Attend a comprehensive seminar exploring advanced topics in geometric and modular representation theory through expert lectures from leading mathematicians. Delve into Broué's Abelian Defect Group Conjecture with detailed presentations by Jay Taylor and Daniel Juteau, examining both theoretical foundations and applications. Explore the intricate connections between finite groups and algebraic groups in both defining and non-defining characteristics through Raphaël Rouquier's analysis. Master the fundamentals of affine Grassmannians and the geometric Satake equivalence with Jize Yu's introduction, followed by Anne Dranowski's examination of derived equivariant cohomology and Bezrukavnikov's contributions. Investigate the derived geometric Satake equivalence, Iwahori-Whittaker categories, and geometric Casselman-Shalika theory through presentations by Yu and Tony Feng. Gain historical perspective on geometric Satake equivalence from George Lusztig's survey while exploring Lefschetz operators, Hodge-Riemann forms, and their representation-theoretic implications with Peter Fiebig. Study Hecke categories through Dima Arinkin's derived convolution formalism and discover Smith theory applications in representation theory. Examine cutting-edge research including New Age Linkage theory, Hecke category actions on principal blocks, K-rings of Steinberg varieties, and equivariantization techniques. Explore Gaitsgory's central sheaves, monoidal colimits in affine Hecke categories, and geometric realizations of anti-spherical modules and affine Hecke algebras. Conclude with advanced topics including noncommutative Springer resolutions, parabolic versions of realization theorems, microlocal sheaves on affine Springer fibers, modular equivalences, and Frobenius exact symmetric tensor categories, providing a comprehensive overview of current research directions in this rapidly evolving field.

Syllabus

Broué’s Abelian Defect Group Conjecture I - Jay Taylor
Broué’s Abelian Defect Group Conjecture II - Daniel Juteau
Broué’s Abelian Defect Group Conjecture II - Daniel Juteau
Finite groups as algebraic groups in defining characteristic - Raphaël Rouquier
Finite groups as algebraic groups in non-defining characteristic - Raphaël Rouquier
An introduction to affine Grassmanians and the geometric Satake equivalence - Jize Yu
Derived Equivariant Cohomology of the affine Grassmannian and Bezrukavnikov…- Anne Dranowski
The Derived Geometric Satake Equivalence of Bezrukavnikov and Finkelberg - Jize Yu
Iwahori-Whittaker category and geometric Casselman-Shalika - Tony Feng
Geometric Satake equivalence: a historical survey - George Lusztig
Lefschetz operators, Hodge-Riemann forms, and representations - Peter Fiebig
Hecke category via derived convolution formalism - Dima Arinkin
Introduction to Smith theory - Tony Feng
New Age Linkage - Daniel Juteau
The Hecke category action on the principal block via Smith theory - Geordie Williamson
The K-ring of Steinberg varieties - Pablo Boixeda Alvarez
Equivariantization and de-equivariantization - Shotaro Makisumi
Gaitsgory's central sheaves - Tom Braden
The affine Hecke category is a monoidal colimit - James Tao
On two geometric realizations of the anti-spherical module - Tsao-Hsien Chen
Two Geometric Realizations of the Affine Hecke Algebra IPablo Boixeda Alvarez
Affine Hecke category and noncommutative Springer resolution - Roman Bezrukavnikov
Parabolic version of the two realizations theorem and applications to modular... - Ivan Loseu
Microlocal sheaves on certain affine Springer fibers - Zhiwei Yun
Towards a modular "2 realizations" equivalence - Simon Riche
Frobenius exact symmetric tensor categories - Pavel Etingof