This is a series seminar organized by MCM on Number Theory, PDE, Geometry, etc. If you have any questions, please feel free to contact us (mcmoffice@math.ac.cn).
Siqi He, Qingtang Su, Daxin Xu, Hongjie Yu
Thursday, 10:30-11:30 am
Venue:
MCM110
Upcoming talks:
Date: December 26, 2024
Speaker: Dr. Huajie Li (Johns Hopkins University)
Title: 关于Guo-Jacquet迹公式
Abstract: 在本报告中，我将介绍与Pierre-Henri Chaudouard正在合作进行的工作。我们建立了粗略版本的Guo-Jacquet迹公式，其为对称空间的相对迹公式的例子，并被期待应用于Waldspurger关于GL(2)的L函数中值公式的高维推广。为了理解公式中的几何项，我们还得到了一个半单下降公式。特别地，正则半单项可被写为具体的加权轨道积分。
Date: January 2, 2025
Speaker: TBA
Title: TBA
Abstract: TBA
Date: January 9, 2025
Speaker: Dr. Yangyu Fan (Beijing Institute of Technology)
Title: TBA
Abstract: TBA
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Date: June 13
Speaker: Prof. Baoping Liu (Peking Univ)
Time: 9:30-10:30 am
Title: Wellposedness of Davey-Stewartson on T^2
Abstract: In this talk, we consider Davey-Stewartson system on T^2. We show that the integrable hyperbolic-elliptic DS is globally wellposed for data in H^s(s>0) with small L^2 norm. This is in contrast to the general hyperbolic-elliptic DS model with is only local wellposed in H^s(s>1/2) and illposed for s=1/2.
Date: June 13
Speaker: Prof. Zehua Zhao (Beijing Institute of Technology)
Time: 10:30-11:30 am
Title: On Many body Schrodinger Equations and related topics
Abstract: In this talk, the speaker will talk about many body Schrodinger Equations and some related topics for both analysis level and PDE level. This research line is wide open and it is related to many others in the area of dispersive equations.
Date: March 14
Speaker: Dr. Xiaomeng Xu (Peking Univ)
Title: The quantization of irregular Riemann-Hilbert-Birkhoff maps
Abstract: This talk gives an introduction to the Stokes phenomenon and the Riemann-Hilbert-Birkhoff (RHB) map of meromorphic connections. It then introduces the quantum Stokes matrices at an arbitrary order pole, and proves that they give rise to a quantization of the RHB map. In the case of second order pole, it becomes a dictionary between the Stokes phenomenon and the theory of quantum groups.
Date: March 21
Speaker: Dr. De Huang (Peking Univ)
Title: Self-similar finite-time blowups of some 1D models for the incompressible Euler equations
Abstract: A series of 1D models have been proposed to study the competition between advection and vortex stretching for the 3D Euler equations, which include the De Gregorio model, the generalized Constantin–Lax–Majda model, and the 1D Hou-Luo model. In this talk, we present some recent results on exact self-similar finite-time blowup solutions of these models. For the 1D De Gregorio model, we show that there exist infinitely many compactly supported, self-similar solutions that are distinct under rescaling, all corresponding to the eigenfunctions of a self-adjoint compact operator. For the generalized Constantin–Lax–Majda model and the 1D Hou-Luo model, we establish the existence of exact self-similar finite-time blowups using a novel fixed-point method. We will also introduce a novel class of asymptotically self-similar blowup that has multi-scale features, which reveals a new potential blowup mechanism for the 3D Euler equations.
Date: March 28 (Thursday)
Speaker: Prof. Jiaqi Liu (Univ of Chinese Academy of Sciences)
Title: Long-Time Asymptotics for the Kadomtsev-Petviashvili I (KP I) Equation
Abstract: We provide uniform time decay estimates for solutions to the KP I equation. The solution is constructed using inverse scattering formalism for small initial data, which excludes lump solutions. This is joint work in progress with Samir Donmazov and Peter Perry.
Date: April 11 (Thursday)
Speaker: Dr. Dong Yan (BIMSA)
Title: Remarks on Iwasawa linearity conjecture for cyclotomic fields
Abstract: For an irregular pair (p,k), the Iwasawa linearity conjecture claims that the lambda-invariant of its corresponding Kubota-Leopoldt p-adic L-function is equal to one. In this talk, we study this conjecture by analyzing the relation between the Eisenstein ideal and the p-th Fourier coefficient of its associated Hida family.
Time: April 19 (Friday) 15:00-16:00
Speaker: Prof. Haowu Wang (Wuhan Univ)
Place: MCM510
Title: Hyperbolization of affine Lie algebras
Abstract: In 1983, Feingold and Frenkel posed a question about possible relations between affine Lie algebras, hyperbolic Kac--Moody algebras and Siegel modular forms. We give an automorphic answer to this question and its generalization. We classify Borcherds--Kac--Moody algebras whose denominators define reflective automorphic products of singular weight. As a consequence, we prove that there are exactly 81 affine Lie algebras which have nice extensions to BKM algebras. We find that 69 of them appear in Schellekens’ list of holomorphic CFT of central charge 24, while 8 of them correspond to the N=1 structures of holomorphic SCFT of central charge 12 composed of 24 chiral fermions. The last 4 cases are related to exceptional modular invariants from nontrivial automorphisms of fusion algebras. This clarifies the relationship of affine Lie algebras, vertex algebras and hyperbolic BKM superalgebras at the level of modular forms. This is based on a joint paper with Kaiwen Sun and Brandon Williams.
