Paperback 9780691193779
186 pp.
6 1/8 x 9 1/4
1 b/w illus.
forthcoming October 2019
Hardcover
ISBN 9780691193786
(AMS-202)
Arithmetic and Geometry presents highlights of recent work in arithmetic algebraic geometry by some of the world's leading mathematicians. Together, these 2016 lectures?which were delivered in celebration of the tenth anniversary of the annual summer workshops in Alpbach, Austria?provide an introduction to high-level research on three topics: Shimura varieties, hyperelliptic continued fractions and generalized Jacobians, and Faltings height and L-functions. The book consists of notes, written by young researchers, on three sets of lectures or minicourses given at Alpbach.
The first course, taught by Peter Scholze, contains his recent results dealing with the local Langlands conjecture. The fundamental question is whether for a given datum there exists a so-called local Shimura variety. In some cases, they exist in the category of rigid analytic spaces; in others, one has to use Scholze's perfectoid spaces.
The second course, taught by Umberto Zannier, addresses the famous Pell equation?not in the classical setting but rather with the so-called polynomial Pell equation, where the integers are replaced by polynomials in one variable with complex coefficients, which leads to the study of hyperelliptic continued fractions and generalized Jacobians.
The third course, taught by Shou-Wu Zhang, originates in the Chowla?Selberg formula, which was taken up by Gross and Zagier to relate values of the L-function for elliptic curves with the height of Heegner points on the curves. Zhang, X. Yuan, and Wei Zhang prove the Gross?Zagier formula on Shimura curves and verify the Colmez conjecture on average.
Gisbert Wustholz is professor emeritus of mathematics at ETH Zurich. Clemens Fuchs is professor of discrete mathematics at the University of Salzburg.
Hardcover
ISBN 9780691197883
Paperback
ISBN 9780691197890
236 pp.
6 1/8 x 9 1/4
1 b/w illus.
forthcoming December 2019
(AMS-203)
This book studies the interplay between the geometry and topology of locally symmetric spaces, and the arithmetic aspects of the special values of L-functions.
The authors study the cohomology of locally symmetric spaces for GL(N) where the cohomology groups are with coefficients in a local system attached to a finite-dimensional algebraic representation of GL(N). The image of the global cohomology in the cohomology of the Borel?Serre boundary is called Eisenstein cohomology, since at a transcendental level the cohomology classes may be described in terms of Eisenstein series and induced representations. However, because the groups are sheaf-theoretically defined, one can control their rationality and even integrality properties. A celebrated theorem by Langlands describes the constant term of an Eisenstein series in terms of automorphic L-functions. A cohomological interpretation of this theorem in terms of maps in Eisenstein cohomology allows the authors to study the rationality properties of the special values of Rankin?Selberg L-functions for GL(n) x GL(m), where n + m = N. The authors carry through the entire program with an eye toward generalizations.
This book should be of interest to advanced graduate students and researchers interested in number theory, automorphic forms, representation theory, and the cohomology of arithmetic groups.
Gunter Harder is professor emeritus of mathematics at the University of Bonn and former director of the Max Planck Institute for Mathematics. He is the author of the two-volume Lectures on Algebraic Geometry. A. Raghuram is professor and chair of mathematics at the Indian Institute of Science Education and Research, Pune.
Hardcover
ISBN
9780691191744
Paperback
9780691191751
200 pp.
6 1/8 x 9 1/4
37 b/w illus.
forthcoming March 2020
(AMS-204)
This book gives a clear introductory account of equivariant cohomology, a central topic in algebraic topology. Equivariant cohomology is concerned with the algebraic topology of spaces with a group action, or in other words, with symmetries of spaces. First defined in the 1950s, it has been introduced into K-theory and algebraic geometry, but it is in algebraic topology that the concepts are the most transparent and the proofs are the simplest. One of the most useful applications of equivariant cohomology is the equivariant localization theorem of Atiyah-Bott and Berline-Vergne, which converts the integral of an equivariant differential form into a finite sum over the fixed point set of the group action, providing a powerful tool for computing integrals over a manifold. Because integrals and symmetries are ubiquitous, equivariant cohomology has found applications in diverse areas of mathematics and physics.
Assuming readers have taken one semester of manifold theory and a year of algebraic topology, Loring Tu begins with the topological construction of equivariant cohomology, then develops the theory for smooth manifolds with the aid of differential forms. To keep the exposition simple, the equivariant localization theorem is proven only for a circle action. An appendix gives a proof of the equivariant de Rham theorem, demonstrating that equivariant cohomology can be computed using equivariant differential forms. Examples and calculations illustrate new concepts. Exercises include hints or solutions, making this book suitable for self-study.
Loring W. Tu is professor of mathematics at Tufts University. He is the author of An Introduction to Manifolds and Differential Geometry, and the coauthor (with Raoul Bott) of Differential Forms in Algebraic Topology.
