EMSW21-RTG: Geometry and Topology

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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.