EMSW21-RTG: Geometry and Topology

该项目旨在促进拓扑学和几何学最高水平的研究和培训。该计划涵盖了几何学和拓扑学的广泛领域。最小表面的理论，它们的结构和更一般的表面演化。黎曼几何，特别是爱因斯坦度规和度规空间。辛拓扑及其与镜像对称的关系，以及低维拓扑。同伦理论及其与数论的紧密联系。规范理论及其在低维拓扑中的应用。参与这个项目的麻省理工学院教师是这些领域的领导者。该计划除了改善研究环境外，还将为各个层次的博士后、研究生和本科生提供培训和指导。此外，这笔拨款将允许数学系为代表性不足的少数族裔开设本科课程。几何和拓扑学仍然是当今数学的一个主要主题，并且积极地与从数论到物理学的其他领域相互作用。最近进入这一学科的新思想有助于解决数学中长期存在的开放性问题。此外，该计划所支持的数学领域影响着许多具有实际重要性的领域。最小表面和表面的演变在材料科学、化学和化学工程中一直很重要。辛拓扑已经应用于理解动力系统，并且是当前高能理论物理在弦理论和镜像对称领域工作的核心。低维拓扑已经应用于生物学问题，特别是量化和理解DNA的打结行为以及物质的拓扑相。

This project aims to promote research and training at the highest level in topology and geometry. The program spans a broad area of geometry and topology. The theory of minimal surfaces, their structure and more generally evolution of surfaces. Riemann geometry, in particular Einstein metrics and the space of metrics. Symplectic topology and relations to mirror symmetry, and low dimensional topology. Homotopy theory and its ever closer ties to number theory. Gauge theory and its applicationsto low dimensional topology. The MIT faculty involved in this program are leaders in these areas. The program, in addition to enhancing the research environment, will provide training and mentoring at all levels, postdocs, graduate students, and undergraduates. In addition the grant will allow the math department to support a undergraduate program for underrepresented minorities.Geometry and topology remains a major theme in present-day mathematics, and actively interacts with other areas ranging from number theory to physics. New ideas that have entered the subject recently have contributed to the solution of long-standing open problems in mathematics. Moreover the areas of mathematics supported by this program impact many areas of practical importance. Minimal surfaces and evolution of surfaces have historically been important for material sciences, chemistry and chemical engineering. Symplectic topology has had applications to understanding dynamical systems and lies at the heart of current work in high energy theoretic physics in the areas of string theory and mirror symmetry. Low dimensional topology has had applications to problems in biology in particular quantifying and understanding the knotted behavior of DNA as well as topological phases in matter.