Research
I’m a low-dimensional topologist with a special interest in dimension four. I’m particularly interested in the development and application of the theory of trisections of four-manifolds. The tools I use most often are trisections, Floer cohomology theories, and their sheaf-theoretic counterparts.
Teaching and mentoring
In the summer of 2021, I participated as a mentor for the topology REU hosted by UVA. My primary responsibilities were two-fold: I wrote an introductory minicourse on classical knot theory, and I supervised a group of six students (four undergraduate partipants and two graduate student mentors) through a research project which studied the Alexander invariants of knotted 2-spheres embedded in four-space.
An introduction to smooth four-manifold topology with an eye towards the applicationss of trisections and the constructions of small exotic four-manifolds.
A continuation of Calc I and II, this course is about functions of several variables. Topics include finding maxima and minima of functions of several variables/surfaces and curves in three-dimensional space/integration over these surfaces and curves. Additional topics: conservative vector fields/Stokes’ and the divergence theorem/how these concepts relate to real world applications. Prerequisite: MATH 1320 or the equivalent.
Experiences
Supervised by Vivek Shende
Supervised by Thomas E. Mark
While a graduate student at UGA, I also enjoyed spending the academic year 2019-2020 in the guest program at the Max Planck Institute for Mathematics in Bonn, Germany.
Publications and preprints
My work is centers on developing the theory of trisections of four-manifolds, as introduced by Gay and Kirby. So far, this has meant applying techniques of modern symplectic geometry in the form of Heegaard Floer and quilted Floer cohomology to this geometric situation.
Given a smooth four-manifold with connected boundary and equipped with a trisected Morse 2-function we demonstrate how to compute the relative invariant of Ozsvath and Szabo in the setting of Heegaard Floer homology.
One can associate to any codimension two embedding its unit conormal lift. Here, we study the microlocal sheaf category of Kashiwara and Schapira associated to tangles and knotted surfaces in four-space, and we prove a homotopy limit formula which computes these categories using a local-to-global approach.
Cimansoni and Conway have constructed an Alexander-Burau 2-functor which takes values in a 2-cospan category of modules over a Laurent polynomial ring. We apply their construction in the setting of bridge-trisected 2-knots. Our main result realizes the Alexander module of a 2-knot as a categorical trace in the sense of Kapranov.
We show that the Lasagna fillings of Morrison-Walker-Wedrich constitute a homotopy operad in the form of a dendroidal space. See, for example:
Equity in Mathematics
I do my best to incorporate research-based teaching practices in support of equity and diversity within mathematics. Listed below are a few links which I’ve found useful and enlightening:
Seminar Invitations
A list of my invited seminar talks. Most recently, I’ve focused on discussing my work with Honghao Gao on applying microlocal sheaf theory to bridge-trisected surface knots.
University of Georgia
We adapt Seidel’s construction from Lefschetz fibrations to broken Lefschetz fibrations which relates Heegaard Floer homology to Perutz’s Lagrangian matching invariants.