Abstract. Let k be an algebraically closed field of characteristic zero. It is well known that oriented 2-dimensional cobordisms recover the Deligne category Rep(St), which interpolates the category of finite-dimensional representations of the symmetric group Sn from natural numbers n to an arbitrary parameter t in k. In this talk, we review this construction and show that an analogous story holds in the unoriented case: unoriented 2-dimensional cobordisms recover the generalized Deligne category Rep(St wreath Z2).

Abstract. (Joint with Ruofan Jiang and Alexei Oblomkov) The Hilbert scheme of points on planar singularities is an object with rich connections (q,t-Catalan numbers, HOMFLY polynomials, Oblomkov–Rasmussen–Shende conjecture). The Quot scheme of points is a higher rank generalization of the Hilbert scheme of points. As our main result, we prove that for the "torus knot singularity" x^a = y^b with gcd(a,b)=1, the Quot scheme admits a cell decomposition: every Birula-Białynicki stratum is “as nice as possible” despite poor global geometry. The proof uses two key properties of the rectangular‑grid poset: an Ext‑vanishing for certain quiver representations and a structural result on the poset flag variety. Time permitting, I will discuss a conjectured Rogers–Ramanujan type identity, whose sum side is a summation on (nested) a x b Dyck paths and product side has modulus a+b.

Abstract. I will give a microlocal description of the kernel of Whittaker averaging. Applying this to the geometric Langlands program, I will generalize some results of Færgeman and Raskin, using the geometry of the global nilpotent cone. By an argument of Bezrukavnikov and Morton-Ferguson, this also has applications to Gaitsgory's central sheaves in the affine Hecke category.

Abstract. We prove a decomposition theorem for the quantum cohomology of variations of GIT quotients. More precisely, for any reductive group $G$ and a simple $G$-VGIT wall-crossing $X_- \dashrightarrow X_+$ with a wall $S$, we show that the quantum $D$-module of $X_-$ can be decomposed into a direct sum of that of $X_+$ and copies of that of $S$. As an application, we obtain a decomposition theorem for the quantum cohomology of local models of standard flips in birational geometry.

Abstract. In geometry, the quantum K theory of Grassmannians is a ring with a product deforming the usual K theory product. In (mathematical) physics, it is the coordinate ring of an affine variety given by the Bethe Ansatz equations. I will discuss a dictionary between these two perspectives, with emphasis on geometric interpretations. In particular, the graphical calculus from a 5-vertex lattice model yields Pieri-type rules, to quantum K multiply Schubert classes by Hirzebruch lambda_y classes of tautological bundles. One may also construct eigenvectors of the previous quantum multiplication operators, called Bethe vectors, which quantize the usual classes of torus fixed points. I will discuss how the existence of these Bethe vectors leads to a theory of quantum equivariant localization for Grassmannians. This is joint work with V. Gorbounov and C. Korff, following earlier work with W. Gu, E. Sharpe, and H. Zou.

Abstract. Hassett showed that there are natural reduction morphisms between moduli spaces of weighted pointed stable curves when we reduce weights. I will discuss some joint work with Ziquan Zhuang which constructs similar morphisms between moduli of stable pairs in higher dimension.

Abstract. We study the cohomology of $C_n(X)$, the moduli space of commuting $n$-by-$n$ matrices satisfying the equations defining a variety $X$. This space can be viewed as a non-commutative Weil restriction from the algebra of $n$-by-$n$ matrices to the ground field. We introduce a "Fermionic" counterpart $S_n(X)$, constructed as a convolution $X^n \times^{S_n} GL_n/T_n$. Our main result establishes that a natural map $\sigma:S_n(X) \to C_n(X)$ induces an isomorphism on $\ell$-adic cohomology under mild conditions on $X$ or the characteristic of the field. This confirms a heuristic derived from the classical theory of Weil restrictions and highlights a version of Boson-Fermion correspondence. Furthermore, we derive explicit combinatorial formulae for the Betti numbers of $C_n(X)$ and a Macdonald-type generating series. Finally we also provide a Hermitian variant of our main result. This is joint work with Asvin G, Yifeng Huang, Ruofan Jiang.

Abstract. We will discuss a Hodge-theoretic B-model for the open mirror symmetry of toric Calabi-Yau 3-folds and its identification with the B-model for the dual toric Calabi-Yau 4-folds in terms of periods of integral cycles and variations of mixed Hodge structures on cohomology. We will also discuss implications for Hosono’s Conjecture on Gamma-integral structures.

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