We discuss a novel multi-phase upscaling technique, which uses rigorous multiscale concepts based on Constraint Energy Minimization (CEM-GMsFEM). CEM-GMsFEM concepts use local spectral problems to design multiscale basis functions, which are supported in oversampled regions. The coarse-grid solution using these basis functions provides first-order accuracy with respect to the coarse-mesh size independent of high permeability contrast. The degrees of freedom in multiscale methods represent the coordinates of the solution in the multiscale space. To design an upscaled model, we modify these basis functions such that the degrees of freedom have physical meanings, in particular, the averages of the solution in each continua.

The dual Minkowski problem asks for the necessary and sufficient conditions on a given measure $/mu$ so that it can be realized as the dual curvature measure of a convex body. Just like the classical Minkowski problem, the dual Minkowski problem reduces to Monge-Ampere type equation. However, in this talk, we shall solve the dual Minkowski problem for $o$-symmetric convex bodies directly for measures using variation of calculus. The solution is closely connected to a measure concentration condition.

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.