第五届全国计算机数学学术会议

报告题目：我在非标准分析方面的一个工作

摘要：

报告题目：分片代数簇中的一些论题

摘要： “代数簇”是代数几何的主要研究对象之一，它研究多元多项式曲面公共零点集的几何性质。由于现代曲面主要采用分片多项式（多元样条）来刻画。 我们从多元样条理论及应用研究出发，提出并研究了由多元分片多项式曲面公共交点集合所界定的“分片代数簇”。它是经典代数簇的推广，并在计 算机科学中曲面表示与设计、曲面拼接，求交以及图像处理等问题上有广泛应用，因而研究“分片代数簇”有重要的理论和实际意义。由于多元分片 多项式严重依赖于剖分的几何性质、使分片代数簇具有异常复杂的几何结构，故在“分片代数簇”研究中有实质性困难。近年来我们研究了分片代数簇， 包括实分片代数簇，得到了分片代数曲线的最大有限交点数定理；实分片代数曲线与四色定理的等价命题；分片代数曲线的N?ther 型定理， Cayley-Bacharach型定理等一些相关结果。本报告还将介绍一些公开问题。

报告题目：On the Problem of Global Grid Systems

摘要：In astronomy, physics, climate modeling, geoscience, computer graphics and many other disciplines, mass of data usually came from the spherical sampling. An efficient and distortion free representation of spherical data is therefore important. This talk introduces a new kind of spherical (global) coordinates which is free of any singularity. Unlike classical coordinates such as Cartesian and spherical polar systems, this coordinate system is naturally defined on the spherical surface. The base idea of this coordinate system is originated from the classical planar barycentric coordinates that describe the positions of points in a plane with respect to the vertices of a given planar triangle, spherical area coordinates (SAC) describe the positions of points on a sphere with respect to the vertices of a given spherical triangle. In particular, by decomposing the globe into several identical triangular regions and constructing local coordinates for each region and then combine them together, one obtains a global coordinate system. While the SACs have been established, the coordinate iso-lines comprise a new class of global grid system. this kind of grid system has some good properties: the grid cells exhaustively cover the globe and without overlapping, grid cells have the same shape, the grid system has congruent hierarchical structure and simple relationship to the traditional coordinates. Since these good behaviors, they are suitable for organizing, representing and analyzing the spatial data.

报告题目：Schubert calculus and cohomologies of Lie groups

摘要：The problem of computing the cohomologies of Lie groups was raised by E.Cartan in 1929, and has been a focus of algebraic topology for the fundamental role of Lie groups playing in geometry and topology. On the other hand Schubert calculus begun with the intersection theory of the 19th century, and clarifying this calculus was a major theme of the 20 century algebraic geometry. In this talk we bring a connection bwteen these two topics both with distinguished historic background, and demonstrate how Schubert calculus is extended as to give a unified construction of the integral cohomologies of all simply connected Lie groups.

报告题目：Computational Conformal Geometry: Theory, Algorithm and Applications

摘要：Computational conformal geometry is an emerging interdisciplinary field, combing modern geometry with computer science. In mathematics, conformal geometry is the intersection of Riemann surface theory, differential geometry, algebraic topology and Partial differential equation; in computer science, conformal geometry plays fundamental roles in computer graphics, geometric modeling, computer vision, networking and medical imaging.
     Conformal geometry studies the invariants under angle-preserving transformations. Fundamental problems in the field include computing the conformal structure of a metric surface, computing the conformal invariants (conformal modules), computing conformal mappings between surfaces, computing uniformization Riemannian metrics, computing extremal quasi-conformal mappings and so on. The solutions to these problems will be introduced.
     Computational strategies including surface harmonic mapping method, holomorphic differential method and discrete surface Ricci flow. Ricci flow is the process to deform the Riemannian metric proportional to the curvature, such that the curvature evolves according to a heat diffusion process and becomes constant eventually. Surface Ricci flow leads to the uniformization metric without curvature blow up. The discrete surface Ricci flow theory and algorithm will be covered in details.
     Computational Conformal geometry has been applied for many engineering fields, including global parameterization in computer graphics, deformable surface registration in computer vision, manifold splines in geometric modeling, efficient routing in networking, brain mapping and virtual colonoscopy in medical imaging. These applications will be briefly introduced.
     This work is collaborated with Professor Shing-Tung Yau, Professor Feng Luo, Professor Ronald Lok Lui and many other mathematicians, computer scientists and medical doctors.