Tsinghua University

One of the fundamental problems in representation theory is determining the representation type of algebras. In this talk, we will introduce the representation type of cyclotomic quiver Hecke algebras, also known as cyclotomic Khovanov-Lauda-Rouquier algebras, especially in affine type A and affine type C. Our main result relies on novel constructions of the maximal dominant weights of integrable highest weight modules over quantum groups. This talk is based on collaborations with Susumu Ariki, Berta Hudak, and Linliang Song.

In his pioneering work in 1860, Riemann proposed the Riemann problem and solved it for isentropic gas in terms of shocks and rarefaction waves. It eventually became the foundation of the theory of one-dimension conservation laws developed in the 20th century. We prove the non-nonlinear structural stability of the Riemann problem for multi-dimensional isentropic Euler equations in the regime of rarefaction waves. This is a joint work with Tian-Wen Luo.

I will discuss a string theory perspective on the Bethe/Gauge correspondence for the XXX superspin chain. I explain how to realize 4d Chern-Simons theory with gauge supergroup using branes, and how the brane configurations for the superspin chain get mapped to 2d N = (2,2) quiver gauge theories proposed by Nekrasov. This is based on my ongoing work with Nafiz Ishtiaque, Faroogh Moosavian and Surya Raghavendran.

The Alfven waves are fundamental wave phenomena in magnetized plasmas and the dynamics of Alfven waves are governed by the MHD system. In the talk,  we construct and study the long time behavior of (viscous and non-viscous) Alfven waves.

As applications, (1) We provide a rigorous justification for the following dynamical phenomenon observed in many contexts: the solution at the beginning behave like non-dispersive waves and the shape of the solution persists for a very long time (proportional to the Reynolds number); thereafter, the solution will be damped due to the long-time accumulation of the diffusive effects;

(2) We prove the rigidity aspects of the scattering problem for the MHD equations: We prove that the Alfven waves must vanish if their scattering fields vanish at infinities.

It is an old problem in number theory to count number fields of a fixed degree and having a fixed Galois group for its Galois closure, ordered by their absolute discriminant, say. In this talk, I shall discuss some background of this problem, and then report a recent work with Stanley Xiao. In our paper, we considered quartic $D_4$-fields whose ring of integers has a certain nice algebraic property, and we counted such fields by their conductor.