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RESEARCH
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].
Support
National Science Foundation Focused Research Group in Algorithmic Randomness (award #0652326), July 2007-June 2010; co-PI, total award $532,000.
Junior Faculty Fellowship, Fall 2009 (extra non-resident term).
Editing
Guest editor (and coauthor of preface, with Doug Cenzer) for Archive for Mathematical Logic 45(2008) dedicated to the Special Year in Logic at the University of Florida and the Conference on Computability and Complexity in Analysis (November 2006, Gainesville, Florida).
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 EΠ. 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.
TEACHING
Thesis advisor to mathematics graduate student Seth Harris.
Recent Courses
| 2012, Winter | Math 89: Seminar in Logic (Set Theory) |
| 2012, Winter | Math 24: Honors Linear Algebra |
| 2011, Fall | Math 8: Calculus II |
| 2011, Spring | Math 29: Computability Theory |
| 2011, Spring | Math 38: Graph Theory |
| 2010, Fall | Math 19/CS 19: Discrete Math for Computer Science |
| 2010, Spring | Math 10: Introductory Statistics |
| 2009, Spring | Math 29: Computability Theory |
| 2009, Winter | Math 13: Vector Calculus |
| 2008, Fall | Math 8: Calculus II |
| 2008, Fall | Math 12: Calculus Plus |
| 2008, Winter | Math 8: Calculus II |
| 2008, Winter | Math 28: Combinatorics |
| 2007, Fall | Math 8: Calculus II |
| 2007, Fall | Math 39: Logic |
DEPARTMENTAL SERVICE
| 2011-2012 | Graduate Program Representative |
| 2005-2012 | Graduate Program Committee |
| 2010-2011 | Graduate Admissions Committee |
| 2006-2007 | Brochure design (rewrite of departmental brochure) |
SELECTED TALKS
| 2011 | Lowness for dimension |
| Dartmouth Logic Seminar | |
| 2011 | Effective dimension |
| Dartmouth Logic Seminar | |
| 2010 | Sets automorphic to low sets |
| Computability Theory Special Session invited talk | |
| American Mathematical Society Sectional Meeting, Notre Dame | |
| 2010 | The 5 W's of computability theory |
| Dartmouth College Mathematics Department Prospective Graduate Student Open House | |
| 2010 | Degree invariance in the Π01 classes |
| Association for Symbolic Logic Annual Meeting, George Washington University, Washington, DC | |
| 2010 | What is computability theory? |
| University of Richmond Mathematics Colloquium | |
| 2010 | Reals that are low for information |
| Southeastern Logic Symposium, University of Florida, Gainesville | |
| 2009 | Mutual information |
| Notre Dame Logic Seminar | |
| University of Wisconsin--Madison Logic Seminar | |
| 2009 | Patterns and definability |
| Notre Dame Working Seminar, 5-talk series | |
| 2009 | Degree invariance in the Π01 classes |
| Midwest Computability Seminar, University of Chicago | |
| 2009 | Do you know how much you know? |
| MIT Logic Seminar | |
| 2009 | What is Computability? |
| MIT Women in Math Lecture Series | |
| 2009 | Do you know how much you know? |
| Connecticut Logic Seminar | |
| 2008 | Hierarchies |
| Dartmouth Mathematics Graduate Open House | |
| 2007 | Randomness of closed subsets of 2ω |
| FRG Workshop: Effective Randomness, University of Chicago | |
| 2007 | Computability and randomness |
| Dartmouth College Mathematics Department Prospective Graduate Student Open House | |
| 2006 | Tutorial: Π01 classes (two parts) |
| Computability and Complexity in Analysis 3rd International Conference, Gainesville, FL | |
| 2006 | Making randomness rigorous |
| Summer Program for Women in Mathematics, George Washington University | |
| 2006 | Randomness and Π02 nullsets |
| Greater Boston Logic Conference, MIT |