在这个报告中，我们首先介绍分形几何中Hausdorff测度与Hausdorff维数的概念并介绍如何求解古典三分Cantor集的Hausdorff维数，然后简单介绍我们近期在Beta展式研究中所取得的部分结果。

In this talk, I will present our recent numerical methodology studies for unsteady moving interface problems and applications to dynamic fluid-structure interaction (FSI) problems. Our numerical methodologies include the arbitrary Lagrangian−Eulerian (ALE) method, the distributed Lagrange multiplier/fictitious domain (DLM/FD) method and their combinations with the mixed finite element method. A fully coupled (monolithic) mixed finite element approximation is developed for all numerical methodologies to unconditionally stabilize numerical computations for moving- interface and FSI problems. Numerical analyses on the well-posedness, stability and convergence are carried out for the proposed monolithic ALE and DLM/FD methods when they are applied to various moving−interface problems.

In this talk, we will introduce two classes of Hamiltonian systems: LV Hamiltonian system in R^4 and Kolmogorov Hamiltonian systems in R^2 . Global dynamics is given for the LV Hamiltonian system in R^4 and all kinds of centers are obtained for a Kolmogorov Hamiltonian system in R^2. Moreover, the topological structures of a class of real hypersurfaces in R_+^4 and real algebraic curves in R^2 are given.

Forecasting and predictive inference are fundamental data analysis tasks. Most studies employ parametric approaches making strong assumptions about the data generating process. On the other hand, while nonparametric models are applied, it is sometimes found in situations involving low signal to noise ratios or large numbers of covariates that their performance is unsatisfactory. We propose a new varying-coefficient semiparametric model averaging prediction (VC-SMAP) approach to analyze large data sets with abundant covariates. Performance of the procedure is investigated with numerical examples.

首先，我将做一个关于机器学习和数据科学的概述，及其在研究风力发电机的最佳选址方案，太阳能发电预测，以及我如何使用机器学习来帮助克莱斯勒来大幅提高铸件生产线成品率方面的应用。此外，我将提出一个全新的用机器学习来快速预报偏微分方程数值解的范例。我们将以机器学习的方法来预报各向异性的椭圆方程在任意几何区域，任意边界条件，任意右端项，和任意各向异性的传导率下的数值解。实时预测材料的失效和断裂是一项具有挑战性的任务。基于peridynamics的LAMMP包提供了一个强大的仿真工具。但计算成本高，不能实现材料失效和断裂的实时预测。在这一挑战的激励下，我们采用了一种深度机器学习工具来预测基于动态模拟数据的材料损伤模式。特别地，我将通过一个磁盘损坏问题来演示我们的方法:使用深度机器学习工具来预测向前问题的磁盘损伤模式

Quantum K-theory of G/P is a K-theoretic version of quantum cohomology of G/P, and has a basis of so-called Schubert classes. The structure constants are combinations of K-theoretic Gromov-Witten invariants, and conjecturally satisfy positivity propery up to a sign depending on the dimension of the corresponding Schubert varieties. In this talk, we will discuss the summation of Schubert structure constants. This is my joint work with Anders S. Buch, Sjuvon Chung, and Leonardo C. Mihalcea.

In this presentation, we will describe the relationship between various positivity notions in differential geometry and algebraic geometry. We shall also introduce a new concept called "RC-positivity" in differential geometry and use it to characterize uniruled and rationally connected projective manifolds. In particular, we confirm a conjecture of Yau that a compact Kahler manifold with positive holomorphic sectional curvature is projective algebraic and rationally connected.

In this talk I will introduce the computation of the mapping class group of a class of 6-dimensional manifold. This class of manifolds includes 3-dimensional complex complete intersections in complex projective spaces, especially the quintic Calabi-Yau 3-fold. I will compare the result with the classical mapping class group of surfaces. This is a joint work with M.Kreck.