The asymptotic distribution theory for coefficients of a polynomial is an active topic in asymptotic analysis. In 1967, Harper proposed a criterion to measure the asymptotic normality of a series of numbers, when he researched the asymptotic behavior of Stirling numbers of the second kind. In this talk, we will discuss some further asymptotic normality criteria of coefficients of a polynomial with all real roots or purely imaginary roots (including 0). These new asymptotic normality criteria turn out to be very efficient and have abundant applications in combinatorics, mainly including the coefficients of a series of characteristic polynomials of adjacency matrix, Laplacian matrix, signless Laplacian matrix, skew-adjacency matrix, chromatic polynomial

New inf-sup stable mixed elements are proposed and analyzed for solving the Maxwell equations in terms of electric field and Lagrange multiplier. Nodal- continuous Lagrange elements of any order on simplexes in two- and three- dimensional spaces can be used for the electric field. The multiplier is compatibly approximated always by the discontinuous piecewise constant elements. A general theory of stability and error estimates is developed; when applied to the eigenvalue problem, by establishing the key property of discrete compactness, we show that the proposed mixed elements provide spectral-correct, spurious-free approximations.

In this talk, I will introduce the Finsler geometry and the background of the isoperimetric problem. By using the variational theory, we give the local solution of the isoperimetric problem in 2-dimensional Finsler space forms. This is the joint work with Mengqing Zhan.

In this talk, I first recalled some knowledge of the Kahler-Ricci flow. Then I will talk about the behavior of the Kahler-Ricci flow on some Fano bundles which is a trivial bundle on one Zariski open set. We show that if the fiber is $/mathbb{P}^{m}$ blown up at one point and the initial metric is in a suitable kahler class, then the fibers collapse in finite time and the metrics converge sub-sequentially in Gromov-Hausdorff sense to a metric on the base. In the talk, I will aim to talk about in the case of product manifolds. This is a joint work with Xin Fu.

Let N be an (n+1)-dimensional complete manifold of Ricci curvature≥-n, and B_2 (p) be the geodesic ball in N. Let M be an area-minimizing hypersurface in B_2 (p) with p∈M and ∂M⊆〖∂B〗_2 (p). In this talk, we will discuss the Sobolev and Neumann-Poincare inequalities on M∩B_1 (p). As an application, we get the gradient estimates for the solutions of the minimal hypersurface equation on an n-manifold with Ric≥-(n-1).

Column LLT polynomials are well-studied symmetric functions defined by Lascoux, Leclerc and Thibon. Column LLT polynomials are Schur positive, and they have many nice combinatorial properties. We conjecture that column LLT polynomials are e-positive when substituting q with 1+q, and the coefficients of the e-basis expansion can be obtained by constructions of "inequality" graphs.

Let $M$ be a connected topological manifold of dimension at least 2 with an effective action of a finite group $G$. Associated with the orbit configuration space of the $G$-manifold $M$, we give the notion of orbit braids and define the orbit braid group. We further study the subgroups of orbit braid group and various relations among these subgroups with the fundamental groups of the configuration spaces given by the manifold $M$ with or without the $G$-action, and the $M/G$. We also calculate the orbit braid groups in the plane with two typical actions. This talk is based upon the joint work with Hao Li and Fengling Li.

In 1995, Victor Andreevich Toponogov authored the following conjecture: “Every smooth strictly convex and complete classical surface of the type of a plane has an umbilic point, possibly at infinity“. In our talk, we will outline a proof, in collaboration with Brendan Guilfoyle. Namely we prove that (a) the Fredholm index of an associated Riemann Hilbert boundary problem for holomorphic discs is negative, which is an outcome of the regularity of the Cauchy-Riemann operator in presence of a symmetry group. Thereby, (b) no such solutions may exist for a generic perturbation of the boundary condition (these form a Banach manifold under the assumption that the Conjecture is incorrect).