My present research focuses on the study of non-probabilistic phenomena with analogues in classical probability. For instance, the central limit theorem describes the limiting behavior of the convolution powers of positive real valued functions on the lattice Zd. When such functions are allowed to be complex valued, new and interesting limiting behavior is observed. Much of my research aims to determine this behavior.

Research Statement

Publications and Preprints

  1. 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.
  2. 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.
  3. Convolution powers of complex functions on Zd (with Laurent Saloff-Coste), Revista Matemática Iberoamericana 33(3) 1045-1121 (2017). http:// Preprint (with high-quality images)
  4. 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). Preprint (with high-quality images)
  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).
  6. A Generalized Polar-coordinate Integration Formula with Applications to the Study of Convolution Powers of Complex-valued Functions on Zd (with Huan Q. Bui, Submitted),
  7. Convolution Powers of Complex Functions on Zd: a study of local limit theorems and sup-norm estimates (In Preparation).


Fourier series notes for I.H.S. Senior Seminar
Self-adjoint operators, semigroups and Dirichlet forms

For fun

The cane problem  (watch the video)