Arithmetic Geometry and Algebraic Groups (25w5323)

The Banff International Research Station will host the “Arithmetic Geometry and Algebraic Groups” workshop in Banff from September 28 to October 3, 2025.

The workshop responds to two recent trends at the interface of the theory of linear algebraic groups and arithmetic geometry and number theory. First, some old problems have been solved by combining the methods of the theory of algebraic and Lie groups with those coming from number theory. As an example, one can mention the resolution of the problem concerning linear groups having bounded generation, which relied on ideas from Diophantine approximation. Second, while the classical theory of algebraic groups over global fields remains an area of active research, a growing number of results in the last 10-15 years have extended some aspects of the theory to more general fields. Particularly significant success has been achieved in the investigation of local-global principles over the function fields of $p$-adic curves and, more generally, over semi-global fields. Recently, a number of finiteness conjectures have been formulated over arbitrary finitely generated fields. Further progress on these and some other related problems will undoubtedly require new methods based on modern techniques from arithmetic geometry, number theory, and related areas; in particular, the study of the arithmetic properties of $K3$ surfaces and related varieties will likely be instrumental in new developments. It is important to point out that results for algebraic groups defined over fields of arithmetic nature are likely to be useful for adjacent subjects, including discrete subgroups of Lie groups and spectral geometry.

The workshop will bring together well-established and young researchers working on different aspects of the theory of algebraic groups, arithmetic geometry, and number theory, and will emphasize the interactions between these areas. The scientific program of the workshop will consist of 45-minute lectures surveying key topics in the area and presenting recent significant results and 20-minute short communications containing research announcements. We hope that the workshop will help to identify new problems in arithmetic geometry that have
potential applications to algebraic groups and promote the development of thearithmetic theory of algebraic groups over general fields.

The Banff International Research Station for Mathematical Innovation and Discovery (BIRS) is a collaborative Canada-US-Mexico venture that provides an environment for creative interaction as well as the exchange of ideas, knowledge, and methods within the Mathematical Sciences, with related disciplines and with industry. The research station is located at The Banff Centre in Alberta and is supported by Canada’s Natural Science and Engineering Research Council (NSERC), the U.S. National Science Foundation (NSF), and Alberta’s Advanced Education and Technology.