Abstract: The standard approach to representation theory of Lie type involves the combinatorics of the reflection representation of the underlying Weyl group, leading ultimately to Kazhdan-Lusztig polynomials and Soergel bimodules. In this minicourse I will talk about a non-standard approach within each of the families of classical groups (general linear, orthogonal, symplectic, ...). Perhaps it is less satisfactory because it has a somewhat different flavor depending on the particular family of classical group, but it is just as rich and has an equally long and distinguished history. The combinatorics that arises is that of integrable representations of certain affine Kac-Moody Lie algebras, connecting to crystal/canonical bases and 2-quantum groups. I will mainly focus on the GL family which is fully developed, with applications to representation theory of symmetric and general linear groups. There has been recent progress in the OSp, P and Q families, although lots of interesting questions remain to be investigated.

Abstract: Two-dimensional conformal field theory is a powerful tool to understand the geometry of surfaces.  Liouville conformal field theory in the classical (large central charge) limit encodes the geometry of the moduli space of Riemann surfaces. I describe an efficient algorithm to compute the Weil--Petersson metric to arbitrary accuracy using Zamolodchikov's recursion relation for conformal blocks, focusing on examples of a sphere with four punctures and generalizations to other one-complex-dimensional moduli spaces. Comparison with analytic results for volumes and geodesic lengths finds excellent agreement. In the case of M_{0,4}, I discuss numerical results for eigenvalues of the Weil-Petersson Laplacian and connections with random matrix theory. Based on work with K. Coleville, A. Maloney, K. Namjou, and T. Numasawa.

Abstract: Nakajima quiver varieties are moduli spaces of representations of quivers. Examples include cotangent bundles of flag varieties, Hilbert schemes of points, and du Val singularities. Realizing a space as a quiver variety gives: (I) a restriction on its types of singularities, (II) a stratification (by symplectic leaves) and (III) a GIT description amenable to variation of stability parameter. 

Bellamy, Craw, and Schedler used these features to classify all projective symplectic resolutions of singularities for certain quiver varieties. In joint work with Schedler, we leverage this to build resolutions for spaces that are (analytically) locally quiver varieties. The main result is a local-to-global procedure: choose local resolutions around the most singular points and then demonstrate that certain (compatible, monodromy-free) choices extend and glue to a global resolution. 

Abstract: Random circuit models describe the process of generating a complex gate by multiplying a sequence of independent and identically distributed (i.i.d.) elementary gates, which can also be interpreted as a random walk on the respective groups. For quantum circuits on n qubits, the group is the unitary group U(2^n); for classical reversible circuits on n bits, it is the symmetric group Sym(2^n). Can this process be compressed? In other words, if L elementary gates are sampled, is it possible to approximate their product using fewer than L 2-qubit gates? This question is fundamentally related to the growth of (robust) circuit complexity in random circuits and is known as the robust Brown-Susskind conjecture. We prove this conjecture. Specifically, we show that for L up to exponentially large values, random quantum circuits cannot be implemented with fewer than L/poly(n) gates.

Based on https://arxiv.org/pdf/2406.07478 and https://arxiv.org/abs/2402.05239

Abstract: Hurwitz moduli spaces of branched G-covers of the affine line are important geometric objects in number theory, especially arithmetic statistics, over function fields. In this talk, I will motivate the study of these spaces, describe them geometrically and topologically, and explain how information about certain objects in quantum algebra may shed light on their homology. Time permitting, I will explain how one may extend this framework to study the homology of generalized Hurwitz spaces over punctured curves of genus 0. 

Abstract: In this talk, I will discuss recent work constructing a partial compactification of the space of Bridgeland stability conditions. This partial compactification has several appealing features, including an explicit description of when a sequence is converging to the boundary as well as explicit "modular" descriptions of the boundary points. Time permitting, I will explain how the boundary strata give a geometric realisation of braid group actions on semiorthogonal decompositions of a triangulated category. This is based on joint work with Daniel Halpern-Leistner. 

Abstract: Moduli spaces of meromorphic connections defined on principal bundles over compact Riemann surfaces have rich geometric structures. In particular they are complex Poisson spaces, identified with character varieties in the regular singular case; but the irregular examples are particularly interesting and involve generalizations of character varieties. Symplectic leaves are then sometimes described by "semisimple" coadjoint orbits of certain non-reductive Lie groups, involving truncated gauge transformations.

