# John Wallbaum

PhD: University of Notre Dame - May 2010

Advisor: Julia Knight

Research Area: Logic, Algebra

Year Started: 2005

*Last updated on: 7/14/2010*

#### Research Interests

My research uses computability theory to study algebraic structures such as groups, orderings, and equivalence structures. I consider questions about when a structure can be presented in an computable way and about the complexity of different algorithms in the structure. In one project, our research group was able to determine how hard it is to compute a basis of a computable free group. We also calculated the most efficient way to determine if a computable group is a free group or not. In another project, I studied limitwise monotonic functions, which tell us when certain abelian p-groups and equivalence structures can be presented computably.

#### Courses Taught

Summer 2008: Calculus 1

#### Talks

"Computability of free groups" Ohio State Logic Seminar, March 2010.

“Primitive elements of free groups” Midwest Computability Seminar (University of Chicago), January 2010.

“Index Sets of Free Groups” Mal’tsev Meeting (Novosibirsk, Russia) August 2009.

“Limitwise Monotonic Functions in Computable Algebra” AMS Regional Meeting (Middle Tennessee State University), November 2007.

“Limitwise Monotonic Functions” Computable Models and Numberings Workshop (Novosibirsk, Russia), August 2007.

“Computable Embeddings” Notre Dame Logic Seminar, January 2007.

#### Papers

“Describing free groups” with J. Carson, V.S. Harizanov, J.F. Knight, K. Lange, C. Maher, A. Morozov, C. McCoy, and S. Quinn, to appear in Transactions of the AMS.

“Describing free groups, part II: Π_4 hardness and no Σ_2 basis” with C. McCoy, to appear in Transactions of the AMS.

“A ∆_2 poset with no positive presentation”, submitted.