Research

Papers & Publications

The Structure of the Spin^h Bordism Spectrum, to appear in Proc. Amer. Math. Soc., DOI: 10.1090/proc/16748 (arXiv:2306.17709)

Spin^h manifolds are the quaternionic analogue to spin^c manifolds. We compute the spin^h bordism groups at the prime 2 by proving a structure theorem for the cohomology of the spin^h bordism spectrum MSpin^h as a module over the mod 2 Steenrod algebra. This provides a 2-local splitting of MSpin^h as a wedge sum of familiar spectra. We also compute the decomposition of H^*(MSpin^h; Z/2Z) explicitly in degrees up through 30 via a counting process.

Almost Complex Structures on Homotopy Complex Projective Spaces, Topology and its Applications 344 (2024) 108810, DOI: 10.1016/j.topol.2023.108810 (arXiv:2201.07176)

We show that all homotopy CP^ns, smooth closed manifolds with the oriented homotopy type of CP^n, admit almost complex structures for 3 ≤ n ≤ 6, and classify these structures by their Chern classes. Our methods provide a new proof of a result of Libgober and Wood on the classification of almost complex structures on homotopy CP^4s.

Conference & Seminar Talks

The Structure of the Spin^h Bordism Spectrum, AMS Contributed Paper Session on Algebraic Topology and Manifolds (II), Joint Mathematics Meetings, January 2024
The Structure of the Spin^h Bordism Spectrum, The South Central Topology Conference III, New Mexico State University, October 2023
Almost Complex Structures on Homotopy Complex Projective Spaces, UO Topology/Geometry Zoom Seminar, University of Oregon, April 2023
Almost Complex Manifolds and Homotopy Projective Spaces, CIRGET Junior Seminar, Université du Québec à Montréal, February 2023
Almost Complex Manifolds and Homotopy Complex Projective Spaces, Graduate Student Geometry and Topology Conference, Georgia Tech, April 2022
Almost Complex Structures on Homotopy Complex Projective Spaces, UMD Faculty Geometry-Topology Seminar, University of Maryland, April 2022

Notes & Exposition

Some notes I've written while trying to understand various topics, and some expository writing that ended up as final papers for some of my coursework. Unless mentioned otherwise, almost none of the material is original; all errors are mine.

My thesis presentation for my dissertation defense in March 2024.
Talk notes for "The Structure of the Spin^h Bordism Spectrum" for a 15-minute talk at JMM 2024 on one of my research projects. A similar set of notes were used for a 12-minute "rodeo talk" on this project at the South Central Topology Conference in October 2023.
Almost Complex and Almost Contact Manifolds and the corresponding talk notes, expository writing on almost complex/contact/symplectic manifolds that came out of writing a final paper (and a 15-minute final presentation) for a topics course on Symplectic Geometry taught by Dr. Daniel Cristofaro-Gardiner in Spring 2022. The talk notes are also linked on the course website's lecture summaries for the last day of class.
The class of almost complex manifolds is closed under connect sum with homotopy spheres, an argument from June 2021 answering a question posed by my advisor. (I don't know if this argument exists elsewhere, but it's one page and not difficult.)
Simplicial Sets and Classifying Spaces, a final paper for a topics course on Scissors Congruences taught by Dr. Christian Zickert in Spring 2020.
h-Cobordism and the Generalized Poincaré Conjecture, notes for a talk I gave in the UMD Student Geometry-Topology Seminar in Spring 2020.
Low-Dimensional Unoriented Cobordism, notes for a talk I gave in the UMD Student Geometry-Topology Seminar in Fall 2019.