In this talk we will recall the construction of a deformation quantization of standard semisimple orbits, and then describe the extension that occurs in the irregular case: this is work with D. Calaque, G. Felder, and R. Wentworth.

(If time allows, we will also recall how to deform moduli spaces in an admissible/isomonodromic way: this is work with P. Boalch, J. Douçot, and M. Tamiozzo.)

Abstract:  Let G be a complex semisimple group with Lie algebra g. Grothendieck-Springer resolutions are distinguished vector bundles over partial flag varieties of G. Each turns out to be an algebraic Poisson variety with a Hamiltonian action of G. The associated moment map to g can be regarded as a "partial resolution" of the Lie-Poisson structure. I will give a Lie-theoretic introduction to Grothendieck-Springer resolutions and their algebro-geometric features. All of the above-mentioned concepts will be defined in this process.  Particular attention will be paid to Grothendieck-Springer resolutions in Lie type A, and examples will be interspersed throughout the presentation. If time permits, I will outline joint work with Mayrand on new applications to the Moore-Tachikawa conjecture.

Abstract: 

Crystal graphs provide combinatorial tools to study the representation theory of Lie algebras. In this talk, I will discuss joint work with Sylvie Corteel, Zajj Daugherty and Anne Schilling investigating the (Type A) crystal skeleton, which is a graph obtained by contracting certain components of a crystal graph. On the representation theoretic level, crystal skeletons model the expansion of Schur functions into Gessel's quasisymmetric functions. Motivated by questions of Schur positivity, we provide a combinatorial description of crystal skeletons, as well as an axiomatic characterization in analogy to the Stembridge axioms for crystals.

Abstract:

Abstract: Shifted Yangians are infinite-dimensional algebras of capital importance in the study of integrable systems, Coulomb branches and cluster categorifications. They admit remarkable quotients, called truncations, which are (graded) quantizations of generalized slices in affine grassmannians. 

In the first part of this talk, we show that a notable family of representations for the truncations, the “generalized prefundamental modules”, satisfy the so-called “extended QQ-relations”. This proves a conjecture of Frenkel—Hernandez. We then explain how our results relate to the study of a cluster algebra that was recently introduced by Geiss—Hernandez—Leclerc.

In the second part, we give an explicit realization of generalized prefundamental modules in the minuscule case and extract combinatorial formulas for their q-characters.

This is joint work with Artem Kalmykov, Joel Kamnitzer and Alex Weekes.

Abstract: The vector space of chord diagrams and the vector space of (open) Jacobi diagrams are important ingredients of the construction of the universal finite type invariant of knots. In my talk, I will introduce these vector spaсes and define the frame of a Jacobi diagram. This concept allows us to connect Jacobi diagrams to chord diagrams in a new way, and this connection leads to an intriguing conjecture. I will formulate this conjecture and describe a possible way of finding a supporting evidence for this conjecture. Some parts of this story are presented in the paper "Matching of frames of open Jacobi diagrams and chord diagrams" by D. Antonovych, V. Makozyuk and V. Tysiachnyi. 

Abstract: A vertex algebra can be thought of as a commutative algebra whose multiplication map need not be globally defined and can have certain singularities. Starting from this, the concept of a quantum vertex algebra is obtained by relaxing the requirement that the algebra is commutative and instead imposing only the weaker hypothesis that its product and opposite product are intertwined by a solution of the quantum Yang-Baxter equation. At the turn of the century, Etingof and Kazhdan formalized this concept and constructed a quantum vertex algebra structure on the vacuum module of the Yangian double of the special linear algebra at some fixed level k. This example provides a deformation of the standard level k affine vertex algebra and to this day remains one of the most important and well understood examples of a quantum vertex algebra. In this talk, I will explain how to uniformly generalize this construction starting from the notion of a graded quantum vertex coalgebra, which naturally captures certain key properties exhibited by quantum groups like Yangians. This is based on joint work with Alex Weekes and Matt Rupert. 

Abstract: In this joint work with Alina Marian, we study the geometric representation theory of Quot schemes of 0-dimensional coherent sheaves on curves: the overall structure we obtain is a categorification of quantum loop sl_2 acting on the derived categories of Quot schemes. We have two uses for this machinery: we produce an explicit semiorthogonal decomposition of the derived categories in question (which we expect match Toda's wall-crossing construction), and we prove conjectures of Krug and Oprea-Sinha on the cohomology of tautological bundles on Quot schemes.