The UC Berkeley Representation theory and tensor categories seminar |
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DATE |
SPEAKER |
TITLE (click to show abstract) |

January 30th | Nicolai Reshetikhin, UC Berkeley |
## Asymptotic distribution of tilting modules in large tensor products.Abstract: This talk is focused on the following problem: find the distribution of tilting (and irreducible) components in \( V^{\otimes N} \) where \( V \) is a finite dimensional representation, when \( N\rightarrow \infty \). This is a typical question in the area that is known as "asymptotic representation theory". The talk will be focused on three examples: \( \mathfrak{sl}_2 \)-modules; modules over quantum \( \mathfrak{sl}_2 \) at roots of unity with divided powers, so called "big quantum \( \mathfrak{sl}_2 \)"; and modules over small quantum \( \mathfrak{sl}_{2} \), a finite dimensional Hopf algebra which is a subalgebra of big quantum \( \mathfrak{sl}_2\). The talk is based on a joint work with A. Lachowska, O. Postnova and D. Soloviev. |

February 6th | Iryna Kashuba, University of São Paulo |
## On free Jordan algebrasAbstract: We study a structure of homogeneous components of the free Jordan algebra \(J(D)\) in \(D\) generators over a field of characteristic zero. It is done by employing the prominent Tits–Kantor–Koecher construction which associates to a Jordan algebra a Lie algebra acted on by \( \mathfrak{sl}_2\) by means of derivations. We conjecture that the homology groups \(H_k, k\geq 0 \), of the TKK\((J(D))\) are trivial, which allows us to describe the character of \(J(D)\) as \(GL(D)\)-module. We will discuss several equivalent versions of the conjecture, numerical evidence which support it and how far have we advance in proving it. |

February 13th | Nicolai Reshetikhin, UC Berkeley |
## Continuation of the previous talkAbstract: This talk is focused on the following problem: find the distribution of tilting (and irreducible) components in \( V^{\otimes N} \) where \( V \) is a finite dimensional representation, when \( N\rightarrow \infty \). This is a typical question in the area that is known as "asymptotic representation theory". The talk will be focused on three examples: \( \mathfrak{sl}_2 \)-modules; modules over quantum \( \mathfrak{sl}_2 \) at roots of unity with divided powers, so called "big quantum \( \mathfrak{sl}_2 \)"; and modules over small quantum \( \mathfrak{sl}_{2} \), a finite dimensional Hopf algebra which is a subalgebra of big quantum \( \mathfrak{sl}_2\). The talk is based on a joint work with A. Lachowska, O. Postnova and D. Soloviev. |

February 20th | No seminar | |

February 27th | Vadim Gorin, UC Berkeley |
## Quantized asymptotic representation theory of classical Lie groups.Abstract: An example of an object of interest in the asymptotic representation theory is the infinite-dimensional group U(\infty), which is defined as the union of finite-dimensional unitary groups naturally embedded into each other. The focus is on two tasks: classification of irreducible representations (or characters) and harmonic analysis (=decomposition of natural representations into irreducible components). In the recent years a rich quantized or q-deformed version of the theory was developed in the combinatorial language of branching graphs and central measures. One could expect that this should correspond to representations of quantum groups of infinite rank. However, despite several attempts, a fully satisfactory translation of the theory into that language was not achieved so far. I will review the key developments and open questions. |

March 6th | Nicolle Gonzalez, UC Berkeley |
## A DAHA Shuffle Theorem and Higher Rank (q,t)-Catalan PolynomialsAbstract: The rational shuffle theorem states that the Frobenius characters of certain representations \(L_{m/n}\) of the rational Cherednik algebra arise under a geometric action of the elliptic Hall algebra on the ring of symmetric functions and are expressible as combinatorial sums over (m,n)-parking functions. In this talk I will describe an analogue of this theorem in the context of the double affine Hecke algebra (DAHA). Namely, we'll discuss how certain actions of the spherical DAHA on symmetric polynomials gives rise to truncations of the characters of \(L_{m/n}\), which we prove are given by new objects called the higher rank (q,t)-Catalan polynomials. These polynomials, which are defined as certain sums over semistandard parking functions, provide a multiparametric generalization to the usual (q,t)-Catalan numbers and interpolate between Dyck paths and (m,n)-parking functions. This is joint work with Jose Simental and Monica Vazirani. |

