Andy Parrish andytparrish at gmail dot com

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Research
Through 2013, I was a PhD student in the Math department of UC San Diego. I worked in combinatorics, and specifically trying to discern necessary structures within large graphs, sets of integers, and anything else I can wrap my head around. This leads to some buzzwords like "extremal combinatorics," "Ramsey theory," and perhaps "the study of unreasonably large numbers." I worked with Ron Graham, with additional guidance from Fan Chung.

Papers

• Adventures in Graph Ramsey Theory This is my PhD dissertation. It mostly covers material from An additive version of Ramsey's theorem and Toward a graph version of Rado's theorem, with updated notation and a more unified framework.
• Toward a graph version of Rado's theorem (2013, Electronic Journal of Combinatorics). This is a follow-up to the previous paper. Call an equation graph-regular if every r-coloring of the complete graph on the natural numbers contains a monochromatic complete subgraph whose vertices solve the equation. This paper gives two Rado-like conditions for an equation which are respectively necessary and sufficient for graph-regularity.
• An additive version of Ramsey's theorem (2011, Journal of Combinatorics). This is a combination of the two flavors of Ramsey theory: additive and graph-theoretic. In short, Ramsey's theorem guarantees that any edge-coloring of a large complete graph will give large monochromatic complete subgraphs. In this paper, We show that we can get more: if the vertex set is the natural numbers up to n, we can guarantee some additive structure of the vertices in the monochromatic subgraph. Specifically, we prove that, for all r and k, there is an n = n(r, k) so that, for any r-coloring of the complete graph on vertex set {1, 2, ..., n}, there must be a solution to x1 + ... + xk = y1 + ... + yk by distinct numbers, so that all edges among those values have the same color.
• Three Proofs of the Hypergraph Ramsey Theorem (An Exposition) (2012, with William Gasarch and Sandow Sinai, submitted). This is an exposition of three proofs of the hypergraph version of Ramsey's theorem, giving successively better bounds.
• Van der Waerden's Theorem: Variants and Applications. I am working on this book with Bill Gasarch and Clyde Kruskal. It is a collection of combinatorial proofs of results in additive Ramsey theorey. This is built on my senior thesis at the University of Maryland. The original motivation for the book is its detailed combinatorial proofs of the Polynomial van der Waerden theorem and the Polynomial Hales-Jewett theorem, expanding on proofs in a short paper by Mark Walters.
• Efficient Computationally Private Information Retrieval from Anonymity or Trapdoor Groups (With Jonathan Trostle, appeared at ISC 2010). We describe a protocol for Private Information Retrieval, designed to be practical, in contrast with previous systems which have impressive communication bounds, but bad computational complexity.
• System and Method for Computationally Private Information Retrieval (With Jonathan Trostle, submitted to USPTO in 2009). Patent application for the above Private Information Retrieval protocol.
• There is no "large van der Waerden theorem" (2009). This is a minor negative result in Ramsey theory from a question of Bill Gasarch: the Large Ramsey theorem does not translate directly to a "Large van der Waerden theorem."
• Exploration of the three-person duel (2006). This was a toy problem I worked on as an undergrad at the University of Maryland. Under the guidance of Bill Gasarch, I formulated the equations governing the probabilities of winning in a three-person duel, and explored the conditions for the three players to be evenly matched.

Past teaching
I have a been teaching assistant for these courses:

Past course webpage
In summer 2009, I studied random geometric graphs with some other students and maintained the course website.

Jobs
Here are some jobs I have held outside of teaching. If you have any jobs you would like to see added to this list, the process begins with my résumé.

• Epic — Epic is a major force in the world of electronic medical records. As of August 2013, I am a full time software developer for its oncology application.
• Google — I worked in Google's Cambridge office in summer 2011. I attempted to optimize local storage for YouTube by understanding video request patterns.
• Johns Hopkins University Applied Physics Lab — I worked for two summers (2007-2008) investigating a new protocol for Private Information Retrieval, with the goal of attaining practical algorithms with low communication costs.