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.

In many longitudinal studies, repeated response and predictors are not directly observed, but can be treated as distorted by unknown functions of a common confounding covariate. Moreover, longitudinal data involve an observation process which may be informative with a longitudinal response process in practice. To deal with such complex data, we propose a class of flexible semiparametric covariate-adjusted joint models. The new models not only allow for the longitudinal response to be correlated with observation times through latent variables and completely unspecified link functions, but they also characterize distorted longitudinal response and predictors by unknown multiplicative factors depending on time and a confounding covariate.

Theta functions are special functions of two complex variables, which is a central subject in the study of mathematics. Theta functions play a key role in many branches of mathematics, including algebraic geometry, number theory, modular forms, partial differential equations and combinatorics. In this talk, I will give a brief introduction to theta functions with emphasis on theta function identities.