I am an analyst working at the intersection of analysis, probability, and partial differential equations. Much of my research is motivated by the question Who needs positivity? For instance, the central limit theorem describes the asymptotic behavior of the convolution powers of positive real-valued functions on the lattice. When such functions are allowed to be complex valued, new and surprising behavior is observed. One goal of my research is to understand this behavior.

Publications and Preprints

1. A note on the completeness of Fourier-based metrics on measures (submitted). https://arxiv.org/abs/2402.03983
2. On-diagonal asymptotics for heat kernels of a class of inhomogeneous partial differential operators (with Laurent Saloff-Coste), Journal of Differential Equations 363 67-125, (2023) https://arxiv.org/abs/2206.05865
3. Local Limit Theorems for Complex Functions on Z^d The Journal of Mathematical Analysis and Applications 5(2) (2023) https://arxiv.org/abs/2201.01319
4. A Generalized Polar-coordinate Integration Formula with Applications to the Study of Convolution Powers of Complex-valued Functions on Z^d (with Huan Q. Bui), The Journal of Fourier Analysis and Applications 28(19), (2022). The final publication is available at Springer. http://arxiv.org/abs/2103.04161
5. Davies' method for heat-kernel estimates: An extension to the semi-elliptic setting (with Laurent Saloff-Coste), Transactions of the American Mathematical Society 373(4) 2525-2565 (2020). https://arxiv.org/abs/1908.00595
6. Convolution powers of complex functions on Z^d (with Laurent Saloff-Coste), Revista Matemática Iberoamericana 33(3) 1045-1121 (2017). http://http://arxiv.org/abs/1507.03501 Preprint (with high-quality images)
7. Positive-homogeneous operators, heat kernel estimates and the Legendre transform (with Laurent Saloff-Coste), Stochastic Analysis and Related Topics: A Festschrift in Honor of Rodrigo Bañuelos. Progress in Probability 72 (2017). http://arxiv.org/abs/1602.08744 Preprint (with high-quality images)
8. On the Convolution Powers of Complex Functions on Z (with Laurent Saloff-Coste), The Journal of Fourier Analysis and Applications 21(4) 754-798 (2015). The final publication is available at Springer. http://arxiv.org/abs/1212.4700
9. Fermi coordinates, simultaneity, and expanding space in Robertson-Walker cosmologies (with David Klein), Annales Henri Poincaré 12 303-28 (2011). The final publication is available at Springer. http://arxiv.org/abs/1010.0588

Expository and Course Notes

Self-adjoint operators, semigroups and Dirichlet forms

The cane problem  (watch the video)