Floer theory and topology in dimension 3.5

Topology is the study of shapes and spaces. One way to study spaces is to cut them up into simpler pieces. Depending on what space you start with, this decomposition might not always be possible. To study when it is possible, topologists use tools called invariants. This project will use algebraic invariants of three-dimensional spaces to answer the question of when a high-dimensional space can be cut into triangles, as well as to study knots and surfaces. The principal investigator (PI) will mentor graduate students and postdocs in topology, and will also organize conferences, workshops, and seminars in the field. The PI plans to use tools from Floer homology to answer questions about topology. One project uses algebraic tools to give a simpler characterization of when a high-dimensional topological manifold is triangulable. Another project deals with knots and ribbon concordances between them. Lastly, the PI studies properties of Heegaard Floer homology of 3-manifolds. These projects will contribute to the field’s understanding of smooth and topological manifolds in a range of dimensions. 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.