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.
The grant supported a large group of graduate and undergraduate students, postdoctoral and senior faculty in the area of geometry and topology and related fields at Columbia University Mathematics Department. It allowed to develop new courses that are now taught at the Mathematics Department, supported two conferences on geometric group theory and a conference on categorification, and sponsored summer research programs for undergraduates. Partial support provided by the grant to postdocs and graduate students led to several key developments in mathematics, including (a) Aaron Lauda's diagrammatical theory of categorified quantum sl(2) and its extension by Khovanov and Lauda. The Lie algebra sl(2) describes infinitesimal structure of the 3-dimensional world. A suitable symmetry braking allows to quantize this structure, making it noncommutative. Lauda's contribution was to lift this quantization one dimension up (such lift is called categorification) and to provide a diagrammatical presentation for the lifting. This construction has found many applications in mathematics and is expected to be of fundamental value for mathematical physics. (b) Jennifer Hom's fundamental work studying four-dimensional interactions (concordances) between knots and links (3-dimensional objects) and her discovery of a vast new family of obstacles to concordance and related invariants. (c) Ben Elias' diagrammatic construction of the Soergel category, leading to the first elementary proof of the famous Kazhdan-Lusztig conjecture. This theory starts with permutations of elements of a set, linearizes it, allowing additions and scalings in the background, then quantizes it, and finally, lifts it one dimension up (categorification), similar to Lauda's work (a) but for a different object. (d) Ellis-Khovanov discovery of odd symmetric functions. Symmetric functions (quantities invariant under permutations of variables) are indispensable in mathematics from early on, and Ellis-Khovanov discovered the analogue of this functions in the situation when variables anticommute rather than commute (moving one variable past the other introduces a minus sign). Some of these and other recent discoveries by researchers supported by the grant have already acquired significant citation volumes on Google Scholar. They are being actively developed by several groups and many individual researchers. Graduate students supported by the grant co-authored 26 papers while at Columbia University. Postdoctoral fellows supported by the grant co-authored 22 papers while at Columbia. The vast majority of these papers are available to the public, free of charge, on the arXiv.
