Schedule

Title:  A new geometry for multitudes, not magnitudes

Abstract:  Aristotle distinguished quantities as either magnitudes (what we measure) or multitudes (what we count). Modern geometry has an elegant language for magnitudes: on a manifold M, one chooses a metric g, and from that choice one extracts invariants such as curvature R_g. But there is no comparably natural geometry on manifolds designed to encode multitudes.

In this talk I introduce a new structure on a smooth manifold M called an arithmic κ. Like a metric, an arithmic is additional geometry on M beyond its smooth structure, but it is tuned to multitudes rather than magnitudes. Fixing κ canonically produces quantum-information-type data for the pair (M,κ) together with genuine invariants, playing a role for arithmics analogous to curvature in the metric setting.

I will illustrate the theory on smooth 2-manifolds via an explicit arithmic invariant built from a normalized Penrose polynomial P_κ, and then describe a deeper package of invariants: filtered n-color homologies associated to (M,κ). The point is not merely that these invariants recover familiar representation-theoretic data in special cases, but that, once the arithmic viewpoint is adopted, they already give answers to classical deep problems in mathematics in a direct and conceptual way. This suggests that arithmics capture a genuinely new layer of structure on manifolds---one that has not previously been visible through the usual geometry of magnitudes.

Title:  What are Lie superalgebras good for?

Abstract:  I will try to answer, as honestly as I can, this question. Lie superalgebras are important in mathematical physics (supersymmetry), in representation theory, in categorification, in quantum topology, but also in classical topology. Namely, they may detect the genus of a smallest spanning surface of a knot. Come and listen about some theorems and experimental evidence, and decide for yourself if this is an accident, a conspiracy theory, or a manifestation of the truth! 

Title:  Double affine Hecke algebra and refined quantum invariant

Abstract:  We discuss a relationship between the skein algebra and the double affine Hecke algebra.

Title:  Parabolic induction for Springer fibers of type C and diagrammatics of entropy

Abstract:  I will discuss the geometry of nilpotent orbits. Nilpotent orbits are important objects in geometric representation theory since they appear in Springer’s construction of Weyl group representations, associated varieties of primitive ideals of enveloping algebras, conical symplectic singularities, and modular representation theory of Lie algebras, to name a few. The theory of Lusztig—Spaltenstein induction is a geometric process for constructing nilpotent orbits in the Lie algebra of a reductive group from the nilpotent orbits in Levi subalgebras. I will discuss the parabolic induction for Springer fibers of type A as well as some progress made involving the parabolic induction for Springer fibers of type C. This is joint with Neil Saunders and Arik Wilbert.

In the second part of my talk, I will discuss how cocycles appear in a graphical network. Furthermore, the Shannon entropy of a finite probability distribution has a natural interpretation in terms of diagrammatics. I will explain the diagrammatics and their connections to infinitesimal dilogarithms and entropy. This is joint work with Mikhail Khovanov.

Title:  Casimir Actions of Parabolic Positive Representations

Abstract:  The parabolic positive representations of U_q(g_R) were previously constructed by quantizing the classical parabolic induction corresponding to arbitrary parabolic subgroups, such that the Chevalley generators act by positive self-adjoint operators on a Hilbert space. This generalizes the (standard) positive representations introduced earlier corresponding to the minimal parabolic (i.e. Borel) subgroup. In this talk, we will show how one can study the scalar actions of the generalized Casimir operators by certain reductions from the standard representations to the parabolic cases.

Title:  Cyclic quantum Teichmuller theory

Abstract:  Based on the ideas of Kashaev, we present a fully explicit construction of a finite-dimensional projective representation of the dotted Ptolemy groupoid when the quantum parameter q is a root of unity. 

We then study the action of the Chekhov—Fock algebra on the space of quantum states. We provide a geometric method to obtain an irreducible decomposition.

We introduce two versions of quantum intertwiners associated with a mapping class: one on the entire space and the other on each irreducible component. Both are given as composites of cyclic quantum dilogarithm operators. We prove that the former gives an intertwiner of local representations of quantum Teichmüller space in the sense of Bai—Bonahon—Liu, and that it coincides with the transpose of the reduced quantum hyperbolic operator of Baseilhac--Benedetti. 

The irreducible intertwiner is conjectured to coincide with the Bonahon—Liu (= Bonahon—Wong) intertwiner.

Overall, I will provide a unified cluster-algebraic perspective on these pioneering works. 

Title:  Exterior algebras with braided Hopf structures, MOY-calculus, and R-matrices.

Abstract:  Braided Hopf algebras with automorphisms give rise to constant solutions of the Yang—Baxter equation. In this talk, I will explain how this construction works in the case of exterior algebras realised as Nichols algebras with non-diagonal braiding. Joint work with Vladimir Mangazeev.

