My research area is mathematical logic and the foundations of mathematics. I work primarily in the overlap between Computability Theory (a.k.a. Recursion Theory) and Set Theory. (Yes, this overlap is not empty!) Broadly speaking, I am interested in analyzing the relative complexity of various mathematical objects and methods. This goal has led me to branch into several areas of mathematical logic.

Degrees of constructibility, or simply c-degrees, are used to measure the relative complexity of objects in the universe of set theory. I am investigating the possible initial segments of this structure. The word possible is essential here since the standard axioms of Set Theory (ZFC) are not enough to completely determine the structure of c-degrees. I use forcing (specifically forcing with Souslin trees) to obtain relative consistency results about the structure of c-degrees. In my thesis, I showed that all constructible, dual-algebraic lattices can be realized as an initial segment of the c-degrees in a forcing extension of the universe of constructible sets. I also showed that constructible, separable, dual-continuous lattices (e.g. the constructible real interval [0,1]) can be similarly realized.

Degrees of computability, or simply T-degrees (T stands for Turing), are another way to measure the relative complexity of mathematical objects. The basic universe for T-degrees, rather than the universe of Set Theory (as is the case for c-degrees), is the natural numbers. Although this is a much smaller universe to work with, researchers in the field of Reverse Mathematics have shown that natural numbers and sets of natural numbers suffice to express many of the basic structures of Analysis, Algebra, and Combinatorics. I am currently investigating the relative strength of certain combinatorial statements along the lines of Ramsey's Theorem using techniques from Computability Theory and Reverse Mathematics.

A key feature of my line of research is that it allows me to dabble in many areas of mathematics. I am a passionate mathematician, interested in all of mathematics and related fields. In my spare time, I like to do some work in Combinatorics, Computer Science, and Number Theory. Recently, I have been involved in an exhaustive search for the very elusive third Wieferich prime. I wrote all the code to efficiently search for such numbers and, together with Dominic Klyve, I spent over a year running it on research clusters at Dartmouth College. We are currently analyzing the data we gathered, sadly no new Wieferich prime was found...

Further details on my research can be found in my Research Statement.

Publications

Ph.D. Thesis