Previous Seminars

Time：December 30, Monday, 16:00-17:00  

Title: Theta correspondence and Springer correspondence

Speaker: Jialiang Zou 邹佳良 (University of Michigan)

Abstract: Let V and W be an orthogonal and a symplectic space, respectively. The action of G=O(V)\times Sp(W) on V\otimes W provides an example of G-hyperspherical varieties introduced by D. Ben-Zvi, Y. Sakellaridis, and A. Venkatesh (BZSV for short). It is the classical limit of theta correspondence from the perspective of quantization.. I will explain a geometric construction motivated by theta correspondence over finite fields, which describes how principal series representations behave under theta correspondence using Springer correspondence. 

BZSV proposed a relative Langlands duality linking certain G-hyperspherical varieties M with their dual G^\vee-hyperspherical varieties M^\vee. A remarkable instance of this duality is that the hyperspherical variety underlying theta correspondence is dual to the hyperspherical variety  underlying the branching problem in the Gan-Gross-Prasad conjecture. I will discuss how these results fit into the broader framework of this relative Langlands duality. This is an ongoing  joint work with Jiajun Ma, Congling Qiu, and Zhiwei Yun.

Time：December 4, Wednesday, 16:00-17:00  

Title: On Recent Progress in the Overconvergence of Étale (ϕ,Γ)-Modules

Speaker: Heng Du 杜衡 (Tsinghua University)

Abstract:  Étale (ϕ,Γ)-modules are a fundamental tool in p-adic Hodge theory. The overconvergence property of these modules is a key aspect that enables the assignment of p-adic differential equations to Galois representations. In this talk, we will revisit the foundational work of Cherbonnier and Colmez and discuss recent advancements in the field. In particular, we will mention a joint work with Tong Liu that extends this result to geometric families.

Time：November 27, Wednesday, 16:00-17:00  

Title: Geometric Langlands for Irregular Theta Connections and Epipelagic Representations

Speaker: Lingfei Yi 易灵飞 (Fudan University)

Abstract: Gradings on semisimple Lie algebras with stable vectors play an important role in representation theory. From stable vectors, Reeder and Yu constructed a family of supercuspidal representations called epipelagic representations. Over the function field of projective line, Yun globalized epipelagic representations using rigid automorphic data and constructed a family of Hecke eigensheaves. On the other hand, in the de Rham setting Yun also constructed an irregular connection on punctured projective line from a stable vector, called a theta connection. I will explain a geometric Langlands correspondence between the theta connections and the Hecke eigensheaves of Yun, based on a joint work with Tsao-Hsien Chen. 

I will also explain an application to the correspondence in the l-adic setting, based on a joint work in progress with Daxin Xu.

Time：November 20, Wednesday, 16:15-17:15

Title: Integral Sen theory and integral Hodge filtration

Speaker: Hui Gao 高辉 (SUSTech)

Abstract: Using the Breuil--Kisin module attached to an integral crystalline representation, one can define an integral Hodge filtration whose behavior is closely related to arithmetic and geometry of the representation. In this talk, we discuss vanishing and torsion bound on graded pieces of this filtration, using a filtered integral Sen theory as key tool. This is joint work with Tong Liu.

Time：November 20, Wednesday, 15:00-16:00

Title: A stacky p-adic Riemann-Hilbert correspondence on Hitchin-small locus

Speaker: Yupeng Wang 王宇鹏 (Beijing International Center for Mathematical Research)

Abstract: Let X be a smooth rigid analytic variety over C with a liftable smooth (or semi-stable) formal model over O_C. We will establish an equivalence between the moduli stack of Hitchin-small de Rham local systems on X_v and the moduli stack of Hitchin-small integrable connections on X_et, based on a new period sheaf with connection on X_v. This is a joint work with Yudong Liu, Chenglong Ma, Xiecheng Nie and Xiaoyu Qu.

Time：November 13, Wednesday, 16:00-17:00

Venue: Lecture Hall

Title: Geometry of affine Springer fibers and generalizations

Speaker: Jingren Chi 迟敬人 (Chinese Academy of Sciences)

Abstract: Affine Springer fibers are analogues of Springer fibers for the loop Lie algebras of reductive groups. They were first studied by Kazhdan and Lusztig and they have played important roles in various problems from geometric representation theory and automorphic representation theory. In this talk I will review the basic geometric properties of affine Springer fibers and report on recent work on some of their generalizations, including the group version and the mixed characteristic analogue.

Time：November 6, Wednesday, 16:00-17:00

Venue: Lecture Hall

Title: : l-Modular Blocks: How far are we from reduction to depth zero?