Hardcover
ISBN
9780691167763
Paperback
ISBN
9780691167770
472 pp.
6 1/8 x 9 1/4
forthcoming May 2020
William Thurston (1946?2012) was one of the great mathematicians of the twentieth century. He was a visionary whose extraordinary ideas revolutionized a broad range of mathematical fields, from foliations, contact structures, and Teichmuller theory to automorphisms of surfaces, hyperbolic geometry, geometrization of 3-manifolds, geometric group theory, and rational maps. In addition, he discovered connections between disciplines that led to astonishing breakthroughs in mathematical understanding as well as the creation of entirely new fields. His far-reaching questions and conjectures led to enormous progress by other researchers. What's Next? brings together many of today's leading mathematicians to describe recent advances and future directions inspired by Thurston's transformative ideas.
Including valuable insights from his colleagues and former students, What's Next? discusses Thurston's fundamental contributions to topology, geometry, and dynamical systems and includes many deep and original contributions to the field. This incisive and wide-ranging book also explores how he introduced new ways of thinking about and doing mathematics, innovations that have had a profound and lasting impact on the mathematical community as a whole.
Dylan Thurston is professor of mathematics at Indiana University, Bloomington. He is an editor of the journal Quantum Topology.
2018 / viii + 115 pages / Softcover / 978-1-611975-41-3
Keywords: singular integral, Hilbert transform, Riesz transforms, Littlewood-Paley theory, spherical harmonics
In just over 100 pages, this book provides basic, essential knowledge of some of the tools of real analysis: the Hardy?Littlewood maximal operator, the Calderon?Zygmund theory, the Littlewood?Paley theory, interpolation of spaces and operators, and the basics of H1 and BMO spaces. This concise text offers brief proofs and exercises of various difficulties designed to challenge and engage students.
An Introduction to Singular Integrals is meant to give first-year graduate students in Fourier analysis and partial differential equations an introduction to harmonic analysis. While some background material is included in the appendices, readers should have a basic knowledge of functional analysis, some acquaintance with measure and integration theory, and familiarity with the Fourier transform in Euclidean spaces.
Jacques Peyriere is an emeritus professor of mathematics at Universite Paris-Sud (Orsay). He has been head of the Equipe d'Analyse Harmonique (a CNRS team) there for 10 years. Professor Peyriere has published two books and more than 60 articles on harmonic analysis and related topics in mathematical journals, including Duke Mathematical Journal, Advances in Mathematics, and Probability Theory and Related Fields. His research interests are harmonic analysis, probability theory, and fractals.
2019 / xiv + 168 pages / Softcover / ISBN 978-1-611975-61-1
Keywords: Nonlocal mechanics, nonlocal diffusion, nonlocal vector calculus, nonlocal variational problems and dynamic systems, numerical methods
Studies of complexity, singularity, and anomaly using nonlocal continuum models are steadily gaining popularity. This monograph provides an introduction to basic analytical, computational, and modeling issues and to some of the latest developments in these areas.
Nonlocal Modeling, Analysis, and Computation includes motivational examples of nonlocal models, basic building blocks of nonlocal vector calculus, elements of theory for well-posedness and nonlocal spaces, connections to and coupling with local models, convergence and compatibility of numerical approximations, and various applications, such as nonlocal dynamics of anomalous diffusion and nonlocal peridynamic models of elasticity and fracture mechanics.
A particular focus is on nonlocal systems with a finite range of interaction to illustrate their connection to traditional local systems represented by partial differential equations and fractional PDEs. These models are designed to represent nonlocal interactions explicitly and to remain valid for complex systems involving possible singular solutions and they have the potential to be alternatives to as well as bridges to existing local continuum and discrete models.
The author discusses ongoing studies of nonlocal models to encourage the discovery of new mathematical theory for nonlocal continuum models and offer new perspectives on existing discrete models and local continuum models and the connections between them.
Nonlocal Modeling, Analysis, and Computation offers applied mathematicians, computational scientists, researchers, and graduate students an illustration of the broad applicability and rich mathematics of nonlocal models and provides motivation for further investigations.
Qiang Du is the Fu Foundation Professor of Applied Mathematics at Columbia University, where he chairs the applied mathematics program in the Department of Applied Physics and Applied Mathematics and is a co-chair of the Center for Foundations of Data Science in the Data Science Institute. Professor Du was the Verne M. Willaman Professor of Mathematics and Professor of Materials Science and Engineering at Penn State University. Recognition for his work includes the SIAM Outstanding Paper Prize and ACM Gordon Bell Prize Finalist in 2016, and the Excellent Paper Prize of Scientia Sinica Mathematica in 2017. He was an invited speaker at the International Congress of Mathematicians (ICM 2018) and is a Fellow of SIAM and of the American Association for the Advancement of Science.