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.