Topology Between Dimensions Three and Four

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Topology is the study of shapes. Low-dimensional topology refers to the study of 3- and 4- dimensional spaces, and curves and surfaces inside of them. Dimensions 3 and 4 are of particular interest because they are high enough to allow sufficiently complex phenomena (unlike dimensions 1 and 2, which are relatively well understood), yet not so high that interesting things become uninteresting (which is the case in dimensions 5 and above). Fundamental questions in low-dimensional topology include: How many times must a knot pass through itself in order to become untangled? What can 3-dimensional spaces tell us about higher dimensional triangulations? The PI plans to investigate such questions, and to mentor graduate students and postdocs in this field of study. She will also organize conferences, workshops, and seminars, with an aim to broadening the participation of women and members of historically underrepresented groups.The PI plans to use Heegaard Floer homology to study low-dimensional topology, in continuation of an established program. Knot Floer homology provides unknotting bounds, and bordered Floer homology is well suited for studying satellite knots. The PI will use these tools in tandem to investigate the unknotting number of satellite knots. She also proposes to use involutive Floer homology to refine Manolescu’s disproof of the high dimensional triangulation conjecture, giving a simple characterization of when a topological manifold is triangulable. Lastly, she plans to study the differences between the integral and rational knot concordance and homology cobordism groups.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.