This book presents selected chapters from the program Zariski Dense Subgroups, Number Theory, and Geometric Applications, held at the International Center for Theoretical Sciences (ICTS) in Bengaluru, Karnataka, India, from January 1 12, 2024. The program encompassed a rich array of topics centered around Zariski-dense subgroups, with connections to algebraic and Lie groups, geometry, and number theory. It highlights the application of Diophantine approximation techniques to questions on linear groups with bounded generation, as well as innovative developments in the Bruhat Tits theory for algebraic groups over local fields. These ideas were explored through four mini-courses alongside numerous research and expository lectures.

Chapters are published in two volumes: Volume 1 features expanded notes from four mini-courses and two expository talks, while Volume 2 comprises twelve original research articles. Collectively, the volumes make recent advances in the theory of Zariski-dense subgroups accessible to a broad mathematical audience. The topic has continued to draw significant interest, building on discussions from earlier meetings such as the MSRI workshop in Berkeley (2012) and the IPAM workshop at UCLA (2015). Over the past two decades, Zariski-dense subgroups of algebraic groups have become a focal point of intense research, yielding a wealth of results with far-reaching applications. Notably, this line of inquiry has contributed to the construction of expander graphs and the study of spectral gaps, developments that culminated in the theory of superstrong approximation.

Autorentext

Gopal Prasad is Raoul Bott Professor Emeritus at the University of Michigan, USA. He made contributions to Lie groups, algebraic groups and p-adic representation theory. His volume formula led to an unexpected solution for determining all fake projective planes in algebraic geometry. He published extensively in top-tier journals like Annals of Mathematics, Inventiones Mathematicae and Publications Mathématiques de l IHES. His book Pseudo-reductive Groups completed the structure theory of algebraic groups over arbitrary fields, while Bruhat-Tits Theory: A New Approach made algebraic groups over local fields more accessible, especially in representation theory. He was Managing Editor of the Michigan Mathematical Journal (1998 2011) and Associate Editor of Annals of Mathematics (2004 2010). He spoke at ICM 1990 and received a Guggenheim Fellowship (1998) and a Humboldt Award (2006). He is a fellow of INSA, IAS, and AMS.

Andrei Rapinchuk is McConnell Bernard Professor of Mathematics at the University of Virginia, USA. His work in the arithmetic theory of algebraic groups includes results on the normal subgroup structure of rational points, the congruence subgroup and metaplectic problems. In collaboration with Gopal Prasad, he used arithmetic group theory to study isospectral and length-commensurable locally symmetric spaces. Recently, he helped resolve a long-standing problem on linear groups with bounded generation. His research, published in over 80 papers in top journals like Publications Mathématiques de l IHES, **

Inhalt

Convolution and Square in Abelian Groups III.- Some Recent Results on the Punctual Quot Schemes.- Properly Discontinuous Actions of Discrete Subgroups of Lie Groups: Lie Theory and Computational Methods.- Word Maps and Random Words.- Things We Can Learn by Considering Random Locally Symmetric Manifolds.- Affne Anosov Representations.- Congruent Elliptic Curves over Some p-adic Lie Extensions.- Rost Injectivity and Local-Global Principle for Classical Groups over Function Fields of Arithmetic Surfaces.- Relative Weyl Character Formula, Relative Pieri Formulas and Branching Rules for Classical Groups.- Totally Ramified Subfields of p-algebras over Discrete Valued Fields with Imperfect Residue.- Residual Finiteness and Discrete Subgroups of Lie Groups.- Characterization of Norm and Quasi-norm Forms in S-adic Setting.