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。该计划跨越几何和拓扑的广泛领域。极小曲面的理论，它们的结构和更一般的曲面演化。黎曼几何，特别是爱因斯坦度量和度量空间。辛拓扑与镜像对称、低维拓扑的关系。 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。 参与该计划的麻省理工学院教师是这些领域的领导者。该计划，除了加强研究环境，将提供培训和指导，在各级，博士后，研究生和本科生。此外，这笔赠款将允许数学系支持一个本科生项目，为代表性不足的少数民族。几何和拓扑仍然是当今数学的一个主要主题，并积极与其他领域，从数论到物理学的互动。最近进入这门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，该计划支持的数学领域影响了许多具有实际重要性的领域。最小表面和表面的演化在材料科学、化学和化学工程中具有重要的历史意义。 辛拓扑已经应用于理解动力系统，并且是弦理论和镜像对称领域中高能理论物理学当前工作的核心。低维拓扑在生物学问题中有应用，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.