Academic projects

Education overview

• Ph.D. candidate, Statistics, UC Berkeley, matriculated August 2014, withdrew March 2020. My focus was probability and stochastic processes. My advisors were Steven Evans and Lisa Goldberg.

• B.S., Mathematics, California Institute of Technology, June 1999

Research interests

• Stochastic processes which exhibit cascades or crashes or contagion

• Point processes, especially branching processes and the Hawkes process. Martingale techniques in point processes. Hawkes processes on graphs.

• 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.

Academic writing

Expository notes

• This a short paper on the connection between diffusions and partial differential equations. The emphasis is on how to cook up a stochastic process whose properties satisfy a given PDE.

• This note is about using lace expansions to show the self-avoiding random walk isn't too different from the random walk.

• This explains the Karlin-McGregor theorem and its applications to no-collosion Brownian motion

• This explains the conditions for the convergence of random fourier series

• These are the slides from my Ph.D. qualifying exam talk on techniques for analyzing the Hawkes process. While preparing this talk, I wrote these notes on Galton-Watson processes and probability generating functionals

• Some incomplete notes on Singular Value Decomposition and related topics like the Least Squares regression, the Moore-Penrose Pseudoinverse and Principal Component Analysis

• This is the method of thinning to embed a point process in a Poisson processes. This is a (google) translation from French of a paper by Julien Chevallier.

LaTeX style sheets

I created this LaTeX .sty file. The macros abstract common probability concepts, including the probability measure, expectation and probability distributions. Separately you can specify a typesetting mode, which allows for easy toggling between the widely different typographic conventions. It also defines some other symbols common in probability but rare in other branches, like the “independence” symbol.

Problem sets

I would like to host a wikipedia-like environment for collaboratively editing exercises, similar to www.grephysics.net. The idea is graduate students could collaborate to post the best solutions to problems in standard text books. For example, solving all the problems in Hartshorne's Algebraic Geometry is not uncommon for students in that field.

• Solutions to David Williams Probability with Martingales. This is good graduate introduction with an emphasis on martingales. It's not quite as comprehensive as Durrett or Kallenberg, but it has a fun quirky style.

• Solutions to Terrance Tao's probability course Math 275A. Tao covers measure theory basics, and fundamental convergence theorems like the law of large numbers and the central limit theorem. He takes a few detours into some topics not usually covered, like random matrices and analytic number theory.

• Solutions to the Math subject GRE, practice test 0568. There are various sources online for the questions, for example here. The exam 0568 is the most recent published practice test and it's a fair bit harder than the other practice tests, and more like the actual exams.