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.

In this talk we discuss some monotonicity questions related to Fredholm matrices and operators. A function $f(x)$ is called completely monotonic if $(-1)^mf^{(m)}(x)>0$. It is known that the expectation of having $m$ eigenvalues of a random Hermitian matrix in an interval is a multiple of $(-1)^{m}$ times the $m$-th derivative of a Fredholm determinant at $/lambda=1$. In this work we extend the positivity to half-real line $(-/infty,1]$, and we also study the completely monotonicity of some special functions which arise as Fredholm determinants.

We survey results on q-analogue of the normal distribution and the corresponding orthogonal polynomials.

In this talk, we will investigate a complex short pulse equation and its two-component generalization. First, we will derive them starting from Maxwell equations. Then we will study their various soliton solutions such as bright, dark, breather and rogue wave solutions by using Hirota’s bilinear method and Darboux transformation method.

We generalize the notions of ordinary points and singularities from linear ordinary differential equations to D-finite systems. Ordinary points and apparent singularities of a D-finite system are characterized in terms of its formal power series solutions. We also show that apparent singularities can be removed like in the univariate case by adding suitable additional solutions to the system at hand. Several algorithms are presented for removing and detecting apparent singularities. In addition, an algorithm is given for computing formal power series solutions of a D-finite system at apparent singularities. This is a joint work with Manuel Kauers, Ziming Li and Yi Zhang.