Floer for Three: Symplectic Methods in Low-Dimensional Topology

University Of Oregon Eugene, Eugene OR

Investigators

Abstract

Research under this grant furthers our understanding of curves, surfaces, 3-dimensional spaces, and 4-dimensional spaces, by focusing on the interplay between these different dimensions. Relationships between different dimensions arise in two ways. One way to study complicated 3- or 4-dimensional spaces is to decompose them into simpler pieces; the decomposition happens along lower-dimensional spaces, like surfaces. These are complicated versions of the principle that you cut a 3-dimensional watermelon by using a flat, 2-dimensional knife. Trying to do this leads one to ask in which ways low-dimensional spaces can sit inside high-dimensional ones. For example, 1-dimensional shoelaces can be knotted or unknotted; it was recently discovered that a 3-dimensional watermelon sitting in 4-space can also be knotted. This involves developing aspects of differential equations and abstract algebra. One long-term goal of the research project is to have a computer program that can compute certain subtle invariants of 4-dimensional spaces coming from counting solutions to differential equations from theoretical physics, the Seiberg-Witten equations. The grant will also support training graduate students in these topics, writing a book introducing some of them to graduate students and advanced undergraduates, outreach activities to share the excitement of geometry and topology with K-12 students, and a website to help other researchers use non-photorealistic raytracing to draw useful figures in their own papers. The research involves a number of specific projects. One is to further develop an extension of bordered Heegaard Floer homology to the "minus" version of Heegaard Floer homology. Bordered Heegaard Floer homology is a version of Heegaard Floer homology for 3-manifolds with boundary; many aspects of the theory are currently only defined for the simpler "hat" version of Heegaard Floer homology. Another is to develop new invariants of embedded non-orientable surfaces in 4-space, using variants on Khovanov homology. A third is to apply Floer homotopy theory to questions in equivariant 3-dimensional topology, and a fourth is to study relationships between Floer homology and the mapping class groups of surfaces. 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.