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Research

I study the complexity of important mathematical constructions, structures, and proofs. To do this, I work with different formal notions of “complexity” such as computability and Kolmogorov complexity. From a foundational point of view, this pursuit often yields a precise understanding of the axioms necessary to prove a given theorem or produce a given structure. This type of venture is familiar to most mathematicians, who have often considered equivalents of the Axiom of Choice, or Euclid's Fifth Postulate and hyperbolic geometry, or have asked the questions “is Lemma X necessary to prove Theorem Y?” or “how many different mathematical arguments are there really?” From a computational perspective this pursuit examines the algorithmic content of mathematics, and attempts to classify those structures that can be obtained via computable constructions, and, more generally, attempts to assign a unique “degree of computability” to every mathematical construction. One of the first people to seriously consider this perspective was Leibniz. Later on Hilbert adopted this perspective when he posed his famous Entscheidungsproblem, which was subsequently solved by Alan Turing and others with the development of the Turing Machine and Computability Theory, my primary field of interest. Adopting the general framework of Mathematical Logic and Computability Theory allows me to examine the complexity of constructions, structures, and proofs from all branches of Mathematics, and to find connections and make comparisons between them. This is the heart of my research.

So far I have applied the tools of Computability Theory and Mathematical Logic to obtain results that determine the complexity of constructions, structures, and proofs from many different branches of Mathematics, including: Model Theory, Set Theory, Computability Theory, Analysis, Measure Theory, Combinatorics, and Commutative/Noncommutative Algebra. More specifically, the topics that I have analyzed thus far include: Prime Models, Forcing, Turing Reducibility, Fractal Dimension, Measure Theory, Compactness, Ramsey Theory, Prime Radicals of Noncommutative Rings, Artinian rings, Infinite Dimensional Vector Spaces, locally nilpotent Groups, structures of finite computable dimension, and Euclidean domains.