Bøger i Progress in Mathematics serien

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.

This book presents the most up-to-date and sophisticated account of the theory of Euclidean lattices and sequences of Euclidean lattices, in the framework of Arakelov geometry, where Euclidean lattices are considered as vector bundles over arithmetic curves.

This book provides an overview of the latest progress on rationality questions in algebraic geometry. It discusses new developments such as universal triviality of the Chow group of zero cycles, various aspects of stable birationality, cubic and Fano fourfolds, rationality of moduli spaces and birational invariants of group actions on varieties, contributed by the foremost experts in their fields.The question of whether an algebraic variety can be parametrized by rational functions of as many variables as its dimension has a long history and played an important role in the history of algebraic geometry. Recent developments in algebraic geometry have made this question again a focal point of research and formed the impetus to organize a conference in the series of conferences on the island of Schiermonnikoog. The book follows in the tradition of earlier volumes, which originated from conferences on the islands Texel and Schiermonnikoog.