Bücher der Reihe Progress in Mathematics

This book is concerned with cardinal number valued functions defined for any Boolean algebra. Examples of such functions are independence, which assigns to each Boolean algebra the supremum of the cardinalities of its free subalgebras, and cellularity, which gives the supremum of cardinalities of sets of pairwise disjoint elements. Twenty-one such functions are studied in detail, and many more in passing. The questions considered are the behaviour of these functions under algebraic operations such as products, free products, ultraproducts, and their relationships to one another.Assuming familiarity with only the basics of Boolean algebras and set theory, through simple infinite combinatorics and forcing, the book reviews current knowledge about these functions, giving complete proofs for most facts. A special feature of the book is the attention given to open problems, of which 185 are formulated.Based on Cardinal Functions on Boolean Algebras (1990) and Cardinal Invariants on Boolean Algebras (1996) by the same author, the present work is much larger than either of these. It contains solutions to many of the open problems of the earlier volumes. Among the new topics are continuum cardinals on Boolean algebras, with a lengthy treatment of the reaping number. Diagrams at the end of the book summarize the relationships between the functions for many important classes of Boolean algebras, including interval algebras, tree algebras and superatomic algebras.

Aims to give an overview of selected topics on the topology of real and complex isolated singularities, with emphasis on its relations to other branches of geometry and topology. This book features topics such as the fibration theorems of Milnor; the relation with 3-dimensional Lie groups; spin structures and 3-manifold invariants; and more.

This book studies structural properties of Q-curvature from an extrinsic point of view by regarding it as a derived quantity of certain conformally covariant families of differential operators which are associated to hypersurfaces.

Since the inception of algebraic topology [217] the study of homotopy classes of continuous maps between spheres has enjoyed a very exc- n n tional, central role.

For about a decade I have made an effort to study quadratic forms in infinite dimensional vector spaces over arbitrary division rings. If one wants to give an introduction to the geometric algebra of infinite dimensional quadratic spaces, a discussion of ~ -dimensional 0 spaces ideally serves the purpose.

Several important problems arising in Physics, Di?erential Geometry and other n topics lead to consider semilinear variational elliptic equations on R and a great deal of work has been devoted to their study.

An exploration of the theory of discrete integrable systems, with an emphasis on the following general problem: how to discretize one or several of independent variables in a given integrable system of differential equations, maintaining the integrability property?

Over the last forty years, David Vogan has left an indelible imprint on the representation theory of reductive groups.

This book presents a consistent development of the Kohn-Nirenberg type global quantization theory in the setting of graded nilpotent Lie groups in terms of their representations.

This volume contains selected papers authored by speakers and participants of the 2013 Arbeitstagung, held at the Max Planck Institute for Mathematics in Bonn, Germany, from May 22-28.

The book will be of interest to researchers in the fields of differential geometry, complex geometry, and several complex variables geometry, as well as to graduate students in mathematics.

Maxim's vision has inspired major developments in many areas of mathematics, ranging all the way from probability theory to motives over finite fields, and has brought forth a paradigm shift at the interface of modern geometry and mathematical physics.

Over the last forty years, David Vogan has left an indelible imprint on the representation theory of reductive groups.

This book features a series of lectures that explores three different fields in which functor homology (short for homological algebra in functor categories) has recently played a significant role.

This book contains selected papers based on talks given at the "Representation Theory, Number Theory, and Invariant Theory" conference held at Yale University from June 1 to June 5, 2015.

This book consists of survey articles and original research papers in the representation theory of reductive p-adic groups.

This volume, a celebration of Anthony Joseph's fundamental influence on classical and quantized representation theory, explores a wide array of current topics in Lie theory by experts in the area.

The present monograph develops a unified theory of Steinberg groups, independent of matrix representations, based on the theory of Jordan pairs and the theory of 3-graded locally finite root systems.The development of this approach occurs over six chapters, progressing from groups with commutator relations and their Steinberg groups, then on to Jordan pairs, 3-graded locally finite root systems, and groups associated with Jordan pairs graded by root systems, before exploring the volume's main focus: the definition of the Steinberg group of a root graded Jordan pair by a small set of relations, and its central closedness. Several original concepts, such as the notions of Jordan graphs and Weyl elements, provide readers with the necessary tools from combinatorics and group theory.Steinberg Groups for Jordan Pairs is ideal for PhD students and researchers in the fields of elementary groups, Steinberg groups, Jordanalgebras, and Jordan pairs. By adopting a unified approach, anybody interested in this area who seeks an alternative to case-by-case arguments and explicit matrix calculations will find this book essential.

This book is an outgrowth of the Workshop on "Regulators in Analysis, Geom etry and Number Theory" held at the Edmund Landau Center for Research in Mathematical Analysis of The Hebrew University of Jerusalem in 1996.

