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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
two-volume monograph develops generalization of various topics in
algebraic geometry in the context of derived algebraic geometry.
Volume I presents the theory of ind-coherent sheaves, which are a
“renormalization” 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.
Volume II 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
inf-schemes. As an application of the general theory, the six-functor
formalism for D-modules in derived geometry is obtained.
Graduate students and researchers interested in new trends in algebraic geometry and representation theory.
This volume contains the proceedings of the conference
String-Math 2015, which was held from December 31, 2015–January 4,
2016, at Tsinghua Sanya International Mathematics Forum in Sanya,
China. Two of the main themes of this volume are frontier research on
Calabi-Yau manifolds and mirror symmetry and the development of
non-perturbative methods in supersymmetric gauge theories. The
articles present state-of-the-art developments in these topics.
String theory is a broad subject, which has profound connections
with broad branches of modern mathematics. In the last decades, the
prosperous interaction built upon the joint efforts from both
mathematicians and physicists has given rise to marvelous deep results
in supersymmetric gauge theory, topological string, M-theory and
duality on the physics side, as well as in algebraic geometry,
differential geometry, algebraic topology, representation theory and
number theory on the mathematics side.
Advanced graduate students, post-docs, and post Ph.D. mathematicians and mathematical physicists interested in string theory.
This is an introductory textbook about
nonlinear dynamics of PDEs, with a focus on problems over unbounded
domains and modulation equations. The presentation is
example-oriented, and new mathematical tools are developed step by
step, giving insight into some important classes of nonlinear PDEs and
nonlinear dynamics phenomena which may occur in PDEs.
The book consists of four parts. Parts I and II are introductions
to finite- and infinite-dimensional dynamics defined by ODEs and by
PDEs over bounded domains, respectively, including the basics of
bifurcation and attractor theory. Part III introduces PDEs on the real
line, including the Korteweg-de Vries equation, the Nonlinear
Schrödinger equation and the Ginzburg-Landau equation. These examples
often occur as simplest possible models, namely as amplitude or
modulation equations, for some real world phenomena such as nonlinear
waves and pattern formation. Part IV explores in more detail the
connections between such complicated physical systems and the reduced
models. For many models, a mathematically rigorous justification by
approximation results is given.
The parts of the book are kept as self-contained as possible. The
book is suitable for self-study, and there are various possibilities
to build one- or two-semester courses from the book.
Graduate students and researchers interested in nonlinear dynamics of PDEs.
This book is about differentiation of
functions. It is divided into two parts, which can be used as
different textbooks, one for an advanced undergraduate course in
functions of one variable and one for a graduate course on Sobolev
functions. The first part develops the theory of monotone, absolutely
continuous, and bounded variation functions of one variable and their
relationship with Lebesgue–Stieltjes measures and Sobolev
functions. It also studies decreasing rearrangement and curves. The
second edition includes a chapter on functions mapping time into
Banach spaces.
The second part of the book studies functions of several
variables. It begins with an overview of classical results such as
Rademacher's and Stepanoff's differentiability theorems, Whitney's
extension theorem, Brouwer's fixed point theorem, and the divergence
theorem for Lipschitz domains. It then moves to distributions, Fourier
transforms and tempered distributions.
The remaining chapters are a treatise on Sobolev functions. The
second edition focuses more on higher order derivatives and it
includes the interpolation theorems of Gagliardo and Nirenberg. It
studies embedding theorems, extension domains, chain rule,
superposition, Poincaré's inequalities and traces.
A major change compared to the first edition is the chapter on
Besov spaces, which are now treated using interpolation theory.
Graduate students and researchers interested in Sobolev spaces, particularly their applications to PDEs.
This volume contains the proceedings of the
Conference on Dynamical Systems, Ergodic Theory, and Probability,
which was dedicated to the memory of Nikolai Chernov, held from May
18–20, 2015, at the University of Alabama at Birmingham, Birmingham,
Alabama.
The book is devoted to recent advances in the theory of chaotic and
weakly chaotic dynamical systems and its applications to statistical
mechanics. The papers present new original results as well as
comprehensive surveys.
Graduate students and research mathematicians interested in dynamical systems, ergodic theory, and probability.
The first systematic theory of generalized functions (also known as distributions) was created in the early 1950s, although some aspects were developed much earlier, most notably in the definition of the Green's function in mathematics and in the work of Paul Dirac on quantum electrodynamics in physics. The six-volume collection, Generalized Functions, written by I. M. Gel′fand and co-authors and published in Russian between 1958 and 1966, gives an introduction to generalized functions and presents various applications to analysis, PDE, stochastic processes, and representation theory.
CHEL/377.H
CHEL/378.H
CHEL/379.H
CHEL/380.H
CHEL/381.H
CHEL/382.H
Graduate students and research mathematicians interested in analysis, differential equations, and representation theory.
This treatise, by one of Russia's leading mathematicians, gives in easily accessible form a coherent account of matrix theory with a view to applications in mathematics, theoretical physics, statistics, electrical engineering, etc. The individual chapters have been kept as far as possible independent of each other, so that the reader acquainted with the contents of Chapter 1 can proceed immediately to the chapters of special interest. Until now, much of the material has been available only in the periodical literature.
The first part (10 chapters; “General theory”) gives in satisfactory detail, and with more than customary completeness, the topics which belong to the main body of the … subjects … The point of view is broad and includes much abstract treatment …
The number of subjects which the book treats well is great … would appeal to a wide audience.
-- Mathematical Reviews
The work is an outstanding contribution to matrix theory and contains much material not to be found in any other text.
-- Mathematical Reviews
This is an excellent textbook.
-- Zentralblatt MATH
This is an excellent and unusual textbook on the application of the theory of matrices. In spite of intensive developments in the theory of matrices and appearance of other significant books, both general and specialized, in the last four decades, this monograph has retained its leading role. This textual matter includes many chapters of interest to applied mathematicians.
-- Zentralblatt MATH
This book is a translation by F. Steinhardt of the last of Carathéodory's celebrated text books, Funktiontheorie, Volume 1.
A book by a master … Carathéodory himself regarded [it] as his finest achievement … written from a catholic point of view.
-- Bulletin of the AMS
This reprint in one volume, includes Volumes 6, 7, and 8 of the monographs of the Steklov Institute of Mathematics in Moscow. The text is in French.
The theory of [systems of linear differential equations] is treated with elegance and generality by the author, and his contributors constitute an important addition to the field of differential equations.
-- Applied Mechanics Reviews
Group Theory occupies the first half of the book; applications to Topology, the second. This well-known book is of interest both to algebraists and topologists. The text is in German.