报告题目：The linear instability of some families of Einstein metrics
报告摘要：I will talk about the linear stability problem of Einstein metrics. We proved the linear instability of some Einstein metrics with positive scalar curvature, in particular, including some families of Riemannian manifolds with real Killing spinors, and low dimensional homogeneous non-symmetric Einstein manifolds. This is based on joint works with Uwe Semmelmann and McKenzie Wang.
报告题目：On the automaticity of the Hankel determinants of a family of automatic sequences
报告摘要：Hankel determinants and automatic sequences are two classical subjects widely studied in Mathematics and Theoretical Computer Science. However, these two topics were considered totally independently, until in 1998, when Allouche, Peyri\`ere, Wen and Wen proved that all the Hankel determinants of the Thue-Morse sequence are nonzero. This property allowed Bugeaud to prove that the irrationality exponents of the Thue-Morse-Mahler numbers are exactly 2. Since then, the Hankel determinants of several other automatic sequences, in particular, the paperfolding sequence, the Stern sequence, the period-doubling sequence, are studied by Coons, Vrbik, Guo, Wu, Wen, Bugeaud, Fu, Han, Fokkink, Kraaikamp, and Shallit. On the other hand, it is known that the Hankel determinants of a rational power series are ultimately zero, and the Hankel determinants of a quadratic power series over finite fields are ultimately periodic. It is therefore natural to ask if we can obtain similar results about the Hankel determinants of algebraic series. In the present paper, we provide a partial answer to this question by establishing the automaticity of the reduced Hankel determinants modulo $2$ of a family of automatic sequences. As an application of our result, we give upper bounds for the irratoinality exponent of a family of automatic numbers.
报告题目: Topics in Milman's Problem.
报告摘要：In this talk, I will present a preliminary report of a problem asked by Vitali Milman. This is if there are two convex bodies K and L such that $K+L=K^o+L^o$, is it true that $K=L^o$.
报告题目：On Fano and weak Fano regular Hessenberg varieties
报告摘要：Regular Hessenberg varieties are a family of subvarieties of the full flag variety G/B. This family contains the full flag variety, Peterson variety and permutohedral variety. In this talk, we discuss the Fano and weak Fano regular semisimple Hessenberg varieties in type A. We give a sufficient and necessary condtion for a regular semisimple Hessenberg variety to be Fano and weak Fano. This is joint work with Hiraku Abe and Naoki Fujita.
报告题目：Obstacle problem for a class of IPDEs and applications to stochastic optimal control problem
报告摘要：In this talk, we consider the existence and uniqueness of minimal mild super solutions to the obstacle problem governed by integro-partial differential equations. We first study the well-posedness and locally Lipschitz regularity of L^p solutions (p≥2) to reflected forward-backward stochastic differential equations (FBSDEs) with jump and lower barrier. Then we show that the solutions to reflected FBSDEs provide a probabilistic representation for the mild super solution via a nonlinear Feynman–Kac formula. Finally, we apply the results to study stochastic optimal control/stopping problems.
报告题目：A formula of conditional entropy and some deformations
报告摘要：We established a formula of conditional entropy and introduce a deformation of this formula under some conditions.
报告题目：Finite dimensionality of uniform attractors for smoothing dynamical systems
报告摘要：In this talk we shall introduce a smoothing approach to bound the fractal dimension of uniform attractors. The upper bound is given by the fractal dimension of the symbol space plus an entropy nunber of a compact embedding. A construction of the finite dimensional symbol space of some kinds of nonperiodic functions will also be given.