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The existence of Kahler Einstein metric with mixed cone and cusp singularity has attracted many attentions in recent years. In this talk, we show that their L2 cohomologies coincide with the de Rham cohomology of a good compactification (under both Dirichlet and Neumann boundary conditions) and prove that their L2-Hodge -Frolicher spectral sequence give the pure Hodge structure on them. This is a work joint with Junchao Shentu.

Intersection cohomology is introduced by Goresky and MacPherson as a cohomogy theory on singular spaces which admit the Poincare duality. Decades of works show that this cohomology theory share a bunch of good properties such as Lefschetz package, Hodge-Riemann bilinear relation and Hodge theoretic purity. Although the theory is well established in the context of topology and algebra, the de Rham theorem remains open. In this talk I will introduce the basic knowledge of intersection cohomology and present a solution of the de Rham theorem for intersection cohomology when the space is algebraic and admits only equisingularities. This is the joint work with Chen Zhao.

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. This talk is focused on the distributed online optimization problem over a multi-agent network subject to local set constraints and coupled inequality constraints, which has a large number of applications in practice, such as wireless sensor networks, power systems and plug-in electric vehicles. We present a primal-dual algorithm which overcomes the limitations of existing works where balanced graph and boundedness of certain algorithm parameter are assumed which are inappropriate.

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). In particular, we focus on the classification of irreducible representations of the Lie algebra of vector fields on a torus.

We introduce a recent (on-going) work joint with Fagui Li on minimal hypersurfaces in unit spheres, where some integral inequalities and their equality characterizations are obtained.

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.