Topology
Current activities
Studying Vidit Nanda's lecture notes on computational algebraic topology. Having already some familiarity with topological data analysis, I would like to follow this up with an investigation into developing applications of algebraic topology to machine learning.
Doctoral work
My doctoral research, conducted under the invaluable direction of my advisor, Tye Lidman, focused on low-dimensional manifold topology and knot theory, and especially on the applications of Floer theory, in these fields. A major motivation of my work is the Atiyah-Floer conjecture, holding roughly that each instanton Floer homology should have an isomorphic Lagrangian Floer counterpart. My dissertation constructs a Lagrangian Floer homology invariant of knots/links in 3-manifolds via traceless character varieties and studies its connection to Kronheimer & Mrowka's instanton knot homology.
Numerical linear algebra
Reading Randomized Rank-Structured Matrix Compression by Tagging by Katherine J. Pearce, Anna Yesypenko, James Levitt, Per-Gunnar Martinsson. By way of complementing this current research with fundamental background, I've reviewed several explanatory articles on the fast multipole method (FMM) and hope to survey more on matrix compression/interpolative decomposition (ID).