Publications and Preprints

1. An index formula for families of end-periodic Dirac operators. Submitted. 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 work I established an index formula for families of end-periodic Dirac operators using heat equation methods and renormalized integrals. As a continuation of that work, I'm currently investigating families Seiberg-Witten invariants via families of end-periodic Dirac operators.

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 study positive scalar curvature metrics and to distinguish smooth structures.