Bøger i Mathematical Surveys and Monographs serien

The classification of finite simple groups is a landmark result of modern mathematics. This work presents critical aspects of the classification. It begins with the proof of a major theorem from the classification grid, namely Theorem $\mathcal{C}_7$. It is suitable for graduate students and researchers interested in group theory.

The Yangians and twisted Yangians are remarkable associative algebras taking their origins from the work of St Petersburg's school of mathematical physics in the 1980s. The book gives an introduction to the theory of Yangians and twisted Yangians, with a particular emphasis on the relationship with the classical matrix Lie algebras.

The Fourier coefficients of modular forms are of widespread interest as an important source of arithmetic information. This book covers the Eichler-Selberg/Hijikata trace formula for the traces of Hecke operators on spaces of holomorphic cusp forms of weight $\mathtt{k}>2$ for congruence subgroups of $\operatorname{SL}_2(\mathbf{Z})$.

Develops an arithmetic analog of the theory of ordinary differential equations, where functions are replaced by integer numbers, the derivative operator is replaced by a 'Fermat quotient operator', and differential equations (viewed as functions on jet spaces) are replaced by 'arithmetic differential equations'.

Self-similar groups (groups generated by automata) initially appeared as examples of groups that are easy to define but have exotic properties like nontrivial torsion, intermediate growth, etc. This book studies the self-similarity phenomenon in group theory and shows its relationship with dynamical systems.

Treats free probability theory, which has been extensively developed since the early 1980s. This work focuses on entropy and the random matrix model approach. It is suitable as a reference for mathematicians interested in free probability theory and can serve as a text for an advanced graduate course.

Presents an introduction to the theory of vertex algebras with a particular emphasis on the relationship with the geometry of algebraic curves. This book contains several topics, in particular an introduction to the Beilinson-Drinfeld theory of factorization algebras and the geometric Langlands correspondence.

Presents a study of polynomial identities by combining methods of ring theory, combinatorics, and representation theory of groups with analysis. This book includes such topics as polynomial rings in one or several variables, the Grassmann algebra, and finite-dimensional algebras. It is suitable for students of polynomial identity algebras.

The questions that have been at the center of invariant theory since the 19th century have revolved around the following themes: finiteness, computation, and special classes of invariants. This book begins with a survey of many concrete examples chosen from these themes in the algebraic, homological and combinatorial context.

Focuses on differential calculus in infinite dimensions and those applications in infinite dimensional differential geometry and global analysis not involving Sobolev completions and fixed point theory. This work discusses the existence of smooth partitions of unity, the foundations of manifold theory in infinite dimensions and differential forms.

Presents a systematic account of the developments in the theory of symmetric manifolds achieved over the past 50 years. The book contains detailed and friendly, but rigorous, proofs of the key results in the theory.

Examines the theory of algebraic geometry codes, a subject related to several domains of mathematics. This book systematically presents meaningful and important connections of coding theory with algebraic geometry and number theory.

Provides a thorough exposition of the main concepts and results related to various types of convergence of measures arising in measure theory, probability theory, functional analysis, partial differential equations, mathematical physics, and other theoretical and applied fields. Particular attention is given to weak convergence of measures.

Concentrates on first boundary-value problems for fully nonlinear second-order uniformly elliptic and parabolic equations with discontinuous coefficients. The authors look for solutions in Sobolev classes, or for viscosity solutions. Most of the auxiliary results are taken from old sources, and the main results were obtained in the last few years.

The theory of bounded cohomology, introduced by Gromov in the late 1980s, has had powerful applications in geometric group theory and the geometry and topology of manifolds, and has been the topic of active research continuing to this day. This monograph provides a unified, self-contained introduction to the theory and its applications.

Since the 1980s, Classification of Finite Simple Groups (CFSG) has had a huge influence on work in finite group theory and in many adjacent fields of mathematics. This book attempts to survey and sample a number of such topics from the very large and increasingly active research area of applications of CFSG.

Contains topics in Hamilton's Ricci flow. This book presents the theory of Ricci solitons, Kahler-Ricci flow, compactness theorems, Perelman's entropy monotonicity and no local collapsing, Perelman's reduced distance function and applications to ancient solutions, and a primer of 3-manifold topology.

Offers a modern, up-to-date introduction to quasiconformal mappings from an explicitly geometric perspective, emphasizing both the extensive developments in mapping theory during the past few decades and the remarkable applications of geometric function theory to other fields, including dynamical systems, Kleinian groups, geometric topology, differential geometry, and geometric group theory.

Ricci flow is a powerful technique using a heat-type equation to deform Riemannian metrics on manifolds to better metrics in the search for geometric decompositions. With the fourth part of their volume on techniques and applications of the theory, the authors of this volume discuss long-time solutions of the Ricci flow and related topics.

The ultimate goal of this book is to explain that the Grothendieck-Teichmuller group, as defined by Drinfeld in quantum group theory, has a topological interpretation as a group of homotopy automorphisms associated to the little 2-disc operad. To establish this result, the applications of methods of algebraic topology to operads must be developed.

Presents three introductory tutorials coming out of three courses given at the CIMPA Research School "Galois Theory of Difference Equations" in 2012. The aim of these tutorials is to introduce the reader to three Galois theories of linear difference equations and their interrelations. Each of the three articles addresses a different galoisian aspect of linear difference equations.

The Dynamical Mordell-Lang Conjecture is an analogue of the classical Mordell-Lang conjecture in the context of arithmetic dynamics. This volume presents all known results of the Dynamical Mordell-Lang Conjecture, focusing mainly on a $p$-adic approach which provides a parametrization of the orbit of a point under an endomorphism of a variety.

Examines toric topology, a new area of mathematics that emerged at the end of the 1990s on the border of equivariant topology, algebraic and symplectic geometry, combinatorics, and commutative algebra. It has quickly grown into a very active area with many links to other areas of mathematics, and continues to attract experts from different fields.

The central theme of this book is the study of rational points on algebraic varieties of Fano and intermediate type - both in terms of when such points exist and, if they do, their quantitative density. The book presents the state of the art in computational arithmetic geometry for higher-dimensional algebraic varieties.

Provides a systematic exposition of the modern theory of Gaussian measures. It presents complete and detailed proofs fundamental facts about finite and infinite dimensional Gaussian distributions. Covered topics include linear properties, convexity, linear and nonlinear transformations, and applications to Gaussian and diffusion processes.