Summer Graduate School

The modular representation theory of finite groups studies how the structure of a group is reflected in its representations over fields of positive characteristic. At the heart of this subject lies the concept of blocks of group algebras, which capture the local behavior of representations. This Summer Graduate School will introduce graduate students to the modern theory of blocks, emphasizing their algebraic, geometric, and combinatorial aspects. Participants will learn how numerical invariants (such as decomposition numbers and Cartan matrices), homological tools (Morita and derived equivalences), categorical enhancements (higher representation theory), and geometric constructions (Deligne–Lusztig varieties) combine to shape our understanding of modular representation theory for finite and finite reductive groups. Classical conjectures such as Donovan’s Finiteness Conjecture and Brou´e’s Abelian Defect Conjecture will serve as guiding examples illustrating the unity of these perspectives.

School Structure

Each day will consist of two teaching blocks, one in the morning and one in the afternoon. Each lecture will be followed by a collaboration session led by the teaching assistants, devoted to exploring the material through problem solving, discussion, or computational work in Julia. This alternation of lectures and sessions will allow participants to consolidate new concepts immediately and to connect algebraic, geometric, and combinatorial viewpoints throughout the school.

In addition, each day will begin with a 20-minute “triage” session, during which participants can ask questions and clarify material from the previous day.

Prerequisites

The school is intended for graduate students with a solid background in algebra, at the level of a first or second year Ph.D. program. A prior exposure to representation theory of finite groups or to basic homological algebra will be helpful but is not required. The material will be introduced from first principles whenever possible, and the two lecture series are designed to be complementary in both style and level.

Lecture Series I (Kessar). — This series will be accessible to participants with a general algebra background, such as courses in ring theory, module theory, and basic homological algebra. The emphasis on blocks, invariants, and equivalences makes it particularly suitable for students interested in algebraic structures and category theory. Concepts such as Morita and derived equivalences will be developed gradually and motivated by explicit examples, so that the course remains approachable to non-specialists.

Lecture Series II (Dudas). — This series will appeal to students with an interest in combinatorics, geometry, or Lie theory. It will introduce finite reductive groups and the combinatorial structures that arise in their representation theory, connecting them to ideas in algebraic combinatorics and geometric representation theory. A large portion of the problem sessions will include computational experiments using the programming language Julia. No prior experience with programming is expected: students will be guided through basic examples illustrating how computational exploration supports theoretical understanding.

Application Procedure

For eligibility and how to apply, see the Summer Graduate Schools homepage.