Determining the behavior of viscous flows at small viscosity is one of the most fundamental problems in fluid mechanics. In this talk, we focus on the incompressible flow and study the vanishing viscosity limit of incompressible Navier-Stokes equations. It mainly involves the following contents:

An example of Apostolovet al. indicates that the condition of K-stability may not be correct one for general polarized manifolds. Szekelyhidi modified definition of K-stability by filtration and stated a variant of the Yau-Tian-Donaldson conjecture. We will talk about our proof of this variant of YTD conjecture for toric manifolds and homogeneous toric bundles. This is jointed with Li An-Min and Lian Zhao.

It has been a longstanding open problem to find a direct conservation law for harmonic mappings into manifolds. In the late 1980s, Chen and Shatah independently found a conservation law for weakly harmonic maps into spheres, which can be interpreted by Noether's theorem. This leads to Helein's celebrated regularity theorem on weakly harmonic maps from surfaces. For general target manifolds, Riviere discovered a direct conservation law in two dimension in 2007, allowing him to solve two well-known conjectures of Hildebrandt and Heinz. As observed by Riviere-Struwe in 2008, due to lack of Wente's lemma, Riviere's approach does not extend to higher dimensions. In a recent joint work with Chang-Lin Xiang, we successfully found a conservation law for a class of weakly harmonic maps into general closed manifolds in higher dimensions.

In this talk, we consider a family of character sums as multiplicative analogues of Kloosterman sums. We establish asymptotic formulae for any real (positive) moments of the above character sum as the character runs over all non-trivial multiplicative characters mod p, confirming a conjecture of Professor Wenpeng Zhang from 2002. Moreover, an arcsine law will be established as a consequence of the method of moments in probability theory. The arguments, amongst other tools, also allow us to obtain asymptotic formulae for moments of such character sums weighted by special L-values (at 1/2 and 1). The tools will include Gauss sums, hyper-Kloosterman sums, as well as a recent estimate for bilinear forms with general algebraic trace functions due to Fouvry, Kowalski and Michel.

Kaehler metrics with large symmetry on a toric manifold correspond to a class of convex functions on the whole euclidean space. This allows us to use more special techniques from convex analysis in the study of Kaehler geometry on toric manifolds. In this talk, we will first discuss various approaches in the study of Kaehler-Einstein metrics on toric Fano manifolds. Then we discuss some recent work, in particular on Kaehler-Ricci flow on G-manifolds by the method used for toric manifolds. Some questions/problems are also discussed.

对数凹(log-concave)这个概念在凸几何与泛函不等式的研究中扮演着重要的角色。报告人将从对数凹测度谈起，结合最优传输理论，介绍随机局部化方法在泛函不等式研究中的应用。

We establish the validity of the Euler+Prandtl approximation for solutions of the Navier-Stokes equations in the half plane with Dirichlet boundary conditions, in the vanishing viscosity limit, for initial data which are analytic only near the boundary, and Sobolev smooth away from the boundary. Our proof does not require higher order correctors, and works directly by estimating an L1 -type norm for the vorticity of the error term in the expansion Navier-Stokes−(Euler+Prandtl). An important ingredient in the proof is the propagation of local analyticity for the Euler equation, a result of independent interest.

In this talk, we will study C^{1, 1} regularity for solutions to the degenerate Lp Dual Minkowski problem and the existence of the smooth solution to the Gauss image.腾讯会议号：131-965-267会议链接：https://meeting.tencent.com/dm/iD9IGmFJEu19 All are ...