respect to the coordinates is played by the Fermat quotients of integers with respect to the primes. The role of metrics is played by symmetric matrices with integer coefficients. The role of connections (respectively curvature) attached to metrics is played by certain adelic (respectively global) objects attached to the corresponding matrices.
One of the main conclusions of the theory is that the spectrum of the integers is gintrinsically curvedh; the study of this curvature is then the main task of the theory. The book follows, and builds upon, a series of recent research papers. A significant part of the material has never been published before.
Graduate students and researchers interested in algebraic geometry, number theory, and algebraic groups.
Mathematical Surveys and Monographs, Volume: 221
2017; Hardcover
Print ISBN: 978-1-4704-3569-1
Derived algebraic geometry is a far-reaching generalization of algebraic geometry. It has found numerous applications in various parts of mathematics, most prominently in representation theory. This volume develops the theory of ind-coherent sheaves in the context of derived algebraic geometry. Ind-coherent sheaves are a grenormalizationh of quasi-coherent sheaves and provide a natural setting for Grothendieck-Serre duality as well as geometric incarnations of numerous categories of interest in representation theory.
This volume consists of three parts and an appendix. The first part is a survey of homotopical algebra in the setting of
-categories and the basics of derived algebraic geometry. The second part builds the theory of ind-coherent sheaves as a functor out of the category of correspondences and studies the relationship between ind-coherent and quasi-coherent sheaves. The third part sets up the general machinery of the (,2)-category of correspondences needed for the second part. The category of correspondences, via the theory developed in the third part, provides a general framework for Grothendieck's six-functor formalism. The appendix provides the necessary background on (,2)
Graduate students and researchers interested in new trends in algebraic geometry and representation theory.
Mathematical Surveys and Monographs, Volume: 221
2017; Hardcover
Print ISBN: 978-1-4704-3570-7
Derived algebraic geometry is a far-reaching generalization of algebraic geometry. It has found numerous applications in other parts of mathematics, most prominently in representation theory. This volume develops deformation theory, Lie theory and the theory of algebroids in the context of derived algebraic geometry. To that end, it introduces the notion of inf-scheme, which is an infinitesimal deformation of a scheme and studies ind-coherent sheaves on such. As an application of the general theory, the six-functor formalism for D-modules in derived geometry is obtained.
This volume consists of two parts. The first part introduces the notion of ind-scheme and extends the theory of ind-coherent sheaves to inf-schemes, obtaining the theory of D-modules as an application. The second part establishes the equivalence between formal Lie group(oids) and Lie algebr(oids) in the category of ind-coherent sheaves. This equivalence gives a vast generalization of the equivalence between Lie algebras and formal moduli problems. This theory is applied to study natural filtrations in formal derived geometry generalizing the Hodge filtration.
Graduate students and researchers interested in new trends in algebraic geometry and representation theory.
Graduate Studies in Mathematics, Volume: 179
2017; Hardcover
ISBN: 978-0-8218-4947-7
The theory of modular forms is a fundamental tool used in many areas of mathematics and physics. It is also a very concrete and gfunh subject in itself and abounds with an amazing number of surprising identities.
This comprehensive textbook, which includes numerous exercises, aims to give a complete picture of the classical aspects of the subject, with an emphasis on explicit formulas. After a number of motivating examples such as elliptic functions and theta functions, the modular group, its subgroups, and general aspects of holomorphic and nonholomorphic modular forms are explained, with an emphasis on explicit examples. The heart of the book is the classical theory developed by Hecke and continued up to the Atkin?Lehner?Li theory of newforms and including the theory of Eisenstein series, Rankin?Selberg theory, and a more general theory of theta series including the Weil representation. The final chapter explores in some detail more general types of modular forms such as half-integral weight, Hilbert, Jacobi, Maass, and Siegel modular forms.
Some ggemsh of the book are an immediately implementable trace formula for Hecke operators, generalizations of Haberland's formulas for the computation of Petersson inner products, W. Li's little-known theorem on the diagonalization of the full space of modular forms, and explicit algorithms due to the second author for computing Maass forms.
This book is essentially self-contained; the necessary tools such as gamma and Bessel functions, Bernoulli numbers, and so on are given in a separate chapter.
Graduate students and researchers interested in modular forms.
Mathematical Surveys and Monographs, Volume: 220
2017; Hardcover
Print ISBN: 978-1-4704-3182-2
Looking at a sequence of zeros and ones, we often feel that it is not random, that is, it is not plausible as an outcome of fair coin tossing. Why? The answer is provided by algorithmic information theory: because the sequence is compressible, that is, it has small complexity or, equivalently, can be produced by a short program. This idea, going back to Solomonoff, Kolmogorov, Chaitin, Levin, and others, is now the starting point of algorithmic information theory.
The first part of this book is a textbook-style exposition of the basic notions of complexity and randomness; the second part covers some recent work done by participants of the gKolmogorov seminarh in Moscow (started by Kolmogorov himself in the 1980s) and their colleagues.
This book contains numerous exercises (embedded in the text) that will help readers to grasp the material.
Graduate students and researchers interested in topics related to an algorithmic approach to complexity and randomness.