2017; 464 pp; Hardcover
Print ISBN: 978-1-4704-2962-1
This book is an introduction to the modern theory of Markov chains, whose goal is to determine the rate of convergence to the stationary distribution, as a function of state space size and geometry. This topic has important connections to combinatorics, statistical physics, and theoretical computer science. Many of the techniques presented originate in these disciplines.
The central tools for estimating convergence times, including coupling, strong stationary times, and spectral methods, are developed. The authors discuss many examples, including card shuffling and the Ising model, from statistical mechanics, and present the connection of random walks to electrical networks and apply it to estimate hitting and cover times.
The first edition has been used in courses in mathematics and computer science departments of numerous universities. The second edition features three new chapters (on monotone chains, the exclusion process, and stationary times) and also includes smaller additions and corrections throughout. Updated notes at the end of each chapter inform the reader of recent research developments.
Undergraduate and graduate students interested in the modern theory of Markov chains.
Collected Works Volume: 26
2017; 785 pp; Hardcover
Print ISBN: 978-1-4704-3655-1
In the period since the original four volumes of Phillip Griffiths's Selecta were published (Selected Works of Phillip A. Griffiths with Commentary), Parts 1?4, Collected Works, Volume 18), Griffiths has continued to produce beautiful and important work. The current two-part publication brings Griffiths's Selecta up to date by including the majority of his recent articles, as well as two older papers on differential geometry whose length had precluded their inclusion in the original Selecta.
The papers are organized along the three main topics: Differential Geometry and Hodge Theory (Part 5) and Algebraic Cycles (Part 6). In addition to his papers, Griffiths has been an author of a number of research monographs. To give the reader an overview of what these monographs contain, introductions to some of these are also included.
Graduate students and researchers interested in algebraic and differential geometry.
Graduate Studies in Mathematics, Volume: 181
2017; Hardcover
Print ISBN: 978-1-4704-2921-8
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, Poincare's inequalities and traces.
A major change compared to the first edition is the chapter on Besov spaces, which is now treated using interpolation theory.
Graduate students and researchers interested in Sobolev spaces, particularly their applications to PDEs.
EMS Series of Lectures in Mathematics
ISBN print 978-3-03719-180-4,
July 2017, 144 pages, softcover, 17 x 24 cm.
Michel Kervaire wrote six papers which can be considered fundamental to the development of higher-dimensional knot theory. They are not only of historical interest but naturally introduce to some of the essential techniques in this fascinating theory.
This book is written to provide graduate students with the basic concepts necessary to read texts in higher-dimensional knot theory and its relations with singularities. The first chapters are devoted to a presentation of Pontrjaginfs construction, surgery and the work of Kervaire and Milnor on homotopy spheres. We pursue with Kervairefs fundamental work on the group of a knot, knot modules and knot cobordism. We add developments due to Levine. Tools (like open books, handlebodies, plumbings, c) often used but hard to find in original articles are presented in appendices. We conclude with a description of the Kervaire invariant and the consequences of the Hill?Hopkins?Ravenel results in knot theory.
Knots in high dimensions, homotopy spheres, complex singularities