My academic research was in computability theory (also called recursion theory), a field of mathematical logic that seeks to understand the basic concept of computability (as established by Turing, Church, Post, Kleene, and others) and its connections to other areas of mathematics. Within that area I was most interested in the connections between algebraic (“static”) and computability-theoretic (“dynamic”) properties in the computably enumerable sets and Π01 classes, such as in degree invariance [1,2,4,7,12]. The other major track of my research was algorithmic randomness and its extension to objects other than binary sequences [5,6,8,9,15], as well as randomness-theoretic weakness and related notions [3,6,9,14,16]. I wrote an undergraduate textbook on computability, published by the American Mathematical Society [13].
I have some introductory material available: slides from an undergraduate introduction to computability I gave at my alma mater, UR (modified from a talk I gave at the MIT Women in Math series); slides with accompanying notes from a tutorial on Π01 classes I gave at the CCA conference at UF in 2006.
I was guest editor for Archive for Mathematical Logic 45 (2008), and have refereed (peer reviewed) for Annals of Pure and Applied Logic, Archive for Mathematical Logic, Journal of Logic and Computation, Journal of Mathematics and the Arts, Journal of Symbolic Logic, Mathematical Foundations of Computer Science, Mathematical Logic Quarterly, Notre Dame Journal of Formal Logic, and Theoretical Computer Science.
Publications
[16.] Lowness for effective Hausdorff dimension, with Steffen Lempp, Joe Miller, Keng Meng Ng, and Dan Turetsky. Journal of Mathematical Logic 14(2)(2014), 22 pages.
[15.] Effective randomness of unions and intersections, with Douglas Cenzer. Theory of Computing Systems 52(2013):48-64.
[14.] Finite self-information, with Denis Hirschfeldt. Computability 1(2012):85-98. PDF
[13.] Computability Theory, American Mathematical Society Student Mathematical Library, May 2012. Amazon, AMS review
[12.] Degree invariance in the Π01 classes. Journal of Symbolic Logic, 76(2011): 1184-1210. PDF
[11.] Immunity and non-cupping for closed sets, with Doug Cenzer, Takayuki Kihara, and Guohua Wu. Tbilisi Mathematical Journal, 2(2009): 77-94. PDF
[10.] Immunity of closed sets, with Doug Cenzer and Guohua Wu. Mathematical Theory and Computational Practice (CIE 2009), eds. K. Ambos-Spies, B. Loewe and W. Merkle, Springer Lecture Notes in Computer Science 5635(2009): 109-117. PDF
[9.] K-triviality of closed sets and continuous functions, with George Barmpalias, Doug Cenzer, and Jeff Remmel. Journal of Logic and Computation, 1(2009): 3-16. PDF
[8.] Algorithmic randomness of continuous functions, with George Barmpalias, Paul Brodhead, Doug Cenzer, and Jeff Remmel. Archive for Mathematical Logic, 45(2008): 533-546. PDF
[7.] Prompt simplicity, array computability and cupping, with Rod Downey, Noam Greenberg, and Joe Miller. In Chong et. al. (eds.), Computational Prospects of Infinity, Lecture Notes Series of the Institute for Mathematical Sciences, NUS, vol. 15, World Scientific (2008): 59-78. PDF
[6.] K-trivial closed sets and continuous functions, with George Barmpalias, Doug Cenzer, and Jeff Remmel. CIE 2007, Computation and Logic in the Real World, Third Conference on Computability in Europe, Siena, Italy, June 2007, S.B. Cooper, B. Loewe and A. Sorbi (Eds.), Springer Lecture Notes in Computer Science 4497(2007): 135-145. PDF
[5.] Algorithmic randomness of closed sets, with George Barmpalias, Paul Brodhead, Doug Cenzer, and Seyyed Dashti. Journal for Logic and Computation, 17(2007): 1041-1062. PDF
[4.] Totally ω-computably enumerable degrees I: bounding critical triples, with Rod Downey and Noam Greenberg. Journal of Mathematical Logic 7(2007): 145-171. PDF
[3.] Lowness and Π02 nullsets, with Rod Downey, Andre Nies, and Liang Yu. Journal of Symbolic Logic, 71(3)(2006): 1044-1052. PDF
[2.] Invariance in E* and EΠ. Transactions of the American Mathematical Society 358(2006): 3023-3059. PDF
[1.] A definable relation between c.e. sets and ideals. Ph.D. thesis under the direction of Peter Cholak, University of Notre Dame, 2004.