EMSW21-RTG: New Techniques in Low-Dimensional Topology and Geometry

Columbia University, New York NY

Investigators

Abstract

Abstract Award: DMS 0739392 Principal Investigator: John W. Morgan, Mikhail G. Khovanov, Walter D. Neumann, Peter S. Ozsvath Low-dimensional topology is undergoing a revolution thanks to an infusion of new techniques from other areas, including mathematical physics, representation theory, holomorphic curve theory, analysis, and algebraic geometry. This fusion of techniques has led to an explosion of results and also new avenues of research. Examples of this include Perelman's recent proof of the Poincare conjecture using methods from geometric analysis; Donaldson's gauge theory approach to construct invariants for smooth four-manifolds, and the subsequent discovery of the Seiberg-Witten equations; invariants of three and four-manifolds defined via holomorphic curves; the quantum polynomial invariants for knots and, more recently, their categorifications. This project aims to train the next generation of researchers in these exciting and vibrant subjects, including graduate students and postdocs; and also to widely disseminate the latest discoveries in these fields. The aim of this proposal is to stimulate the research training environment at Columbia University in the new developments in low-dimensional topology and geometry. Many of these developments have been pioneered by faculty at Columbia. The objectives of this program are: (i) to increase the number of undergraduates who will pursue graduate studies in this exciting and active area of mathematics, and to ensure that their training equips them well for graduate studies; (ii) to broaden and strengthen the research background of the graduate students at Columbia, so that they are better equipped to fulfill their research potential, and contribute meaningfully to this rapidly-changing subject; (iii) to prepare the postdoctoral associates for more independent research in this area; (iv) to foster communication among all strata in the research group at Columbia, and in fact to train the associated students and postdocs in expository skills, teaching, and research; and (v) to foster the dissemination of knowledge in this field from Columbia University to the wider mathematical community.