Speaker: Peiyi Cui 崔沛仪 (Morningside Center of Mathematics)

Abstract:  Reduction to depth zero is a promising approach for understanding l-modular blocks of p-adic groups when l differs from p. In this talk, I will introduce l-modular blocks of SL_n from this perspective. We will explore the technical challenges in associating an l-modular block with a depth-zero block and consider a natural candidate for this potential connection. A conjecture on l-modular blocks of general groups will be proposed. Toward the end, I will also discuss related topics on both the automorphic and Galois sides.

Time：October 23, Wednesday, 15:00-16:00

Venue: Lecture Hall

Title: A locally analytic p-adic Langlands correspondence for GLn(Qp) in the crystabelline case

Speaker: Yiwen Ding 丁一文 (Peking University)

Abstract: We build a one-to-one correspondence between n-dimensional generic non-critical crystabelline Gal(Qpbar/Qp)-representations of regular Hodge-Tate weights and certain locally analytic representations of GLn(Qp). We show the correspondence can be realized in subspaces of p-adic automorphic representations.

Time：October 16, Wednesday, 16:00-17:00

Venue: Lecture Hall

Title: Eisenstein cocycles for imaginary quadratic fields

Speaker: Emmanuel Lecouturier (Westlake University)

Abstract: Sharifi constructed a beautiful explicit map $\varpi_N$ from the first singular homology group of the modular curve $X_1(N)$ to the second algebraic K-group of the cyclotomic field $\Q(\zeta_N)$. He conjectured that this map is annihilated by a certain Eisenstein ideal, which was proved recently by Sharifi and Venkatesh using the motivic cohomology of the square of the torus. In this talk, we explain a construction of an analogue of $\varpi_N$ in the setting of Bianchi 3-folds attached to imaginary quadratic fields. This map is not as explicit as in the case of modular curves due to the lack of a canonical and explicit presentation of the first homology group of Bianchi 3-folds in general. Our construction, and the proof of the Eisenstein property, relies on techniques similar to the ones of Sharifi and Venkatesh but replacing the square of a torus by products of CM elliptic curves. This is a joint work in progress with Romyar Sharifi, Sheng-Chi Shih, and Jun Wang.

Time：September 18, Wednesday, 16:00-17:00

Venue: Lecture Hall

Title: Fourier Transforms and Langlands Conjecture

Speaker: Dihua Jiang 江迪华 (University of Minnesota)

Abstract: I will discuss relations between nonabelian Fourier transforms and the Langlands Conjecture on automorphic L-functions and associated local factors in the framework of Braverman-Kazhdan-Ngo.

Time：September 11, Wednesday, 16:00-17:00

Venue: Lecture Hall

Title: Arithmetic Level Raising for U(2r, 1)

Speaker: Ruiqi Bai 白瑞祺 (University of Cambridge)

Abstract: I will talk about the ongoing work with Hao Fu to show an arithmetic level-raising result ffor the special fiber of U(2r, 1) Shimura variety at an inert prime. We exhibit elements in the higher Chow group of the supersingular locus and use this to prove the surjectivity of the Abel–Jacobi map. A key ingredient of the proof is to show a form of Ihara’s lemma. This work is inspired by the work of Rong Zhou on quaternionic Shimura varieties, and it can be viewed as an even-dimensional analogue of the ongoing work of Yifeng Liu, Yichao Tian, and Liang Xiao. We would like to express our sincere gratitude to IASM for the assistance provided during our visit in 2022.

Time：September 4, Wednesday, 16:00-17:00

Venue: Lecture Hall

Title: 100 years  of metric Diophantine approximation

Speaker: Daodao Yang 杨道道 (Université de Montréal)

Abstract: The metric theory of Diophantine  approximation investigates whether typical real numbers can be well approximated by rational numbers. In 1924, Khintchine proposed the first 0-1 law in metric Diophantine approximation: assuming that the approximation function has a certain monotonicity, this law provides a necessary and sufficient condition for typical real numbers to be well approximated by infinitely many rational numbers.  In 1941, Duffin and Schaeffer attempted to generalize Khintchine's theorem without assuming that the approximation function has monotonicity. Their proposed new 0-1 law, known as the Duffin–Schaeffer conjecture, became a central problem in Diophantine approximation. After considerable efforts by many number theorists, the Duffin–Schaeffer conjecture was finally resolved by Koukoulopoulos and Maynard in 2019. A remaining challenge is to establish a quantitative law, similar to the strengthening of Khintchine's theorem by Erdős and Schmidt, to estimate the number of reduced fractions that can approximate the given real number sufficiently well. In 2022,  Aistleitner, Borda and Hauke established a quantitative version of the Duffin–Schaeffer conjecture, but the error term they obtained only achieved a logarithmic power-saving.  An open problem is to establish an asymptotic formula with a power-saving error term. In a recent joint work with Koukoulopoulos and Maynard, we  prove a quantitative version of the Duffin–Schaeffer conjecture with an almost sharp error term. 

