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).

**Organizers:**

Siqi He, Qingtang Su and Daxin Xu

**Time:**

Thursday, 10:30-11:30 am

**Venue:**

MCM110

**Upcoming talks:**

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: June 6

Speaker: Dr. Zipeng Wang (Westlake Univ)

Title: TBC

Abstract: TBC

Date: June 13

Speaker: Prof. Zehua Zhao (Beijing Institute of Technology)

Time: 9:30-10:30 am

Title: TBC

Abstract: TBC

Date: June 13

Speaker: Prof. Baoping Liu (Peking Univ)

Time: 10:30-11:30 am

Title: TBC

Abstract: TBC

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.

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