Education
- Ph.D. candidate, Statistics — UC Berkeley, matriculated 2014, left 2020. Focus on probability and stochastic processes. Advisors: Steven Evans and Lisa Goldberg.
- B.S., Mathematics — California Institute of Technology, 1999.
Research Interests
- Stochastic processes exhibiting cascades, crashes, and contagion
- Point processes — branching processes, the Hawkes process, and Hawkes processes on graphs. Martingale techniques in point process theory.
- Principal component analysis and factor analysis in high dimensions. Statistical learning of high-dimensional features.
- Gaussian processes for machine learning. Gaussian processes as generating processes for point processes.
- Formal verification of mathematics — currently participating in a working group formalizing stochastic calculus (Itô's lemma) in Lean.
Expository Notes
- Diffusions and partial differential equations — How to construct a stochastic process whose properties satisfy a given PDE.
- Lace expansions and the self-avoiding random walk — Using lace expansions to show the self-avoiding random walk isn't too different from the ordinary random walk.
- The Karlin-McGregor theorem — Applications to non-collision Brownian motion.
- Random Fourier series — Conditions for convergence.
- Qualifying exam slides — Techniques for analyzing the Hawkes process. Also I wrote some accompanying notes on Galton-Watson processes and probability generating functionals.
- Singular Value Decomposition — SVD and related topics: least squares regression, the Moore-Penrose pseudoinverse, and PCA.
- Thinning of point processes — Embedding a point process in a Poisson process. Translation of a paper by Julien Chevallier.
LaTeX Tools
A LaTeX probability.sty
file with macros for common probability concepts — the probability measure, expectation, distributions,
and symbols like the independence symbol $\perp\!\!\!\perp$. Supports toggling between typographic conventions.
Problem Sets
Solutions to problem sets from graduate probability courses, hosted on Martingale AI:
- Probability with Martingales — David Williams. A graduate introduction emphasizing martingale theory.
- Terence Tao's Math 275A — Measure theory basics, law of large numbers, central limit theorem, with detours into random matrices and analytic number theory.
- Math GRE practice test 0568 — Solutions a published practice examzx