Time：June 19, Wednesday, 16:00-17:00

Venue: Lecture Hall

Title: The local twisted Gan-Gross-Prasad conjecture over nonarchimedean fields

Speaker: Le Nhat Hoang (National University of Singapore)

Abstract: The Gan-Gross-Prasad (GGP) conjecture studies a family of restriction problems for classical groups and proposes precise answers to these problems using the local and global Langlands correspondences. It also has a twisted variant in the equal-rank Fourier-Jacobi case, which is called the twisted Gan-Gross-Prasad conjecture. In this talk, I will report my progress on the local twisted GGP conjecture for tempered representations in (GL(V),U(V)-case. The strategy is to adapt Waldspurger and Beuzart-Plessis's method to develop a local relative trace formula as well as a twisted trace formula and compare their elliptic parts. Although the geometric sides of both trace formulae have not been developed, one can use a partial comparison and an instance of a discrete series representation to prove the statement. Moreover, I will propose a geometric multiplicity formula for the local twisted GGP period and necessary ingredients to prove it.

Time：June 12, Wednesday, 16:00-17:00

Venue： Lecture Hall

Title: Kudla-Rapoport conjecture at bad reduction primes

Speaker： Qiao He 贺乔 (Columbia University)

Abstract:The Kudla-Rapoport conjecture is a local analogue of arithmetic Siegel-Weil formula which relates arithmetic intersections of special cycles with derivatives of local densities. The original conjecture was formulated for unitary Rapoport-Zink space over unramified primes with good reduction and proved by Chao Li and Wei Zhang. However, it is a mysterious problem for a long time to formulate a precise conjecture when the RZ has bad reduction. In this talk, I will motivate the original Kudla-Rapoport conjecture first and explain how we can modify the original conjecture to incorporate the bad reduction cases. Then I will talk about the proof strategy and highlight some striking new phenomenon in the bad reduction cases. If time permitted, I would also mention some speculation and progress for the orthogonal case. This talk is based on several joint works with a few collaborators, including Sungyoon Cho, Chao Li, Yu Luo, Yousheng Shi, Tonghai Yang, Zhiyu Zhang and Baiqing Zhu.

Time：June 5, Wednesday, 14:30-15:30

Venue： Lecture Hall

Title: On the trace formula for covering groups

Speaker： Yuanqing Cai 蔡园青 (Kanazawa University)

Abstract: We discuss the trace formula for the even-fold covering group of SL(2). Along the way, we also revisit the work of Flicker on the trace formula for covering groups of GL(2).

Time：May 29, Wednesday, 16:00-17:00

Venue: Lecture Hall

Title: Gross and Zagier’s work revisited

Speaker: Tonghai Yang 杨同海（University of Wisconsin）

Abstract: In 80s, Gross and Zagier discovered and proved a deep and well-known formula between derivative of an L-function and the height of some `CM’ point on an elliptic curve---Gross-Zagier formula, which gives partial answer the Birch and Swinnerton-Dyer conjecture. In the process, they also proved a beautiful factorization formula for the difference of CM values of the $j$-function. In addition, they also gave a conjecture about the algebraicity of the CM values of higher Green functions.  In this talk, we will look at the these work from different point of view (regularized theta lifting), and give a proof of their algebraicity conjecture if time permits. This talk is based on joint work with a few collaborators, including J. Bruinier, S. Kudla, Yingkun Li, Hongbo Yin, and Peng Yu among others.

Time：May 15, Wednesday, 16:00-17:00

Venue: Lecture Hall

Title:The sign of linear periods

Speaker: Nadir Rafi Matringe (NYU Shanghai)

Abstract: Let $G$ be a group with subgroup $H$, and let $(\pi,V)$ be a complex representation of $G$. The natural action of the normalizer $N$ of $H$ in $G$ on the space $\Hom_H(\pi,\BC)$ of $H$-invariant linear forms on $V$, provides a representation $\chi_{\pi}$ of $ \frac{N}{H} $, which is a character when $\Hom_H(\pi,\BC)$ is one dimensional. If moreover $G$ is a reductive group over a $p$-adic field, and $\pi$ is smooth irreducible, it is an interesting problem to express $\chi_{\pi}$ in terms of the possibly conjectural Langlands parameter $\phi_\pi$ of $\pi$. We will consider the following situation: $G=GL_m(D)$ for $D$ a central division algebra of dimension $d^2$ over a $p$-adic field $F$, $H$ is the centralizer of a non central element $\delta\in G$ such that $\delta^2$ is in the center of $G$, and $\pi$ has generic Jacquet-Langlands transfer to $GL_{md}(F)$. In this setting the space $\Hom_H(\pi,\mathbb{C})$ is at most one dimensional. When $\Hom_H(\pi, \mathbb{C} )\simeq \mathbb{C}$ and $H\neq N$, we prove that the value of the $\chi_{\pi}$ on the non trivial class of $\frac{N}{H}$ is $(-1)^m\e(\phi_\pi)$ where $\e(\phi_\pi)$ is the root number of $\phi_{\pi}$. This is a joint work with U.K. Anandavardhanan, H. Lu, V.  Sécherre  and C. Yang.

