During the 2020-2021 academic year, the School will have a special program on Geometric and Modular Representation Theory. Geordie Williamson of the University of Sydney will be the Distinguished Visiting Professor.
Representation theory began with the work of Frobenius in the late 19th century and soon grew to play an important role in the development of modern mathematics. The second half of the last century saw the introduction of powerful new geometric techniques. Some of the deepest results in representation theory are obtained via geometric means, via the passage to algebraic geometry and the use of D-modules, perverse sheaves and weights. More recently, techniques of higher representation theory have provided new techniques and impetus from algebra and higher category theory.
A focus of this special year will be on modular representations (i.e. representations in positive characteristic). Here experience suggests that simple questions (e.g. understanding simple representations) can be extremely difficult. The subject has been dominated for the last thirty years by conjectures stating that the story should be "the same" as over the complex numbers, where "classical" tools of geometric representation theory provide the answer. However recent results suggest that the story is more complicated, and one is in need of new conjectures. It seems likely that both algebraic and geometric tools will be necessary to make progress. One might hope that a better understanding of pure characteristic p phenomena (e.g. Frobenius twist, Steenrod operations, relatively simple structure of motives...) will become essential to further understanding.
Another focus of this special year will be to achieve a better understanding of derived equivalence. This notion has grown into a unifying principle throughout representation theory: from attempts to categorify counting conjectures in finite group theory, through the representation theory of real Lie groups, to the local geometric Langlands program. A better understanding of these equivalences and their consequences seems guaranteed to lead to further progress.