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. Uniformly positive-homogeneous operators with measurable coefficients (with Laurent Saloff-Coste) (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)