Time: May 8, Wednesday, 16:00-17:00

Venue: Lecture Hall

Title: Drinfeld Lemma for F-isocrystals

Speaker: Daxin Xu 许大昕 (Chinese Academy of Sciences)

Abstract: Drinfeld's lemma for l-adic local systems is a fundamental result in arithmetic geometry. It plays an important role in the Langlands  correspondence for a reductive group over the function field of a curve over a finite field, pioneered by Drinfeld for GL_2 and subsequently extended by L. Lafforgue and then V. Lafforgue. In this talk, we will discuss Drinfeld's lemma for p-adic local systems: overconvergent/convergent F-isocrystals. This is based on a joint work with Kiran Kedlaya. 

Time：April 24, Wednesday, 16:00-17:00

Venue: Lecture Hall

Title: The global unramified geometric Langlands equivalence

Speaker: Lin Chen 陈麟 (Tsinghua University)

Abstract: Recently, Gaitsgory's school (to which I am honoured to belong) announced their proof of the global unramified geometric Langlands conjecture. I will explain the history, motivation and statement of this conjecture and the main ingredients used in this proof. If time permits, I will also introduce some questions in this field that remain open after this proof.

Time：April 10, Wednesday, 16:15-17:15

Venue: Lecture Hall

Title: Gelfand-Kirillov dimension in p-adic Langlands program

Speaker: Yongquan Hu 胡永泉 (Chinese Academy of Sciences)

Abstract: Gelfand-Kirillov dimension is an important concept in the study of admissible smooth representations of p-adic Lie groups. 

In this talk, I will explain how to control the Gelfand-Kirillov dimension for mod p representations coming from mod p cohomology in the case of GL_2. This is joint work with Breuil, Herzig, Morra, Schraen, and with Wang.

Time: April 10, Wednesday, 15:00-16:00

Venue: Lecture Hall

Title: How do generic properties spread?

Speaker: Yu Fu 付裕 (Caltech)

Abstract: Given a family of algebraic varieties, a natural question to ask is what type of properties of the generic fiber, and how those properties extend to other fibers. Let's explore this topic from an arithmetic point of view by looking at the scenario: Suppose we have a 1-dimensional family of pairs of elliptic curves over a number field $K$,  with the generic fiber of this family being a pair of non-isogenous elliptic curves. Furthermore, suppose the (projective) height of the parametrizer is less than or equal to $B$. One may ask how does the property of being isogenous extends to the special fibers. Can we give a quantitative estimation for the number of specializations of height at most $B$, such that the two elliptic curves at the specializations are isogenous? 

Time：March 6, Wednesday, 16:00-17:00

Venue: Lecture Hall

Title: The Brun--Titchmarsh Theorem

Speaker: Ping Xi 郗平(Xi’an Jiaotong University)

Abstract：It is fundamental to understand the distribution of primes in arithmetic progressions. With the aid of Brun’s sieve, Titchmarsh gave the first upper bound, which is of correct order of magnitude, on the number of such primes in an individual arithmetic progression. This gives the so-called Brun--Titchmarsh theorem. We will discuss our recent work on sharpening this theorem with better constants for general moduli and for special moduli, and the tools include Dirichlet polynomials, character/exponential sums, moments of L-functions, $\ell$-adic cohomology and spectral theory of automorphic forms. This is a joint work with Junren Zheng.

Time：January 3, Wednesday, 16:00-17:00

Venue： Lecture Hall

Title: Generating series of complex multiplication cycles

Speaker： Andreas Mihatsch (University of Bonn)

Abstract：Let $c_n$ be the number of isomorphism classes of pairs $(E, x)$  consisting of an elliptic curve $E$ over $\mathbb{C}$ and an  endomorphism $x$ that satisfies $x^2 = -n$. A classical theorem of  Zagier states that the series $\sum_{n = 1}^\infty c_n q^n$ is the  positive part of the $q$-expansion of a non-holomorphic modular form.  Its arithmetic version, due to Kudla--Rapoport--Yang, states that the  generating series of complex multiplication (CM) divisors on the  integral modular curve has a similar modularity property.  In my talk, I will define CM cycle generating series for symplectic and  unitary Shimura varieties, and present first results on their  modularity. This adds a new facet to the Kudla program, which aims to  systematically relate special cycles on Shimura varieties with Fourier  expansions of automorphic forms. My talk is based on joint work with  Lucas Gerth, Siddarth Sankaran, and Tonghai Yang.