Research

I study group theory and mathematical logic, in particular complexity notions with a computability-theoretic or combinatorial flavor in finitely generated groups. These questions are naturally related to combinatorial group theory, computability theory, and model theory. Some specific (but still quite broad) topics that I am interested in include:

• The computable structure theory of finitely generated groups, especially decision problems in group theory

• The first-order theory of random groups and random structures

• Formal languages and their application in group theory

• Word maps, especially on finite nilpotent groups

• Left-distributive algebras

Research opportunities for students:

I am currently recruiting students to conduct research with me on the abovementioned topics as paid research assistants. If you are interested, please contact me directly via email, and include a list of related courses (and grades) you have taken and what your favorite topics and/or courses are. 

My research is partially supported by NSF grant no. 2054558.

My collaborators: Uri Andrews, Bogdan Chlebus, Seongjun Choi, Laura Ciobanu, William Cocke, Matthew Cordes, Moon Duchin, Yen Duong, Alex Evetts, Johanna N. Y. Franklin, Matthew Harrison-Trainor, Stephen C. Jackson, Julia Knight, Steffen Lempp, Russel G. Miller, Mark Pengitore, Ayla P. Sánchez, Noah Schweber.

You can also see my publications on Arxiv, MathSciNet, or Google Scholar.

1. Countable and finitary reductions on equivalence relations, with Stephen C. Jackson, Steffen Lempp, Russell G. Miller, and Noah D. Schweber, 2024, submitted.

2. Torsion-free abelian groups of finite rank and fields of finite transcendence degree, with Julia Knight and Russell Miller, submitted.

3. Two results on complexities of decision problems of groups, with Uri Andrews and Matthew Harrison-Trainor, submitted. 

4. Algorithmically finite, universal, and ∗-universal groups, with Uri Andrews, arXiv preprint arXiv:2402.01882 (2024), submitted.

5. Free structures and limiting density, with Johanna N. Y. Franklin and Julia Knight, arXiv preprint arXiv:2209.04069 (2022), submitted.

6. Enumerating word maps in finite groups, with Bogdan S. Chlebus and William Cocke, Int. J. Group Theory, 13 (2024), no. 3, 307–318.

7. Rational growth in torus bundle groups of odd trace, with Seongjun Choi and Mark Pengitore, Proc. Edinb. Math. Soc. (2) 65 (2022), no. 4, 1080–1132.

8. Finitely generated groups are universal, with Matthew Harrison-Trainor, Ann. Pure Appl. Logic, 172 (2021), no. 1, 102855.

9. The conjugacy growth of the soluble Baumslag-Solitar groups, with Laura Ciobanu and Alex Evetts, New York J. Math. 26 (2020), 473-495.

10. Word maps in finite simple groups, with William Cocke, Arch. Math. (Basel), 113 (2019), no. 6, 565-570.

11. Characterizations of cancellable groups, with Matthew Harrison-Trainor, Proc. Amer. Math. Soc., 147 (2019), no. 8, 3533–3545.

12. The probability distribution of word maps on finite groups, with William Cocke, J. Algebra, 518 (2019), 440–452.

13. The word problem of Z^m is a multiple-context free language, Groups Complex. Cryptol. 10 (2018), no. 1, 9–15.

14. On optimal Scott sentences of finitely generated algebraic structures, with Matthew Harrison-Trainor, Proc. Amer. Math. Soc. 146 (2018), no. 10, 4473–4485.

15. On the symmetry of images of words maps in groups, with William Cocke, Comm. Algebra 46 (2018), no. 2, 756–763.

16. Random nilpotent groups I, with Matthew Cordes, Moon Duchin, Yen Duong, and Ayla P. Sánchez, Int. Math. Res. Not. (2018), no. 7, 1921–1953.

17. Describing groups, Proc. Amer. Math. Soc. 145 (2017), no. 5, 2223-2239.

Book Chapter:

1. Group Theory, with William Cocke, a chapter in the book Mathematics in Cyber Research, edited by Paul L. Goethals, Natalie M. Scala, and Daniel T. Bennett, CRC Press, 2022