Motives and Invariants: Theory and Applications to Algebraic Groups and their Torsors (23w5146)

The Banff International Research Station will host the "Motives and Invariants: Theory and Applications to Algebraic Groups and their Torsors" workshop in Banff from October 8 to October 13, 2023.

Algebraic groups, both classical and exceptional, play a central role in many parts of modern algebra and number theory. For instance, the celebrated Langlands program, a vast web of unifying conjectures driving much of contemporary number theory, concerns the investigation of the complex representations of reductive algebraic groups over local and global fields. Over general fields, it was discovered some 60 years ago by Andre Weil that reductive algebraic groups of classical type can be described concretely in terms of algebras with involution, solidifying long-known connections to quadratic forms and division algebras. Today, these objects and their associated algebraic groups are investigated using an array of methods that draw from many areas of mathematics, ranging from the classical theory of Galois cohomology to recent developments in abstract algebraic geometry related to the theory of motives and motivic homotopy theory. In turn, some of the central problems in the study of quadratic forms and related algebraic structures have also directly inspired major developments in these areas, the most striking example being given by Voevodsky's spectacular proof of the Milnor Conjecture in the mid-1990s.

In this workshop, we want to bring together researchers from both sides of this development: those mathematicians specializing in the application of modern algebraic and algebro-geometric methods to the study of algebraic groups and their torsors, as well as those working on relevant aspects of the motivic and cohomological approaches from a more abstract viewpoint. We hope that bringing this mix of researchers together will have a synergistic impact, leading to concrete new applications of the abstract machinery to outstanding problems on quadratic forms, algebras with involution and algebraic groups, but also inspiring progress on the general development of the motivic and cohomological methods.

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), Alberta's Advanced Education and Technology, and Mexico's Consejo Nacional de Ciencia y Tecnología (CONACYT).