Research

I work on topology, the large-scale study of shapes. If you are not a mathematician, these pages are unlikely to have much meaning to you, but others have written many nice introductions to topology, including:

Overview

I work on low-dimensional topology, using tools from partial differential equations (J-holomorphic curves), homological algebra (Khovanov homology), and homotopy theory. Much of my work is focused on developing these tools rather than applying them to specific questions in low-dimensional topology. Most of my work can be divided into a few themes:

  • Work on Heegaard Floer homology, particularly bordered Heegaard Floer homology, an extension of Heegaard Floer homology to 3-manifolds with boundary.
  • Developing and applying the Khovanov stable homotopy type, a refinement of Khovanov homology.
  • Studying equivariant Floer homology and Khovanov homology, in the presence of symmetries.

These topics are not disjoint; for example, the Khovanov stable homotopy type is useful for studying equivariant Khovanov homology, and bordered Heegaard Floer homology can be used to study equivariant Heegaard Floer homology.

More information

See the dedicated pages giving a list of papers, code used in those papers, and my recent and upcoming talks.

Funding

My research has been made possible by generous support for basic research. Specifically it has been supported by a National Science Foundation (NSF) Graduate Research Fellowship, an NSF Mathematical Sciences Postdoctoral Research Fellowship, a Sloan Research Fellowship, a Simons Foundation Fellowship in Mathematics, a Simons Foundation Travel Grant for Mathematicians, and NSF Grants DMS-0905796, DMS-1149800, DMS-1560783, DMS-1642067, DMS-1810893, DMS-2204214, and DMS-2505715. The support specific to each paper is listed in that paper.

Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.