Kleinian singularities are remarkable singular affine surfaces. They arise as quotients of C^2 by finite subgroups of SL_2(C). The exceptional loci in the minimal resolutions of Kleinian singularities are in 1-to-1 correspondence with simply-laced Dynkin diagrams. In this talk, I will introduce certain singular Lagrangian subvarieties in minimal resolutions of Kleinian singularities that appear in the classification of irreducible Harish-Chandra (g, K)-modules. These singular Lagrangian subvarieties have irreducible components given by P^1's and A^1's and contain the exceptional locus as a subvariety. I will describe how these irreducible components intersect with each other through the realization of Kleinian singularities as Nakajima quiver varieties.

I will give a brief review on how to count curves on algebraic manifolds. I will emphasise more on the evolution of counting curves, especially on the importance of change of viewpoints at various stages. I will mention some key new concepts or methods introduced whenever we meet some conceptual or computational difficulties.  Near the end, I will present a method to calculate Gromov–Witten invariants of the Calabi–Yau quintic three-fold.

Let G be a reductive group over the p-adic numbers with P = MU a parabolic subgroup. A basic fact in smooth representation theory is that parabolic induction preserves the property of being admissible.  In this talk, we will discuss the analogue of this in the geometrization of the local Langlands program. In particular, smooth representations will be replaced by sheaves on Bun_G, the moduli stack of G-bundles on the Fargues–Fontaine curve, and parabolic induction will be replaced by a geometric Eisenstein functor carrying sheaves on Bun_M to sheaves on Bun_G. The property of being admissible translates into the rather bizarre property of being ULA over a point, which is a new phenomenon native to analytic variants of the geometric Langlands program. The main result we will discuss is that the geometric Eisenstein functor sends sheaves which are ULA over a point on Bun_M to sheaves which are ULA over a point on Bun_G. This generalizes the basic fact on admissibility mentioned at the beginning, and much more interestingly shows that various gluing functors on Bun_G send admissible representations to admissible representations.  Along the way, we hope to explain some of the similarities and differences between the usual geometric Langlands programs and the Fargues–Scholze geometric Langlands program, mostly stemming from the differences between l-adic sheaves on algebraic and p-adic analytic spaces, respectively. 

I will discuss two computations involving representations of finite Chevalley groups in equal characteristic. The first one reduces by an argument of Bao Le Hung and Tony Feng to description of a class in top homology of a certain affine Springer fiber (performed jointly with Bao Le Hung, Tony Feng and Pablo Boixeda Alvarez). The second one (joint with Michael Finkelberg, David Kazhdan and Calder Morton-Ferguson) involves a new basis for the ring O(T) of regular functions on the maximal torus T as a module over the ring O(T)^W of invariant regular functions, different from but related to the basis constructed by Steinberg in 1975.

For a general algebraic stack X, we will present combinatorial structures underlying the connected components of the stack of filtrations of X. This allows us to define analogues of the Hall algebra in different flavors, except that we do not get algebras but a more general kind of structure. Classically these Hall algebras were only defined when X parametrizes objects in an abelian category, while our construction is general. We will then discuss applications of the theory. 

In the motivic setting, we define a notion of Euler characteristic for a stack and, in the (-1)-shifted symplectic case, we give an intrinsic definition of Donaldson–Thomas invariants. The construction relies on a no-pole theorem. The invariants depend on the choice of a so-called stability measure. The space of such measures is a unipotent algebraic group that governs how invariants change under wall-crossing.

In the cohomological setting, we get an explicit form of the decomposition theorem for the map from the stack to its good moduli space, in the smooth, 0-symplectic, and (-1)-symplectic case, assuming tangent space representations at closed points are orthogonally symmetric.

This is joint work over different projects with Chenjing Bu, Ben Davison, Daniel Halpern-Leistner, Tasuki Kinjo and Tudor Pădurariu.

In the quantum theory of angular momentum, the Racah–Wigner coefficient, often known as the 6j symbol, is a numerical invariant assigned to a tetrahedron with half-integer edge-lengths. The 6 edge-lengths may be viewed as representations of SU(2) satisfying certain multiplicity-one conditions. One important property of the 6j symbol is its hidden symmetry outside the tetrahedral ones, originally discovered by Regge.

In this talk, we explore a generalized construction, dubbed the tetrahedral symbol, in the context of rank-1 semisimple groups over local fields, and explain how the extra symmetries may be explained by relative Langlands duality. Joint work with Akshay Venkatesh.

Although vertex algebras originated in 2D conformal field theory, recent developments show that they also arise in 3D and 4D quantum field theories. Such vertex algebras often appear as chiralizations of symplectic singularities.

To construct new examples of these chiralizations, we study chiral differential operators on the basic affine space G/U, where G is a simple, simply connected algebraic group of type ADE, and U is a maximal unipotent subgroup of G. We show that the associated variety of these vertex algebras is isomorphic to the affine closure of G/U, which is a symplectic singularity, as conjectured by Ginzburg and Kazhdan and later proved by Jia and Gannon.

This is a joint work in progress with Xuanzhong Dai and Bailin Song.

In the first lecture, I will briefly review the classical story for BGG category O. Then I will give an introduction to the quantum category O with motivations.

I will continue in the second lecture by giving basic properties of quantum category O. Then I will introduce the affine Hecke algebra and the periodic Hecke module, and explain a categorification of the former using coherent sheaves on Steinberg variety.

In the third lecture, I will introduce the main result on an equivalence between quantum category O and affine Hecke category. Then I will explain the corresponding t-structure on the affine Hecke category, and its relation to the periodic Hecke module.

I will explain some results and conjectures relating the representation theory of simple affine vertex algebras to the geometry of homogeneous elliptic affine Springer fibres associated with the Langlands dual affine Lie algebra. I will also explain generalizations of this relation to chiral algebras associated with an n-punctured Riemann surface of genus g, constructed by T. Arakawa, and the geometry of the Hitchin moduli space on the same Riemann surface.