Research

I like working on problems arising from the interaction between algebraic number theory, Galois theory and group theory. Much of my work centers on questions about the structure of the Galois groups associated to certain field extensions with restricted ramification. What kinds of groups can arise? Can they be classified in some way? Given a particular group, can one say anything about how frequently this group should occur in certain natural families of extensions?

I also like thinking about computational problems in symbolic algebra, both theoretically and in terms of getting practical answers to questions that arise in my other work. I often make use of computational algebra systems like Magma, GAP and PARI/GP.

Research with Undergraduates

I enjoy supervising undergraduate research projects. Recent projects have focused on investigating the periodicity of sequences defined by recurrences over various finite algebraic structures.

If you are a W&L student and are potentially interested in working on a project with me, then please stop by my office some time. Be aware that most of my current projects require some background in abstract algebra (Math 321/392), although I’m also open to student suggestions. Most projects take place over the summer. Funding is available through the Summer Research Scholars program.

Publications and Preprints

* indicates co-author was an undergraduate when work was completed.

15. On certain quotients of p-class tower groups of quadratic fields, (arXiv)
submitted.

14. Heuristics for 2-class towers of cyclic cubic fields, (arXiv)
with N. Boston,
accepted subject to some revision by Experimental Mathematics.

13. Period sets of linear recurrences over finite fields and related commutative rings, (arXiv)
with D. Quijada*,
Involve, 14 (2021), No. 3, 361–376.

12. Heuristics for p-class towers of real quadratic fields, (arXiv)
with N. Boston and F. Hajir,
J. Inst. Math. Jussieu, 20 (2021), No. 4, 1429–1452.

11. Galois groups of iterates of some unicritical polynomials, (arXiv)
with W. Hindes and N. Looper,
Acta Arithmetica, 181 (2017), 57–73.

10. Heuristics for p-class towers of imaginary quadratic fields, (arXiv)
with N. Boston and F. Hajir,
Math. Annalen, 368 (2017), 633–669.
Data files: README, ipad_freq_imag (5 MB).

9. 3-class field towers of exact length 3, (arXiv)
with D. C. Mayer,
J. Number Theory, 147 (2015), 766–777.

8. Different Partners, Different Places: Mathematics applied to the construction of four-couple folk dances, (pdf)
with G. M. Roodman,
Journal of Mathematics and the Arts, 7 (2013), 17–28.

7. Mild 2-relator pro-p-groups, (pdf)
with J. Gärtner, J. Labute, and D. Vogel,
New York J. Math., 17 (2011), 281–294.
Published version is freely available online from the journal.

6. An irreducibility lemma, (pdf)
with F. Hajir,
J. Ramanujan Math. Soc., 23, No. 1 (2008), 1–9.

5. Mild pro-p Galois groups with 4 generators, (arXiv)
with J. Labute,
J. Algebra, 308 (2007), 828–839.

4. Maximal unramified 3-extensions of imaginary quadratic fields and SL_2(Z_3), (arXiv)
with L. Bartholdi,
J. Number Theory, 124 (2007), 159–166.

3. Computation of the Galois groups associated to the 2-class towers of some quadratic fields, (arXiv)
J. Number Theory, 100 (2003), 313–325.

2. Computing left Kan extensions,
with M. Leeming, R. F. C. Walters,
J. Symbolic Comput. 35 (2003), 107–126.

1. Integral equation approximation for inhomogeneous fluids: functional optimization,
with M. Booth, A. D. J. Haymet and A. G. Schlijper,
Molecular Physics, Vol. 95, No. 3 (1998), 601–619.

Other Notes and Materials