Zariski Dense Subgroups, Number Theory and Geometric Applications

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.

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, Journal of the AMS and Inventiones Mathematicae, explores arithmetic properties of linear algebraic groups over general fields. His book Algebraic Groups and Number Theory has been a key reference for 30+ years. He advised 12 PhD students, spoke at ICM 2014, and received a Humboldt Award (2004) and Simons Fellowship (2017).

Balasubramanian Sury is Full Professor at the Indian Statistical Institute, Bangalore, with research interests spanning algebraic groups, arithmetic subgroups, division algebras and number theory—including Diophantine equations, elliptic curves and combinatorial number theory. At TIFR Mumbai (1981–1999), he focused on the congruence subgroup problem. Since 1999, at ISI Bangalore, he has pursued new directions in group and number theory, notably determining the structure of the congruence kernel in rank one over global fields of positive characteristic. Author of over 75 research papers and 3 books, he has advanced bounded generation in arithmetic groups and addressed classical problems in number theory. His book The Congruence Subgroup Problem: An Elementary Approach Aimed at Applications remains the only dedicated volume on the subject. He served as President of the Indian Mathematical Society (2020–2021), National Olympiad Coordinator (2016–2021), and is Chief Editor of Resonance.

Aleksy Tralle is Full Professor at the University of Warmia and Mazury, Olsztyn, Poland. His research spans differential geometry, Lie groups and their discrete subgroups. His book Symplectic Manifolds with no Kähler Structure (Springer, 1997) is a widely cited reference in symplectic topology. In collaboration with Y. Rudyak and V. Muñoz, he made key contributions to the topology of symplectic, Kähler and Sasakian manifolds, including solving the formality problem for Sasakian manifolds. His current work focuses on proper actions of discrete subgroups of Lie groups on homogeneous spaces and pseudo-Riemannian geometry. With over 80 research papers, he supervised 6 PhD dissertations. He held long-term visiting positions at Max Planck Institute (Bonn) and IHES (France). He organized numerous conferences, including 7 at the Banach Center, and serves on the CAST Programme Steering Committee. He is Associate Editor of the Journal of Fixed Point Theory and Applications and a recipient of awards from the Polish Academy of Sciences and the Ministry of Higher Education and Science of Poland.