Lin Chen 陈麟 (YMSC, Tsinghua)

Localization for affine Lie algebras

We will talk about the localization functor sending representations of affine Lie algebras to D-modules on the moduli stack of G-torsors on a projective curve. In particular, we will discuss its compatibility with the constant term functor and mention some applications.

Henry Liu 刘华昕 (IPMU)

Wall-crossing for invariants of equivariant CY3 categories

I will describe a "universal" wall-crossing formula for enumerative invariants, particularly in equivariant K-theory, associated to certain moduli problems of 3-Calabi-Yau type. This is joint work with Nikolas Kuhn and Felix Thimm, based on ideas of Joyce. In particular, we quantize Joyce-Song's previous wall-crossing formulas for generalized Donaldson-Thomas invariants. As an immediate application, I give an explicit Donaldson-Thomas/Pandharipande-Thomas correspondence for descendent vertices.

Daxin Xu 许大昕 (AMSS)

Frobenius structures on rigid connections

In this talk, we review some rigid connections including hypergeometric connections, Frenkel-Gross connection etc. We then construct the Frobenius structure on these connections using the geometric Langlands program. We deduce some results in the arithmetic Langlands program based on the Frobenius structure. The talk is based on a series of joint works with Xinwen Zhu, Yichen Qin and Lingfei Yi.

Zeyu Wang 王泽宇 (MIT)

Higher Rankin-Selberg integrals over function fields

Yun and Zhang proved a “higher Gross-Zagier formula” relating intersection numbers of Heeger-Drinfeld cycles on the moduli of \(PGL_2\)-shtukas to higher derivative of L-functions. Since special cycles on moduli of shtukas are rarely compact in higher-dimensional case, a generalization of the formula to higher-dimensional case was not known for a long time. In this talk, I will introduce a definition of “higher periods integrals” and a formula relating the higher periods integrals to higher derivatives of L-functions in the Rankin-Selberg case \(G=GL_n \times GL_{n-1}\). These higher periods integrals are constructed from the RTF-algebra action of the period sheaf introduced by Ben-Zvi, Sakellaridis, and Venkatesh, and are closely related to special cycles on the moduli of shtukas. The talk will be based on my joint work with Shurui Liu ( arXiv2504.00275 ).

Ruotao Yang 杨若涛 (AMSS)

On the Gaiotto conjecture and its Iwahori version

The Gaiotto conjecture says that the category of finite-dimensional representations (resp, category O) of the quantum supergroup can be realized as a certain sheaf category on the affine Grassmannian (resp, affine flags). It is the quantum extension of (a particular case of) the local relative Langlands conjecture of Ben-Zvi-Sakellaridis-Venkatesh. In the talk, we will begin by explaining the background and the precise statement of the Gaiotto conjecture. Then, we will review the known cases due to the former works of Braverman, Finkelberg, Gaitsgory, Ginzburg, Travkin, Yang, etc. Finally, we will focus on the ongoing and in-preparation works joint with Finkelberg and Travkin.

Lingfei Yi 易灵飞 (Fudan University)

A local geometric Langlands for irreducible isoclinic formal connections

We establish a correspondence between irreducible isoclinic formal connections with a family of supercuspidal representations originally constructed by Adler. The correspondence uses the notion of opers and the Feigin-Frenkel isomorphism, and can be formulated for much more general supercuspidal representations using Yu's data. As an application, we confirm a conjectural Langlands correspondence between Airy G-connections and a family of rigid Hecke eigensheaves constructed by Jakob-Kamgarpour-Yi.

Penghui Li 李鹏辉 (Tsinghua)

Graded sheaves and applications

We provide a uniform construction of "mixed versions" or "graded lifts" in the sense of Beilinson-Ginzburg-Soergel which works for arbitrary Artin stacks. Our theory associates to each Artin stack of finite type \(Y\) over \(F_q\) a symmetric monoidal DG-category of graded sheaves on \(Y\), along with the six-functor formalism, a perverse t-structure, and a weight (or co-t-) structure in the sense of Bondarko and Pauksztello. These structures are compatible with the six-functor formalism, perverse t-structures, and Frobenius weights on the category of (mixed) \(\ell\)-adic sheaves. We give two applications of this theory. One is the identification of the HOMFLY-PT invariant as certain coherent sheaves on the Hilbert scheme, as conjectured by Gorsky-Negut-Rasmussen. The other is the relative Serre duality in the Hecke category of Soergel bimodules, as conjectured by Gorsky-Hogancamp-Mellit-Nakagane. This is a joint work with Quoc P. Ho.

Michael McBreen (CUHK)

Symplectic duality and the Tutte polynomial

The Tutte polynomial was introduced in the 1940s as a two-variable generalisation of the chromatic polynomial of a graph. It is the universal matroid invariant satisfying a deletion-contraction relation, and is the subject of much recent work. I will give a geometric realisation of the Tutte polynomial via the cohomology of a symplectic dual pair of hypertoric varieties, and deduce a monotonicity property for the coefficients. Joint work with Ben Davison.