I'm currently an Assistant Professor at Durham University.
I obtained my Ph.D. in 2014 under the supervision of Alex Gamburd at U.C. Santa Cruz, and had postdoctoral positions at I.A.S. Princeton and Yale University.
I'm interested in problems in number theory, dynamics and geometry.
I am from Northern Ireland.
Papers and preprints
11. Surface words are determined by word measures on groups.
(With Doron Puder)
10. Kesten-McKay law for the Markoff surface mod p.
(With Matthew de Courcy-Ireland)
9. Counting saddle connections in a homology class modulo q.
(With Rene Rühr)
Contains an appendix by Rodolfo Gutiérrez-Romo
8. Unitary Group Integrals, Surfaces, and Mapping Class Groups.
(With Doron Puder)
Two 45 minute talks:
(Background and basic results)
(Overview of main results and issues involved)
7. The cycle structure of a Markoff automorphism over finite fields.
(With Alois Cerbu, Elijah Gunther and Luke Peilen)
6. On Selberg's Eigenvalue Conjecture for moduli spaces of abelian differentials.
5. An asymptotic formula for integer points on Markoff-Hurwitz varieties.
(With Alex Gamburd and Ryan Ronan)
One hour talk.
4. Counting one sided simple closed geodesics on Fuchsian thrice punctured projective planes.
3. Uniform congruence counting for Schottky semigroups in SL_2(Z).
(With Hee Oh and Dale Winter).
Appendix to above: Thermodynamic expansion to arbitrary moduli.
(With Jean Bourgain and Alex Kontorovich).
Crelle's Journal, 2017.
2. Arithmetic, zeros, and nodal domains on the sphere.
Communications in Mathematical Physics, 2015.
1. Quantitative spectral gap for thin groups of hyperbolic isometries.
Journal of the European Mathematical Society, 2015.
Work not intended for publication
(i) Word measures on unitary groups.
(With Doron Puder).
No longer intended to be published arXiv
The results of this preprint have now been significantly improved in the paper 'Unitary Group Integrals, Surfaces, and Mapping Class Groups'. On the other hand, this paper contains useful extra background on the more recent results.
Complex Analysis Epiphany 2018, Michaelmas 2018, Epiphany 2019.
Group expansion and number theory Spring 2017.
Number Theory Spring 2017.
Linear Algebra with Applications Fall 2016.
Introduction to Functional Analysis Spring 2016.
Spectral Geometry Spring 2016.