Research
Research Interests
My primary interests lie in effective algebra, computability theory, reverse mathematics, and mathematical logic. Effective algebra endeavors to understand results of algebra and combinatorics from a viewpoint of computability theory. My Ph.D. dissertation concerns the effective content of ordered fields. For more information on the types of questions I like to study, as well as my approach to research, check out my papers below or my research statement
Papers
The following papers, though copyrighted, may be downloaded for personal and non-profit educational use.
- Computable Dimension for Ordered Fields Submitted for publication.
Abstract:We study computable dimension for various classes of computable ordered fields. We show that computable ordered fields with finite transcendence degree are computably stable. We then build computable ordered fields of infinite transcendence degree which have infinite computable dimension, but also computably categorical fields with infinite transcendence degree. Additionally, we use similar techniques to build computable ordered fields which possess transcendence bases which are complicated in two senses: every transcendence basis can compute the halting problem, or alternatively, every transcendence basis is immune.
- Embeddings of Computable Structures with Asher M. Kach and Reed Solomon. Notre Dame J. Formal Logic Volume 51, Number 1 (2010), 55-68.
Abstract: We study what the existence of a classical embedding between computable structures implies about the existence of computable embeddings. In particular, we consider the effect of fixing and varying the computable presentations of the computable structures.
- Computability Theory, Reverse Mathematics and Ordered Fields Ph.D. thesis, UConn, 2009. Adviser: Reed Solomon.
Abstract: The effective content of ordered fields is investigated using tools of computability theory and reverse mathematics. Computable ordered fields are constructed with various interesting computability theoretic properties. These include a computable ordered field for which the sums of squares are reducible to the halting problem, a computable ordered field with no computable set of multiplicatively archimedean class representatives, and a computable ordered field every transcendence basis of which is immune. The question of computable dimension for ordered fields is posed, and answered for archimedean fields, fields with finite transcendence degree, and some purely transcendental fields with infinite transcendence degree. Several results from the reverse mathematics of ordered rings and fields are extended.
Recent and Not-So-Recent Math Talks
Counting Liars and Truth-tellers - presented at the Joint Mathematics Meeting in Boston on January 7, 2012
To Infinity and Beyond - presented at the 31st Anual Coastal Carolina University Highschoo Math Contest, 2010
Computable Dimension of Ordered Fields - presented at the Joint Mathematics Meeting in Washington, D.C. on January 8th, 2009.