A standard invariant of a module over a ring is its annihilator, the collection of elements in the ring that multiply as zero on the module. Some geometry can be added by considering the collection of all prime ideals or maximal ideals of the ring that contain the annihilator. Using group cohomology, similar structures can be made to give rough classifications of modular representations of finite groups. Going further, it leads to a classifiction of some distinctve structures in the category of modules over the finite group. We will present an overview of this subject

Group cohomology is a powerful tool in group representation theory.

To a group action on a vector space, one associates a geometric object called its support variety that is defined using group cohomology. Hopf algebras generalize groups and include many important classes of algebras such as Lie algebras and quantum groups. The theory of varieties for modules generalizes to Hopf algebras to some extent, but there are many open questions.

In this introductory talk, we will define Hopf algebras, their cohomology, and the corresponding varieties for modules. We will discuss known and unknown properties and recent and current research on open problems

Let $V$ be a finite-dimensional (complex) vector space, and $Sym(V)$ be the symmetric algebra on this vector space. We can consider the multiplication map $Sym(V) \otimes V \to V$ as a complex of $GL(V)$-representations of length $2$. I this talk, I will describe how tensor powers of the above complex define interesting complexes of representations of the symmetric group $S_n$, which were studied by Deligne in the paper "La Categorie des Representations du Groupe Symetrique $S_t$, lorsque $t$ n’est pas un Entier Naturel". I will then explain how computing the cohomology of these complexes helps establish a relation between the Deligne categories and the representations of $S_{\infty}$, which are two natural settings for studying stabilization in the theory of finite-dimensional representations of the symmetric groups. This is joint work with D. Barter and Th. Heidersdorf

Morita Frobenius numbers were introduced by Radha Kessar in 2004 in the context of Donovan’s Conjecture in block theory. The Morita Frobenius number of an algebra is the number of Morita equivalence classes of its Frobenius twists. I will present results from joint work with Radha Kessar where we aim to bound the Morita Frobenius numbers of blocks of quasi-simple finite groups. I will explain the connection to Donovan's Conjecture, and illustrate how recent results of Bonnafé-Dat-Rouquier play a crucial role in our methods.

Representation theory of Lie superalgebras  was originally motivated by applications in physics and topology. In recent years duality and categorification unraveled new connections of superalgebras with other branches of representation theory. 

This lecture will be an introduction to the subject with emphasis  on geometric methods and applications to tensor categories. I will also formulate some open problems.