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

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AbstractAward: DMS 0739392Principal Investigator: John W. Morgan, Mikhail G. Khovanov,Walter D. Neumann, Peter S. OzsvathLow-dimensional topology is undergoing a revolution thanks to aninfusion of new techniques from other areas, includingmathematical physics, representation theory, holomorphic curvetheory, analysis, and algebraic geometry. This fusion oftechniques has led to an explosion of results and also newavenues of research. Examples of this include Perelman's recentproof of the Poincare conjecture using methods from geometricanalysis; Donaldson's gauge theory approach to constructinvariants for smooth four-manifolds, and the subsequentdiscovery of the Seiberg-Witten equations; invariants of threeand four-manifolds defined via holomorphic curves; the quantumpolynomial invariants for knots and, more recently, theircategorifications. This project aims to train the nextgeneration of researchers in these exciting and vibrant subjects,including graduate students and postdocs; and also to widelydisseminate the latest discoveries in these fields.The aim of this proposal is to stimulate the research trainingenvironment at Columbia University in the new developments inlow-dimensional topology and geometry. Many of these developmentshave been pioneered by faculty at Columbia. The objectives ofthis program are: (i) to increase the number of undergraduateswho will pursue graduate studies in this exciting and active areaof mathematics, and to ensure that their training equips themwell for graduate studies; (ii) to broaden and strengthen theresearch background of the graduate students at Columbia, so thatthey are better equipped to fulfill their research potential, andcontribute meaningfully to this rapidly-changing subject; (iii)to prepare the postdoctoral associates for more independentresearch in this area; (iv) to foster communication among allstrata in the research group at Columbia, and in fact to trainthe associated students and postdocs in expository skills,teaching, and research; and (v) to foster the dissemination ofknowledge in this field from Columbia University to the widermathematical community.