Lie Groups: Structures, Actions, and Representations, In Honor of Joseph A. Wolf on the Occasion of his 75th Birthday consists of invited expository and research articles on new developments arising from Wolf's profound contributions to mathematics. Due to Professor Wolf's broad interests, outstanding mathematicians and scholars in a wide spectrum of mathematical fields contributed to the volume.  Algebraic, geometric, and analytic methods are employed. More precisely, finite groups and classical finite dimensional, as well as infinite-dimensional Lie groups, and algebras play a role. Actions on classical symmetric spaces, and on abstract homogeneous and representation spaces are discussed. Contributions in the area of representation theory involve numerous viewpoints, including that of algebraic groups and various analytic aspects of harmonic analysis. Contributors D. Akhiezer                         T. OshimaA. Andrada                         I. PacharoniM. L. Barberis                    F. RicciL. Barchini                            S. RosenbergI. Dotti                                  N. ShimenoM. Eastwood                     J. TiraoV. Fischer                            S. TreneerT. Kobayashi                       C.T.C. WallA. Korányi                           D. WallaceB. Kostant                           K. WiboontonP. Kostelec                          F. XuK.-H. Neeb                          O. YakimovaG. Olafsson                         R. ZierauB. Ørsted

This bookprovides an overview of the latest developments concerning the moduli of K3surfaces. It is aimed at algebraic geometers, but is also of interest to numbertheorists and theoretical physicists, and continues the tradition of relatedvolumes like ¿The Moduli Space of Curves¿ and ¿Moduli of Abelian Varieties,¿which originated from conferences on the islands Texel and Schiermonnikoog andwhich have become classics.K3 surfacesand their moduli form a central topic in algebraic geometry and arithmeticgeometry, and have recently attracted a lot of attention from bothmathematicians and theoretical physicists. Advances in this field often resultfrom mixing sophisticated techniques from algebraic geometry, lattice theory,number theory, and dynamical systems. The topic has received significantimpetus due to recent breakthroughs on the Tate conjecture, the study ofstability conditions and derived categories, and links with mirror symmetry andstring theory. At the same time, the theory of irreducible holomorphicsymplectic varieties, the higher dimensional analogues of K3 surfaces, hasbecome a mainstream topic in algebraic geometry.Contributors:S. Boissière, A. Cattaneo, I. Dolgachev, V. Gritsenko, B. Hassett, G. Heckman,K. Hulek, S. Katz, A. Klemm, S. Kondo, C. Liedtke, D. Matsushita, M.Nieper-Wisskirchen, G. Oberdieck, K. Oguiso, R. Pandharipande, S. Rieken, A. Sarti, I.Shimada, R. P. Thomas, Y. Tschinkel, A. Verra, C. Voisin.

The contributions in this book explore various contexts in which the derived category of coherent sheaves on a variety determines some of its arithmetic. This setting provides new geometric tools for interpreting elements of the Brauer group. With a view towards future arithmetic applications, the book extends a number of powerful tools for analyzing rational points on elliptic curves, e.g., isogenies among curves, torsion points, modular curves, and the resulting descent techniques, as well as higher-dimensional varieties like K3 surfaces. Inspired by the rapid recent advances in our understanding of K3 surfaces, the book is intended to foster cross-pollination between the fields of complex algebraic geometry and number theory.Contributors:┬╖ Nicolas Addington┬╖ Benjamin Antieau┬╖ Kenneth Ascher ┬╖ Asher Auel┬╖ Fedor Bogomolov┬╖ Jean-Louis Colliot-Th├⌐l├¿ne┬╖ Krishna Dasaratha┬╖ Brendan Hassett┬╖ Colin Ingalls┬╖ Mart├¡ Lahoz┬╖ Emanuele Macr├¼┬╖ Kelly McKinnie┬╖ Andrew Obus┬╖ Ekin Ozman┬╖ Raman Parimala┬╖ Alexander Perry┬╖ Alena Pirutka┬╖ Justin Sawon┬╖ Alexei N. Skorobogatov┬╖ Paolo Stellari┬╖ Sho Tanimoto┬╖ Hugh Thomas┬╖ Yuri Tschinkel┬╖ Anthony V├írilly-Alvarado┬╖ Bianca Viray┬╖ Rong Zhou

Maxim's vision has inspired major developments in many areas of mathematics, ranging all the way from probability theory to motives over finite fields, and has brought forth a paradigm shift at the interface of modern geometry and mathematical physics.

This volume collects chapters that examine representation theory as connected with affine Lie algebras and their quantum analogues, in celebration of the impact Vyjayanthi Chari has had on this area. The opening chapters are based on mini-courses given at the conference ΓÇ£Interactions of Quantum Affine Algebras with Cluster Algebras, Current Algebras and CategorificationΓÇ¥, held on the occasion of ChariΓÇÖs 60th birthday at the Catholic University of America in Washington D.C., June 2018. The chapters that follow present a broad view of the area, featuring surveys, original research, and an overview of Vyjayanthi ChariΓÇÖs significant contributions. Written by distinguished experts in representation theory, a range of topics are covered, including:String diagrams and categorificationQuantum affine algebras and cluster algebrasSteinberg groups for Jordan pairsDynamical quantum determinants and PfaffiansInteractions of Quantum Affine Algebras with Cluster Algebras, Current Algebras and Categorification will be an ideal resource for researchers in the fields of representation theory and mathematical physics.

The Hardy-Littlewood circle method was invented over a century ago to study integer solutions to special Diophantine equations, but it has since proven to be one of the most successful all-purpose tools available to number theorists.

This book is an outgrowth of the conference "Regulators IV: An International Conference on Arithmetic L-functions and Differential Geometric Methods" that was held in Paris in May 2016.