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:

Schmerl decompositions in first-order arithmetic, with Z. Evans, M. Groszek, and T. Slaman, in preparation.

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

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

Recent Presentations:

Logic Seminar, Rutgers University New York Logic Workshop, CUNY Graduate Center New England Recursion and Definability Seminar Joint Mathematics Meetings, Atlanta, GA RISE Talk Series, Drew University Logic Seminar, Dartmouth College Graduate Student Seminar, Dartmouth College