The UC Berkeley Representation theory and tensor categories seminar 

DATE  SPEAKER  TITLE (click to show abstract) 
September 3  Ilia Nekrasov , UC Berkeley 
Oligomorphic Tensor Categories IAbstract: Let me introduce you to a new kind of tensor categories in this crashcourse (2 talks at the seminar). These categories provide a plethora of new unexpected examples and connections to other fields.The process of cooking up such a category consists of three steps: 1. Take an oligomorphic group \(G\) and construct a category GhatSets 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 GhatSets 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! 
September 10  Ilia Nekrasov, UC Berkeley 
Oligomorphic Tensor Categories IIAbstract: Let me introduce you to a new kind of tensor categories in this crashcourse (2 talks at the seminar). These categories provide a plethora of new unexpected examples and connections to other fields.The process of cooking up such a category consists of three steps: 1. Take an oligomorphic group \(G\) and construct a category GhatSets 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 GhatSets 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! 
September 11 (Wednesday) 3PM4:30PM.  Anton Alekseev , University of Geneva 
BV structures on moduli of flat connections and the Lie supergroup Q(N)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 2dimensional oriented manifold. It is a classical result of AtiyahBott 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 BatalinVilkovisky (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. 
September 17  Alexandra Utiralova, UC Berkeley 
Representations of the general linear group in the Verlinde category.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. 
September 24  Catharina Stroppel, University of Bonn 
Semistrict monoidal 2categories from chain complexes of Soergel bimodules, braidings and dgcentralizersAbstract: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 2categories. They should however arise as categorifications of braided monoidal categories. The goal of this talk is to give an idea what a monoidal 2category is with the main example given by Soergel bimodules. The motivation of this construction comes from (quantum) SchurWeyl 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 2categories following KapranovVoevodsky, BaezNeuchl, 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 2categories and indicate roughly how to construct a braided monoidal 2category using Soergel bimodules. 
October 1 


October 8  Cris Negron, University of Southern California 
Quantum Frobenius kernels at arbitrary roots of 1Abstract: I will discuss constructions of quantum Frobenius kernels at (completely) arbitrary roots of 1. As one output, we associate a finitedimensional (nonsemisimple) modular tensor category to any pairing of a simplyconnected reductive algebraic group with an even order root of 1. I will explain the field theoretic motivations for this work, where quantum groups at "bad" roots of unity appear naturally. 
October 15 


October 22  Theo JohnsonFreyd , Dalhousie University 
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October 29 


November 5 
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November 12 
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November 19 
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November 26 
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December 3 
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