Publications and Preprints

1. An index formula for families of end-periodic Dirac operators. Preprint: arXiv:2508.06029.
2. On a General Class of A-functions. With J. Burnett. Preprint: arXiv:2202.13512.

Talks

• Families index for end-periodic Dirac operators and gravitational anomalies, GLaMP 2025, University of Kentucky, June 6, 2025. Slides
• An index formula for families of end-periodic Dirac operators, Geometric Analysis Seminar, University of Illinois Urbana-Champaign, May 6, 2025. Slides
• A new end-periodic Bismut-Cheeger eta form, GSTGC 2025, University of Indiana Bloomington, April 13, 2025.

Research Summary

My current research is centered around the index theory of Dirac operators on non-compact spaces. In my thesis I established an index formula for families of end-periodic Dirac operators using heat equation methods and renormalized integrals. In the future I would like to investigate gauge-theoretic applications of index theory, and I am also interested in branching out to other problems in the intersection between geometry and analysis.

Broadly speaking, my research is driven by the principle that one can understand geometric and topological structure by studying differential equations, either by analyzing the solution space directly or through invariants such as the Fredholm index. Recently we have seen a proliferation of interesting geometric invariants constructed from PDEs, including spectral invariants like the eta invariant, and gauge-theoretic invariants like Seiberg-Witten, which have been used to distinguish smooth structures and study positive scalar curvature metrics. Along these lines, I am currently investigating gauge-theoretic applications of the index formula for families of end-periodic Dirac operators.

In addition to these topics, I am also interested in branching out to other problems at the interface between geometry and analysis. Index theory has connections with and leads naturally into several other areas. To name a few: pseudodifferential operators and noncommutative trace residues, differential K-theory, Hochschild cohomology and noncommutative geometry, and applications of Ricci flow to the 11/8-conjecture.