Mathematical Sciences: Topology, Geometry and Physics

9626698自由这笔赠款支持德克萨斯大学奥斯汀分校的全球分析和微分几何小组。拟议的研究集中在数学物理的分析和代数方面。研究课题包括：Chern-Simons理论和拓扑量子场论的结构；无限维可积系统和非线性薛定谔方程；符号动力学和分支流形；非相对论量子力学中的几何结构。该项目还包括涉及本科生和高中生的教育活动。量子场论是理论和数学物理学中的一个活跃领域：最终，该领域的研究人员希望能提出一个严格而统一的框架来解释各种基本的自然力。可积系统构成了动力系统理论的一个重要分支--动力系统由常微分方程组给出；在这样的系统中，系统随时间演化的速率与时间无关。动力系统被用来模拟各种自然现象，如人口增长和天气系统。

9626698 Freed This grant supports the global analysis and differential geometry group at the University of Texas at Austin. The proposed research concentrates on the analytic and algebraic aspects of mathematical physics. Research topics to be pursued include: Chern-Simons theory and the structure of topological quantum field theory; infinite dimensional integrable systems and nonlinear Schrodinger equations; symbolic dynamics and branched manifolds; geometrical structures in nonrelativisitic quantum mechanics. The project also includes educational activities involving undergraduate students and high school students. Quantum field theory is an active area within theoretical and mathematical physics: ultimately, the researchers in this field hope to come up with a rigorous and unifying framework explaining various basic forces of nature. Integrable systems constitute an important subarea within the theory of dynamical systems - a dynamical system is given by a system of ordinary differential equations; in such a system the rate at which the system evolves over time is independent of time. Dynamical systems are used to model various natural phenomena such as population growth and weather systems.