Research
My research interests are in logic, computability theory, and reverse mathematics.

What is Reverse Mathematics?:

How do we prove any new theorem? We start with a set of assumptions (axioms) and, using a sequence of logical deductions, we prove the theorem. Reverse mathematics "reverses" this procedure, starts with a theorem, and asks the question: "What axioms did we need to assume to prove this theorem?"

We work in second-order arithmetic, where all objects are natural numbers or sets of natural numbers. This is not really that limiting, since there are easy ways to encode real numbers, functions, and certain sets of real numbers (e.g. Borel sets) in second-order arithmetic, as well as combinatorial objects like graphs. So we can express quite a few familiar theorems (of analysis, discrete math, etc.) in the language of second-order arithmetic.

A surprising number of familiar theorems are equivalent to one of the "Big Five" axiom subsystems, which form a hierarchy of strength, the weakest being our base theory RCA0.

Π11 –CA0 → ATR0 → ACA0 → WKL0 → RCA0

For just one example, the theorem "Every countable vector space has a basis" is equivalent to ACA0. What this really means is that the ACA0 axioms are sufficient to prove the theorem, but also conversely, if you assume "Every countable vector space has a basis" as well as the weak base theory RCA0, you can prove all axioms of ACA0. Not all theorems fall into this neat five-system hierarchy; for instance, Liu proved in 2012 that the infinite Ramsey's Theorem for pairs is incomparable to WKL0, and thus falls outside the hierarchy.

My Research:

My Ph.D. thesis, On-Line Algorithms and Reverse Mathematics, relates the on-line solvability of a problem to the reverse-mathematical strength of its sequential version. A problem is on-line solvable if it can be solved according to a two-player game. For example, if we're solving a graph coloring problem, 2-coloring a forest is solvable, but is not on-line solvable.

James Schmerl (UConn) observed a relationship between the on-line colorability of a class of graphs and the reverse-mathematical strength of the coloring problem; notably, if we're in a model where Weak König's Lemma (WKL0) is false, then there is an infinite forest that is not 2-colorable. My research generalizes Schmerl's results: any finite problem's on-line solvability determines the reverse-mathematical strength of its (infinite) sequential form. Even problems that are rarely thought of as a two-player game (e.g., the pigeonhole principle) can be formulated and classified in this way.

Conferences:

I regularly attend the New England Recursion and Definability Seminar (NERDS). (Appropriate, I know.) The New York Logic seminar, hosted at the CUNY Graduate Center, is a regional gathering for logicians in greater New York City.

Publications:

Evasion, prediction, and on-line algorithms, with F. Dorais, in preparation.

Schmerl decompositions in first-order arithmetic, with Z. Evans, M. Groszek, and T. Slaman, to appear in Annals of Pure and Applied Logic .

On-line algorithms and reverse mathematics, Ph.D. thesis, 2017.

Recent Presentations:

American Mathematical Society Eastern Sectional Meeting, Hartford, CT
• Schmerl decompositions in first-order arithmetic, April 2019
Logic Seminar, Rutgers University
• On-Line Algorithms and Reverse Mathematics, April 2018 (slides)
New York Logic Workshop, CUNY Graduate Center
• On-Line Algorithms and Reverse Mathematics, March 2018
New England Recursion and Definability Seminar
• On-Line Algorithms and Reverse Mathematics, Dartmouth College, April 2017
• Evasion, Prediction, and On-Line Graph Problems, University of Connecticut, May 2015
Joint Mathematics Meetings, Atlanta, GA
• On-Line Algorithms and Reverse Mathematics, January 2017
RISE Talk Series, Drew University
• Reverse Mathematics and Computability, November 2016
Logic Seminar, Dartmouth College
• On-Line Algorithms and Reverse Mathematics, January 2017
• Bushy-Tree Forcing and Friedberg's Lemma, January 2015
• Graph Colorings With And Without WKL0, April 2014
• On the strength of proving a finite combinatorial statement infinitely often, October 2013
• The Complexity of the Finite Jordan Curve Theorem, April 2013
• Ramsey's Theorem for Trees, February 2013