March 13th | Paul Wedrich, University of Hamburg |
## A Kirby color for Khovanov homologyAbstract: The Jones polynomial of a knot can be computed relatively straightforwardly using the Temperley-Lieb algebras, which admit a diagrammatic presentation. Surprisingly, closely related diagrammatic algebras, the dotted Temperley-Lieb algebras (a.k.a. nil-blob algebras), appear when extending Khovanov homology to an invariant of smooth 4-dimensional manifolds. I will define these algebras, assemble them into a monoidal category, and explain how their representation theory encodes colored Khovanov homology. Finally, I will introduce a special ind-object, the eponymous Kirby color for Khovanov homology discovered in joint work with Hogancamp and Rose, which allows a concrete description of a certain 2-handle gluing rule for smooth 4-manifold invariants constructed in joint work with Morrison and Walker. |

March 20th | Emily Bain, UC Berkeley |
## On the mesoscopic limit for dimer modelsAbstract: A dimer model is a probability distribution on the set of perfect matchings on a planar graph. Often, we want to study the correlations between dimers. It has been shown that there are three distinct phases (solid, liquid and gas) found in dimer models, characterized by the rate of decay of the two-point correlation functions. A dimer model often exhibits more than one of these phases, with distinct boundaries between phases, where some interesting behavior can be found. In this talk, we focus on the two-periodic weighted Aztec diamond, which is one of the simplest models to exhibit a gas phase. We will talk about what happens as we shrink the gas region to a point in a specific limit that we call the "mesoscopic limit". |

March 27th | Spring break, no seminar | |

April 3rd | ||

April 10th | Natasha Rozhkovskaya, Kansas State University |
## On central elements of universal enveloping algebras of Lie superalgebrasAbstract: Centers of universals enveloping algebras of Lie(super)algebras are studied through applications of the Schur-Weyl-like dualities, construction of Capelli elements and their eigenvalues. In this talk we will review a construction of distinguished trace-like central elements of universal enveloping algebras of a Lie superalgebra gl(m|n) or q(n) using a superanalogue of special symmetrization, which is a vector space isomorphism of a symmetric and universal enveloping algebras. Special symmetrization was introduced in early nineties by G.Olshanski for the study of classical infinite-dimensional Lie groups and corresponding analogues of universal enveloping algebras. Later special symmetrization along with its generalizations found a number of remarkable applications in representation theory. |

April 17th | Nate Harman, University of Michigan |
## Pre-Tannakian Categories and Oligomorphic GroupsAbstract: The notion of a pre-Tannakian category is an axiomatization of the additional structure that the category of finite dimensional representations of a group (or supergroup) has -- and until recently we only had a handful of examples of pre-Tannakian categories in characteristic zero which were not just categories of representations. I will discuss a construction of pre-Tannakian categories using oligomorphic groups, a class of groups arising in model theory, and then explain a new result that says that these oligomorphic groups necessarily arise when studying pre-Tannakian categories, and were not just an quirk of our approach. |

April 24th | Tom Gannon, UCLA |
## A Proof of the Ginzburg-Kazhdan ConjectureAbstract: The main theorem of this talk will be that the affine closure of the cotangent bundle of the basic affine space (also known as the universal hyperkähler implosion) has symplectic singularities for any reductive group, where essentially all of these terms will be defined in the course of the talk. First, we'll discuss the universal hyperkähler implosion and review some basic examples. Afterwards, we'll define and motivate the notion of symplectic singularities. We will then survey some of the basic facts that are known about the universal hyperkähler implosion and discuss how they are used to prove the main theorem. Time permitting, we'll also discuss how the main theorem relates to the symplectic duality program. |

May 1st | Matt Hogancamp, Northeastern University |
## The nilpotent cone for \( \mathfrak{sl}_2\) and annular link homologyIn this talk I will discuss an equivalence of categories relating SL(2)-equivariant vector bundles on the nilpotent cone for \(\mathfrak{sl}_2\) and the annular Bar-Natan category (this latter category appears in the context of Khovanov homology for links in a thickened annulus). Indeed, both categories admit a diagrammatic description in terms of the same "dotted" Temperley-Lieb diagrammatics, as I will explain. Under this equivalence, Bezrukavnikov's quasi-exceptional collection on the nilcone (in the SL(2) case) has an elegant description in terms of some special annular links. In recent joint work with Dave Rose and Paul Wedrich, we constructed a very special Ind-object in the annular Bar-Natan category which is a categorical analogue of a "Kirby element" from quantum topology; I will conclude by sketching a neat "BGG resolution" afforded by our categorified Kirby element. This is based on joint work with Rose and Wedrich. |