The process of cooking up such a category consists of three steps:

1. Take an oligomorphic group \(G\) and construct a category G-hat-Sets of sets with infinitesimal \(G\)-action;

2. Construct a measure \(\mu\) on this category (each such set is given a measure, a number);

3. Construct a tensor category \(Perm(G, \mu)\).

In the first lecture, I will explain how the topology of oligomorphic groups produces G-hat-Sets and how to get \(Perm(G, \mu)\) (focus is on 1 and 3). The second lecture* will be about measures (focus is on 2).

The story is close to many areas: rep. theory, category theory, model theory, topological groups, \(C^*\)-algebras, and combinatorics. Everyone is warmly welcomed!

*The open problems of the second lecture are accessible to grad.students. Happy to chat after!

The process of cooking up such a category consists of three steps:

1. Take an oligomorphic group \(G\) and construct a category G-hat-Sets of sets with infinitesimal \(G\)-action;

2. Construct a measure \(\mu\) on this category (each such set is given a measure, a number);

3. Construct a tensor category \(Perm(G, \mu)\).

In the first lecture, I will explain how the topology of oligomorphic groups produces G-hat-Sets and how to get \(Perm(G, \mu)\) (focus is on 1 and 3). The second lecture* will be about measures (focus is on 2).

The story is close to many areas: rep. theory, category theory, model theory, topological groups, \(C^*\)-algebras, and combinatorics. Everyone is warmly welcomed!

*The open problems of the second lecture are accessible to grad.students. Happy to chat after!

Abstract: Let \(G\) be a Lie group with Lie algebra \(Lie(G)\) carrying an invariant scalar product (e.g. \(G\) a semisimple Lie group with the Killing form on \(Lie(G)\)), and \(S\) be a compact 2-dimensional oriented manifold. It is a classical result of Atiyah-Bott that the moduli space of flat connections \(M(S, G)\) carries a canonical Poisson structure. Goldman defined a Lie bracket on the space \(F(S)\) of free homotopy classes of loops on \(S\), and a Lie algebra homomorphisms \(p_N\) from \(F(S)\) to functions on \(M(S, GL(N))\). The maps \(p_N\) are surjective for all \(N\), and the intersection of their kernels is trivial.

In this talk, we will discuss moduli spaces \(M(S, G)\) for \(G\) being a Lie supergroup with an odd invariant scalar product on the Lie superalgebra \(Lie(G)\). In this case, we show that \(M(S,G)\) carries a canonical Batalin-Vilkovisky (BV) structure. Furthermore, for the Lie supergroup \(Q(N)\) we construct a counterpart \(q_N\) of the Goldman map using the odd trace on \(Q(N)\). The maps \(q_N\) are BV maps, and we show that \(q_1\) is surjective (if one adds odd determinant functions). The surjectivity question for \(N>1\) and the injectivity question remain open.

The talk is based on a joint work with F. Naef. J. Pulmann, and P. Severa.

Abstract: Verlinde categories \(Ver_p\) are defined as the semisimplification of the category of representations of \(\mathbb Z/p\mathbb Z\) in characteristic \(p\). As shown by Coulembier, Etingof and Ostrik in arXiv:2107.02372, these categories play the role of the target category for the fiber functor for a large class of symmetric tensor categories (Frobenius exact, of moderate growth) in char \(p\) (usually played by the category of super vector spaces in char \(0\)).

Consequently, Tannakian reconstruction tells us that any category with a fiber functor to \(Ver_p\) (and hence any Frobenius exact category of moderate growth) is equivalent to the category of representations of some group scheme in \(Ver_p\).

I will talk about the classification of representations of the general linear group \(GL(X)\) for \(X\in Ver_p\) and the related combinatorics.

Braided monoidal categories are in many ways crucial player in representation theory. Whereas they are quite easy to understand and show up everywhere, this is not the case for braided monoidal 2-categories. They should however arise as categorifications of braided monoidal categories. The goal of this talk is to give an idea what a monoidal 2-category is with the main example given by Soergel bimodules. The motivation of this construction comes from (quantum) Schur-Weyl duality.

We will address the problem of strictness and illustrate why it is difficult to find concrete examples by showing what kind of structure has to be provided. We will shortly discuss the notion of semistrict monoidal 2-categories following Kapranov-Voevodsky, Baez-Neuchl, Crans and address the question why Soergel bimodules do not provide an example. But will also discuss a possible fix. Finally we address the difficulty of braidings on 2-categories and indicate roughly how to construct a braided monoidal 2-category using Soergel bimodules.

Abstract: There is a way to associate a topological space (Balmer spectrum) to a tensor triangulated category. This notion is similar to the spectrum of prime ideals in algebraic geometry. I will start with the definition of the Balmer spectrum and discuss its connection with cohomological support variety.

Then we consider two examples: stable categories of representations of finite groups in positive characteristic and quasireductive algebraic supergroups over C.

The second part of the talk is based on a joint work with J. Pevtsova and A. Sherman.

A toric variety is an (irreducible, normal) algebraic variety \(X\) equipped with an action of an algebraic torus \(T\) and a \(T\)-equivariant open immersion \(T \to X\). The category of toric varieties is equivalent to that of polyhedral fans, inducing a rich interplay between algebraic geometry and combinatorics.

Borrowing the notion of quasitoral supergroups ("supertori") from super representation theory, I will describe a natural generalization of toric varieties wherein \(T\) is replaced by a supertorus. These "toric supervarieties" admit an analogous combinatorial description via decorated fans. I will illustrate this equivalence of categories and show how geometric information can be obtained from the fans and their decorations. Many pictures will be drawn, some worse than others.