(joint with Gerard Ben Arous, Song Mei, Andrea Montanari) In this project we study the energy landscape of a certain random function on the N dimensional hyper-sphere. This energy landscape is a natural model for certain problems in machine learning. We find a certain critical signal-to-noise ratio in this problem below which it becomes infeasable to detect the original signal.

My PhD thesis research was on multi-layer random polymers which are constructed from non-intersecting random processes (see above picture!). A summary of my results are available as a poster or as seminar slides.

Intermediate Disorder Limits for Multi-layer Semi-discrete Directed Polymers

I show convergence for semi-discrete directed polymers (also known as the O'Connell-Yor polymer) to the corresponding partition function for multi-layer continuum polymers. This convergence verifies, modulo a previously hidden constant, an outstanding conjecture proposed by Corwin and Hammond in their construction of the KPZ line ensemble.

Intermediate Disorder Directed Polymers and the Multi-layer Extension of the Stochastic Heat Equation

(Joint with Ivan Corwin) A research project about the scaling limits of multiple non-intersecting directed polymers. We study the partition function of several non-intersecting walks in a random envirnoment in the limit where the 1) the length of the walks goes to infinity 2) the randomness in the disorder goes to zero. If the rate of these two limits is tuned correctly, there is a non-trivial limit. The limit turns out to be related to the "multi-layer extension" of the stochastic heat equation. This whole construction can also be thought of as a limit for the geometric RSK process.

Decorated Young Tableaux and the Poissonized Robinson-Schensted Process

In this project, I generalized the definition of a Young Tableau to include real entries. By applying a generalization of the RS algorithm to a Poisson point process, we get a pair of random such generalized Tableau whose law is related to Schur processes and non-crossing Poisson walks. Published in Stochastic Processes and their Applications .

Optimal Strategy in "Guess Who?": Beyond Binary Search

"Guess Who?" is a popular two player game where players ask "Yes" or "No" questions to search for their opponent's secret identity from a pool of possible candidates. Common wisdom is that using a binary search approach to narrow down candidates is the best stategy. In this work, I showed that this is NOT the case for the player who is behind. Instead, the player who is behind should make certain bold plays to maximize their chance to win. I was able to find an exact formula for these bold plays and prove their optimality. Published in Probability in the Engineering and Informational Sciences .

Stabilization Time for a Type of Evolution on Binary Strings

(with Mike Noyes and Jacob Funk) In this project, we considered a type of non-random TASEP-like evolution for particles on a finite line where particles determinstically move to the left under a simultaneous update rule. (This was presented in terms of binary strings in the paper). We found the limit law for the "stabilization time" for this update rule if you start with a random initial configuration. The limit law is Gaussian unless you start with a near-equal number of holes and particles...in which case an entirely different limit law is found. Published in the Journal of Theoretical Probability

This is a survey of some basic results for PDEs that was published in the Waterloo Mathematics Review.

Limit Theorems II by Professor McKean

My notes from the course "Limit Theorems II" taught by Professor McKean in Spring 2012 at CIMS

Superconcentration by Professor Chatterjee

My notes from the mini-course on Superconcentration and related topics taught by Sourav Chatterjee in Summer 2012 at the Cornell Probability Summer School. Professor Chatterjee has since released a published version of these lectures here .

My oral exam notes

These are the notes I took while studying for the general topics of my oral exam in October 2013. I spent all summer studying for this and wrote a few hundred pages of notes, mostly based on textbooks. My topics were PDEs, Functional Analysis, Real Analysis, Complex Analysis and Probability Theory.

My PhD thesis

Link to my thesis and a breakdown of the chapters. I also have some more assorted notes I have taken on probability and other things posted here.