Kazhdan-Lusztig cells in the weighted Coxeter groups was first introduced systematically by Lusztig in 2003. He propose a bundle of conjectures on the cells in order to generalize some results from the equal parameter case to the unequal parameter case. In this talk, I intend to give a brief introduction on the topic and report some recent achievements concerning the description of cells.

We present some results concerning the solutions of $$ /Delta u +K(x) e^{2u}=0 /quad{/rm in}/;/; /mathbb{R}^2 $$ with $K/le 0$. We introduce the following quantity: $$/alpha_p(K)=/sup/left/{/alpha /in /R:/, /int_{/R^2} |K(x)|^p(1+|x|)^{2/alpha p+2(p-1)} dx<+/infty/right/}, /quad /forall/; p /ge 1.$$ Under the assumption $({/mathbb H}_1)$: $/alpha_p(K)> -/infty$ for some $p>1$ and $/alpha_1(K) > 0$, we show that for any $0 < /alpha < /alpha_1(K)$, there is a unique solution $u_/alpha$ with $u_/alpha(x) = /alpha /ln |x|+ c_/alpha+o/big(|x|^{-/frac{2/beta}{1+2/beta}} /big)$ at infinity and $/beta/in (0,/,/alpha_1(K)-/alpha)$.Furthermore, we show an example $K_0 /leq 0$ such that $/alpha_p(K_0) = -/infty$ for any $p>1$ and $/alpha_1(K_0) > 0$, for which we prove the existence of a solution $u_*$ such that $u_* -/alpha_*/ln|x| = O(1)$ at infinity for some $/alpha_* > 0$, but does not converge to a constant at infinity.

In this talk, we firstly prove that every hyper-Lagrangian submanifold L^{2n}(n > 1) in a hyperkaehler 4n-manifold is a complex Lagrangian submanifold. Secondly, we study the geometry of hyper-Lagrangian surfaces and demonstrate an optimal rigidity theorem with the condition on the complex phase map of self-shrinking surfaces in R^4 . Last but not least, we show that the mean curvature flow from a closed surface with the image of the complex phase map contained in S^2/(S^1_{+}) in a hyperkaehler 4-manifold does not develop any Type I singularity. This is a joint work with Dr. Linlin Sun.

In this talk, we first show that the focal submanifolds of isoparametric hypersurfaces with g＝4 distinct principal curvatures in the unit sphere are Willmore submanifolds of the sphere. Furthermore, we classify which of them are Einstein or Einstein-like. As a byproduct, we give simply connected examples of the Besse problem.

We introduce a recent joint work with Prof. Zizhou Tang on nonnegative polynomials induced from isoparametric polynomials. We completely solve the question that whether they are sums of squares of polynomials, giving infinitely many explicit examples to Hilbert's 17th problem as well as some applications.

We will introduce the L_p dual surface measure (recently defined by Huang- Lutwak-Yang-Zhang in Adv. Math. 2018). Then we will discuss the related L_p dual Minkowski problems in integral and convex geometry.

In this talk, we recall how to solve Minkowski problems by using geometric flows, such as Gauss curvature flow. In particular, a recent joint work, the regularity of Lp dual Minkowski problem with Chuanqiang Chen, Yiming Zhao will be particularly discussed.

We develop a constructive approach to estimating sparse, high-dimensional linear regression models. The approach is a computational algorithm motivated from the KKT conditions for the -penalized least squares solutions. It generates a sequence of solutions iteratively, based on support detection using primal and dual information and root finding. We refer to the algorithm as SDAR for brevity. Under a sparse Rieze condition on the design matrix and certain other conditions, we show that with high probability,the estimation error of the solution sequence decays exponentially to the minimax error bound in steps; and under a mutual coherence condition and certain other conditions, the estimation error decays to the optimal error bound in $O(/log(R))$ steps,where is the number of important predictors, is the relative magnitude of the nonzero target coefficients.