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).

τ-tilting theory was introduced by Adachi, Iyama and Reiten, which completes the classical tilting theory from the viewpoint of mutation. They constructed a class of Λ modules named (support) τ-tilting modules, where Λ is a finite dimensional basic algebra over an algebraically closed field. We call Λ a τ-tilting finite algebra if there are finitely many isomorphism classes of basic τ-tilting Λ-modules. In this talk, we explain τ-tilting theory by discussing the τ-tilting finiteness of two-point algebras. More precisely ,we will show the poset structure on the set of support τ-tilting modules, the bijection between support τ-tilting modules and two-term silting complexes, etc,.

在本报告中，我将给出九维流形上存在切触结构的充分必要条件。该结果是与墨尔本大学Diarmuid Crowley合作完成的。

Hamilton's conjecture in the Ricci flow says that the scalar curvature of the finite-time Ricci flow blows up. In this talk under a curvature condition we give a proof of this conjecture.

Kahler tensor calculus could be formulated in terms of directed graphs. We use to it explore the algebraic relations of invariant differential operators on bounded symmetric domains. I will also talk about the Feynman diagram formulas for the coefficients in the asymptotic expansion of Bergman and heat kernels.

We develop a Fulton-style intersection theory in motivic homotopy categories, which is expressed in terms of the fundamental class in twisted bivariant groups. By introducing a general notion of Euler classes of vector bundles, we prove a formula of excess intersection in this framework; as an application, we deduce a motivic Gauss-Bonnet formula computing the Euler characteristic. This is a joint work with F. Déglise and A. Khan.

In this talk, we report the results of robust tensor completion using tubal singular value decomposition, and its applications. Several applications and theoretical results are discussed. Numerical examples are also presented for demonstration.