Title:  Foams and link homology

Abstract:  This is a review talk on foams and their uses in link homology and categorification.

Title:  Monoidal categorification of the bosonic extension associated with a positive braid

Abstract:  In this talk, I will describe our recent construction of a monoidal categorification of the (quantum) cluster algebra structure on the algebra $\hat A(b)$ associated with an arbitrary positive braid b. Here $\hat A(b)$ is the subalgebra of the bosonic extension $\hat A$ generated by the PBW vectors associated with b. The construction takes place inside the category of finite-dimensional modules over a quantum affine algebra. We introduce a full monoidal subcategory generated by  the so-called  affine cuspidal modules and show that its Grothendieck ring is isomorphic to $\hat A(b)$.  Moreover, it admits a cluster algebra structure and cluster monomials correspond to real simple modules. This is a joint work with Masaki Kashiwara, Se-jin Oh, and Euiyong Park.

Title:  q-Difference Connections and p-Curvature

Abstract:  I will describe how computations in quasimap quantum K-theory are modified when the curve-counting parameter is sent not to unity, but to a primitive root of unity ζ_p instead. In particular, this leads to appearance of Z/pZ symmetry. Upon a reduction of the quantum difference equation for a Nakajuma quiver variety X to the quantum differential equation over a certain field of finite characteristic, we arrive at the Grothendieck–Katz p-curvature and prove that it is isospectral to a standard curvature operator precomposed with Frobenius.

Title:  Q-deformed Howe duality for orthosymplectic Lie superalgebras

Abstract:  In this talk, we introduce a q-analogue of Howe duality associated to a pair (g,G), where g is an orthosymplectic Lie superalgebra and G=O_l, Sp_{2l}. We define explicitly commuting actions of a quantized enveloping algebra of g and the iquantum group of type AI and AII on a q-deformed supersymmetric space. Then we describe its semisimple decomposition whose classical limit recovers the (g,G)-duality. This is a joint work with Jeong Bae.

Title:  Nonequilibrim thermodynamic, information entropy and contact geometry

Abstract:  Both statistical phase space (SPS), T*R^{3N} of N body particle system and kinetic theory phase space (KTPS), and the cotangent bundle T*P(Γ) of the probability space P(Γ) thereon, carry canonical symplectic structures. Starting from this first principle, we provide a canonical derivation of thermodynamic phase space (TPS) of nonequilibrium thermodynamics as a contact manifold. We apply the Marsden—Weinstein reduction and obtain a mesoscopic phase space in between KTPS and TPS as an (infinite dimensional) symplectic fibration. We then show that the relative information entropy defines a generating function that provides a covariant construction of thermodynamic equilibrium as a Legendrian submanifold. This Legendrian submanifold is not necessarily graph-like. We interpret Maxwell's construction via the equal area law as the procedure of finding a continuous, not necessarily differentiable, Gibbs potential and explain the associated phase transition. Our derivation complements the previously proposed contact geometric description of thermodynamic equilibria and explains the origin of phase transition and the Maxwell construction in this framework. This is a joint work with Jinwook Lim.

Title:  Nilpotent Quiver Representations

Abstract:  This talk will explore representation theory of the cyclic quivers. It turns out that there is a surprising analogy between the case of the 2-cycle quiver C_2, and the 1-cycle C_1, which is just the study of linear endomorphisms. We will demonstrate various structural results for representations of C_2, which lead to effective enumerations in the case of finite fields. We conjecture similar results for the more general k-cycle quiver, C_k, as well as consequences in the direction of Springer correspondences.

Title:  Knots as point clouds

Abstract:  Quantum knot invariants (Jones-type invariants coming from quantum groups, and their categorified friends) produce large, high-dimensional datasets when sampled across knots and links, and we view these outputs as point clouds studied via topological data analysis (TDA). This "geometry of invariants" makes correlations, degeneracies, and hidden structure hard to ignore, suggesting new conjectures (sometimes quick proofs) and a concrete way to score how informative an invariant is for questions in quantum topology and representation theory.

Title:  Categorifying Jacobi—Trudi

Abstract:  We discuss a way to categorify the famous Jacobi—Trudi identity by constructing a quasi-hereditary algebra A with a map kS_n to A; the “dominant simple” of A admits a BGG resolution, and after restricting to kS_n, we obtain a resolution of a simple module of kS_n by permutation modules. The algebra A is constructed as a certain quotient of KLR, and to prove the BGG resolution we use the very general technique of "reconstruction from stratification". We will briefly explain this general technique and its relation to Koszul duality.