Hua Luogeng Youth Forum on Mathematics 2018

Hua Luogeng Youth Forum on Mathematics 2018

Content

Quantization and representation theory

Speaker:  Prof. Shilin Yu(Texas A&M University)

Time:2018.03.19,10:00-11:00

Place:N913

Abstract :

Representation theory of Lie groups plays important roles in geometry, number theory and mathematical physics. The problem of classifying all irreducible representations is challenging and closely related to many different branches of mathematics, such as harmonic analysis, symplectic geometry and algebraic geometry. In this talk, I will explain how representations can be understood as deformation from commutative geometric data to noncommutative objects, in a similar way as quantization of classical mechanical systems.

 

Quantization and representation theory

Speaker:  Prof. Shilin Yu(Texas A&M University)

Time:2018.03.20,09:00-11:00

Place:N902

Abstract :

Beilinson and Bernstein generalized the Borel-Weil-Bott theorem and showed that representations of a (noncompact) reductive Lie group G can be realized as D-modules on flag variety. In this talk, I will show that such D-modules live naturally in families, which explains a mysterious analogy between representation theory of the group G and that of its Cartan motion group, due to Mackey, Higson and Afgoustidis. The talk is based partially on the joint work with Qijun Tan and Yijun Yao.

 

Quantization and representation theory

Speaker:  Prof. Shilin Yu(Texas A&M University)

Time:2018.03.21,09:00-11:00

Place:N913

Abstract :

Kirillov's coadjoint orbit method suggests that (unitary) irreducible representations can be constructed as geometric quantization of coadjoint orbits of the group. Except for a lot of evidence, the quantization scheme meets strong resistance in the case of noncompact semisimple groups. I will give a new perspective on the problem using deformation quantization of symplectic varieties and their Lagrangian subvarieties. This is joint work in progress with Conan Leung.

 

Soliton resolution for energy critical wave equations

Speaker:Prof. Hao Jia(University of Minnesota)

Time:2018.05.22, 10:00-12:00

Place:N212

Abstract:

In this talk we will discuss some recent progresses on the study of dynamics of energy critical wave equations, specifically on the soliton resolution conjecture (SRC). SRC predicts that for many dispersive equations, generic solutions should asymptotically de-couple into solitary waves and radiation as time goes to infinity. The conjecture is open for most equations except integrable ones, but is better understood in the case of energy critical wave equations. We will give a sketch of the proof of this conjecture for a sequence of times, in the case of semilinear wave equations. The proof uses many ideas, including optimal perturbation theory, monotonicity formula, unique continuation property for elliptic equations, and most interestingly a channel of energy argument for outgoing waves.

 

 

 

Soliton resolution in dispersive equations

Speaker: Prof. Hao Jia(University of Minnesota)

Time:2018.05.24,10:30-11:30

Place:N913

Abstract:

A remarkable feature for dispersive equations is the ``simplification" of solutions at large times. For linear dispersive equations, this is well understood. But for nonlinear equations where there are complicated solitary waves, the mechanism by which the solution ``de-couple" into the solitary waves plus radiation is still mysterious, except for integrable systems. We will review some history on this fascinating topic, and explain some recent progress in the energy critical wave equations, such as defocusing wave with potential, focusing wave equations, and wave maps.

 

On the De Gregorio model for 3D Euler equations

Speaker:Prof. Hao Jia(University of Minnesota)

Time:2018.05.25,10:00-12:00

Place:N913

Abstract:

The global regularity problem for 3D Euler equations is an important open problem in PDEs. The main issue is to control vorticity, which could grow due to a stretching term in the equation. The main difficulty is to understand the interplay between the vorticity transportation and vorticity stretching. De Gregorio proposed a one dimensional model, based on a modification of the famous Constantin-Lax-Majda model, to gain insight on this effect. It turns out that this one dimensional model is very interesting. Numerical simulations show global existence, but we do not have a proof. In this talk, we will give a proof of global existence in the perturbative regime near the ground state. The proof reveals some interesting features which are relevant in the large data case as well. It also reveals the distinction between several notions of ``criticality" for some quasilinear equations: critical space for well-posedness, persistence of regularity, and the critical space for global existence and