Piñata by Steve, a student in my S07 Math 29 class.	Mr. Maths, prepare to meet your maker!

My CV is available in pdf or html format and was last updated Jan 16.

My research is 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 am 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 [4,7,12,16]. The other major track of my research is algorithmic randomness and its extension to objects other than binary sequences [5,6,8,9,14], as well as randomness-theoretic weakness and related notions [3,6,9,15]. I have written an undergraduate textbook on computability to be published by the AMS [13].

I have course notes (draft, spring 2011) for Math 29, Computability Theory. Note this class has no prerequisites beyond high school math, so the text aims to be self-contained re: background material.

Papers:

[16.] On sets automorphic to low sets, with Peter Cholak. In preparation.

[15.] Reals that are low for information, with Denis Hirschfeldt. Submitted.

[14.] Effective randomness of unions and intersections, with Douglas Center. Submitted.

[13.] Computability Theory, American Mathematical Society Student Mathematical Library. Anticipated publication April 2012.

[12.] Degree invariance in the Π⁰₁ classes. Journal of Symbolic Logic, 76(2011): 1184-1210.

[11.] Immunity and non-cupping for closed sets, with Doug Cenzer, Takayuki Kihara, and Guohua Wu. Tbilisi Mathematical Journal, 2(2009): 77-94.

[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.

[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.

[8.] Algorithmic randomness of continuous functions, with George Barmpalias, Paul Brodhead, Doug Cenzer, and Jeff Remmel. Archive for Mathematical Logic, 45(2008): 533-546.

[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.

[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.

[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.

[4.] Totally ω-computably enumerable degrees I: bounding critical triples, with Rod Downey and Noam Greenberg. Journal of Mathematical Logic 7(2007): 145-171.

[3.] Lowness and Π⁰₂ nullsets, with Rod Downey, Andre Nies, and Liang Yu. Journal of Symbolic Logic, 71(3)(2006): 1044-1052.

[2.] Invariance in E* and E_Π. Transactions of the American Mathematical Society 358(2006): 3023-3059.

[1.] A definable relation between c.e. sets and ideals. Ph.D. thesis under the direction of Peter Cholak, University of Notre Dame, 2004.

Other:

• Slides from my talk at the 2010 ASL Annual Meeting, GWU (Degree invariance in the Π⁰₁ classes).
• Slides from an introduction to computability given at my alma mater, University of Richmond (slightly modified from a talk given at the MIT Women in Math series).
• CCA 2006 tutorial on Π⁰₁ classes: slides 1-up, slides 4-up, accompanying notes (one page).
• Handout from my Dartmouth logic seminar on bounding critical triples.
• My thesis relied heavily on the paper "The Δ⁰₃ automorphism method and noninvariant classes of degrees" by Leo Harrington and Bob Soare. While reading it I put together a glossary of terms to keep track of everything (or at least sections 2-5).
• Ancient work: a graceless Π⁰₁ class is an uncountable Π⁰₁ class with no perfect Π⁰₁ subclass (though the tree it comes from has to have a perfect subtree). Joe Miller and I got interested in the idea when thinking about whether uncountability could be definable in the lattice of Π⁰₁ classes (we later learned the question had been resolved - negatively - much earlier; graceless classes are witnesses that uncountability is not definable via perfect classes). Reed Solomon helped me construct such a class (thereby losing my bet with Joe), and the writeup is here. (It is also a snapshot of my ever-evolving TeX skills, and the lack of evolution in my Xfig skills.)

Last modified January 16, 2012

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