Associated with a Coxeter group W, we may define a Hecke algebra H(W) and an endomorphism algebra E(W), called the Hecke endomorphism algebra. We use the defining relations for Hecke algebras to give a presentation for E(W) over a certain ring and speculate an extension to the integral case which is the key ingredients to categorify E(W) via singular Soergel bimodules.

We are going to recall the definition of partial entropy along expanding foliation, and its regularity with respect to maps.

Consider the approximation of Navier-Stokes equations for a Newtonian fluid by Euler type systems with relaxation. This requires to decompose the second-order derivative terms of the velocity into first-order terms. We use Hurwitz-Radon matrices for this decomposition. We prove the convergence of the approximate systems to the Navier-Stokes equations locally in time for smooth initial data and globally in time for initial data near constant equilibrium states.

Online optimization where the cost function is gradually revealed has a lot of applications in machine learning. Recently, there has been interest in distributed online optimization due to the widespread wireless connection and the emergence of internet of things.

We will discuss recent developments in the representation theory of Lie algebras of polynomial vector fields on algebraic varieties (based on joint results with Y.Billih, J.Nilssen and A.Zaidan).

We present some work on analysis of elliptic PDEs of non-local types. And we give a brief introduction on: some basic maximum principles, existence and uniqueness, classi_cations and Liouville type theorems, Bocher type theorems and the related maximum principles for singular solutions. The focus is on the ideas and related basic estimates behind these research work. We will also discuss some basic estimates for linear equations.

I will survey some recent progress on differential calculus of quivers with potentials and give some applications of it to algebraic geometry and representation theory.

We propose an inexact proximal augmented Lagrangian framework with explicit inner problem termination rule for composite convex optimization problems.