Time: April 26 (Friday) 14:00-15:00
Speaker: Dr. Chenglong Yu (YMSC)
Title: Commensurabilities among lattices in $\mathrm{PU}(1,n)$
Abstract: In simple Lie groups, except the series $\mathrm{PU}(1,n)$ with $n>1$, either all lattices are all arithmetic, or mathematicians constructed infinitely many nonarithmetic lattices. So far there are only finitely many nonarithmetic lattices constructed for $\mathrm{PU}(1, 2)$ and $\mathrm{PU}(1, 3)$ and no examples for $n>3$. One important construction is via monodromy of hypergeometric functions. The discreteness and arithmeticity of those groups are classified by Deligne and Mostow. Thurston also obtained similar results via flat conic metrics. However, the classification of those lattices up to conjugation and finite index (commensurability) is not completed. When $n=1$, it is the commensurabilities of hyperbolic triangles. The cases of $n=2$ are almost resolved by Deligne-Mostow and Sauter's commensurability pairs, and commensurability invariants by Kappes-Möller and McMullen. Our approach relies on the study of some period integrals of higher dimensional varieties instead of complex reflection groups. We obtain some commensurability relations and explicit indices for higher $n$ and also give new proofs for existing pairs in $n=2$. This is based on joint work with Zhiwei Zheng.
Date: May 9
Speaker: Dr. Yang Lan (YMSC)
Title: Asymptotic dynamics near ground state for mass critical Zakharov-Kuznetsov equations in dimension two
Abstract: We consider the focusing mass critical Zakharov-Kuznetsov equation in 2D. We will provide a complete classification of the long time behavior of solutions with initial data near the ground and with a suitable decay on the first variable. We will show that only three behaviors are possible: 1. converging to a traveling wave, 2. blowing up in finite time, 3. linear behavior. Our result is an extention of the work of Martel-Merle-Raphael for mass critical gKdV equations. This work is joint with G. Chen and X. Yuan.
Date: May 16
Speaker: Dr. Lin Chen (YMSC)
Title: The global unramified geometric Langlands equivalence
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.
Date: May 23
Speaker: Dr. Jingbang Guo (Shanghai Center for Mathematical Sciences)
Title: Some Simple Computations in Prismatic Cohomology
Abstract: In this talk, we will briefly introduce the theory of prismatic cohomology and its geometrization, in the sense of realizing prismatic cohomology as the coherent cohomology of certain stack; inspired by this theoretical framework, we will present some simple computations in prismatic cohomology through descent.
Date: May 30
Speaker: Dr. Yigeng Zhao (Westlake Univ)
Title: A fibration formula for characteristic classes of constructible étale sheaves
Abstract: For a constructible étale sheaf, we first review the constructions of cohomological characteristic classes in classical and relative cases. We then prove a fibration formula to calculate them, which involves a new cohomological class, so-called non-acyclicity classes. This is a joint work with Enlin Yang.
Date: November 28, 2024
Speaker: Prof. Jingren Chi (MCM)
Title: Geometry of affine Springer fibers and generalizations
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.
Date: December 5
Speaker: Dr. Pengcheng Li (Tsinghua Univ)
Title: James' row and column removal rules for finite unitary groups
Abstract: James gave the construction of the (modulo l) simple unipotent representations of the finite general linear groups in the 1980s. He showed that row and column removal rules hold for the decomposition matrices of GLn(q). In 2015, Dudas and Malle determined the decomposition matrices of GUn(q) for n smaller than 10, up to a few unknown entries. They conjectured that row and column removal rules should also hold for GUn(q).
In ongoing joint work with Olivier Dudas, we provide the construction of the simple unipotent representations of GUn(q) using modular theta correspondence and generalized Gelfand-Graev representations. We show that the row removal rule for GUn(q) is equivalent to the modular theta descent problem while the column removal rule for GUn(q) is equivalent to the modular Gan-Gross-Prasad descent problem. We show a duality between the modular theta descent and the modular Gan-Gross-Prasad descent, which can be regarded as a finite field instance of relative Langlands duality of Ben-Zvi-Sakellaridis-Venkatesh.
If time permits, I will also discuss the duality of finite symplectic and orthogonal groups.
Date: December 12
Speaker: Prof. Ning Guo (Harbin Institute of Technology)
Title: Geometric Presentation and applications
Abstract: Various problems concerning cohomology sets and K-theory, including Gersten's injectivity and the Grothendieck-Serre conjecture, have affirmative answers in the equi-characteristic case. A helpful tool for dealing with mixed characteristic cases is geometric presentation theorems. Roughly speaking, these theorems realize the local ring under consideration as a relatively smooth (or étale) object over a regular DVR or a field with a low relative dimension such that problems reduce to the DVR or field cases. In this talk, we will discuss several geometric presentation theorems (due to Gabber-Quillen, Colliot-Thélène, Ojanguren, etc) and see their application to the Grothendieck-Serre conjecture in the mixed characteristic case. In particular, I will introduce the notion of weak elementary fibrations, which traces back to Artin's "bon voisinage" in SGA4. These are individual joint works with Ivan Panin, Fei Liu and Yisheng Tian.
Date: December 19
Speaker: Dr. Chuyun Wang (Westlake Univ)
Title: Formal manifolds and Formal Lie groups
Abstract: In the setting of algebraic geometry, formal schemes are intensively studied in the literature. Motivated by applications to representation theory of Lie groups, we study similar objects, called formal manifolds, in the setting of differential geometry. In this talk, we will give a brief introduction to the basic theory of formal manifolds.
Moreover, we will introduce some work of formal Lie groups and represensations of formal Lie groups.
