Gauge Theory, Floer Homology, and Topology

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This award provides travel support to U.S. participants in the conference Tehran Topology 2018, held at the Institute for Research in Fundamental Sciences in Tehran, Iran, on June 26-28, 2018. The conference focuses on recent applications of gauge theory and Floer homology to three- and four-dimensional topology, concordance, and contact topology. It features twelve research lectures, as well as a problem session and a panel discussion on graduate programs and careers in mathematics in the U.S. and Europe. The topic, namely, the large-scale geometry of three- and four-dimensional spaces, is a key area of basic research in pure mathematics, with a highly international research community. All completed research in the area is publicly available through peer-reviewed journals, and most is freely available world-wide on arXiv.org. Conferences such as this facilitate in-person discussions and collaborations between geographically disparate research groups, and U.S. participation is essential to maintain leadership in the field. The meeting also serves as an opportunity to identify and potentially recruit top young researchers from other nations. The topics of Floer-theoretic invariants, concordance, smooth four-dimensional topology, and contact topology have become inextricably entwined in modern mathematics. There continues to be dramatic progress in these areas. One area of recent progress, stemming from the disproof of the hundred-year-old Triangulation Conjecture, is using equivariant Floer theory to study problems in concordance and homology cobordism. These techniques have made accessible questions about homology cobordism that seemed far out of reach ten years ago. Another active area, targeted by the conference, is the topological meaning of Floer homology. Related to concordance is the topology of smooth, closed 4-manifolds, in which recent progress includes both beautiful new obstructions and constructions. In a related direction but in the contact category, the last five years have also seen breakthroughs in understanding Lagrangian fillings of Legendrian knots, via Floer theory, contact topology, and open books. The conference will present new results in these areas and innovative ideas for approaching these problems. The conference website is http://math.ipm.ir/gt/TT2018.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.