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My research is centered around discrete groups acting on hyperbolic space of dimensions 2 and 3; or, in other words, discrete subgroups of PSL2(ℝ) and PSL2(ℂ). My thesis work was conducted at the University of Texas at Austin, under the supervision of Alan Reid, and focused on arithmetic reflection groups.

Publications

Hyperbolic punctured spheres without arithmetic systole maximizers. [pdf] [arXiv]
(with C. Young)
to appear, La Matematica.

Cheeger constants of hyperbolic reflection groups and Maass cusp forms of small eigenvalues. [pdf] [arXiv]
(with B. Benson and H. Then)
Proc. Amer. Math. Soc. 149 (2021), 417--438. [journal link]

Hyperbolic triangular prisms with one ideal vertex. [pdf] [arXiv]
(with C. Roth)
Involve, 13-3 (2020), 361--379. [journal link]

The mod 2 cohomology rings of congruence subgroups in the Bianchi groups. [pdf] [arXiv]
(with E. Berkove and A. Rahm, and with an appendix by B. A. Tuan and S. Schönnenbeck)
J. Algebr. Comb., 52-4 (2020), 527--560. [journal link]

Strict contractions and exotic SO0(d,1) quotients. [pdf] [arXiv]
(with C. Leininger)
J. Lond. Math. Soc., 96-3 (2017), 642--662. [journal link]

Equivalent trace sets for arithmetic Fuchsian groups. [pdf] [arXiv]
Proc. Amer. Math. Soc. 145 (2017), 445--459. [journal link]

Systoles and Dehn surgery for hyperbolic 3-manifolds. [pdf] [arXiv]
(with C. Leininger)
Algebr. Geom. Topol. 14 (2014), 1441--1460. [journal link]

Dirichlet--Ford domains and arithmetic reflection groups. [pdf] [arXiv]
Pacific J. Math. 255-2 (2012), 417--437. [journal link]

Arithmetic reflection groups and congruence subgroups. [pdf]
Ph.D. thesis, University of Texas at Austin, May 2012.

Other Writings

Matrix realizations of the hyperbolic tetrahedral groups in PSL2(ℂ).
[pdf]
These calculations were used by Mehmet Haluk Şengün to compute torsion in homology, and by Neil Hoffman for computations involving regular ideal dodecahedra.

Class notes

Notes from a class on Covering Spaces of 3-Manifolds, taught by Alan Reid at UT Austin, Spring 2011, and TeXed by the class. Any errors, omissions, typos etc. are entirely my own.
[pdf]

Invited Talks

Systoles of hyperbolic punctured spheres.
(University of Wisconsin Milwaukee, Topology seminar, 11/15/21)

Geometry of maximal arithmetic hyperbolic reflection groups.
(AIM Workshop: Arithmetic reflection groups and crystallographic packings, 12/18/20)
[video]

Fuchsian groups and geometry.
(AMS Sectional Meeting, Riverside, CA, Special Session on Computational Methods in Hyperbolic Geometry, 11/9/19)

Cheeger constants and first eigenvalues for hyperbolic reflection groups.
(AMS Sectional Meeting, Honolulu, HI, Special Session on Spectral Geometry: The Length and Laplace Spectra of Riemannian Manifolds, 3/23/19)

Cheeger constants of arithmetic hyperbolic surfaces.
(Oklahoma State University, Topology seminar, 7/17/17)

Cheeger constants of arithmetic hyperbolic surfaces.
(University of Illinois Urbana-Champaign, Geometry, Groups and Dynamics/GEAR seminar, 2/23/16)
[video]

Equivalent trace sets for arithmetic Fuchsian groups.
(AMS Sectional Meeting, East Lansing, Special Session on Interactions Between Geometry, Group Theory, and Number Theory, 3/14/15)

Systoles and Dehn surgery for hyperbolic 3-manifolds.
(University of Warwick Geometry and Topology Seminar, 6/5/14)

Systoles and Dehn surgery for hyperbolic 3-manifolds.
(Groups in Galway, 5/24/14)

Systoles and Dehn surgery for hyperbolic 3-manifolds.
(Eastern Illinois University Mathematics Colloquium, 4/11/14)

Equivalent trace sets for arithmetic Fuchsian groups.
(Purdue University, Geometry Seminar, 4/7/14)

Tiling by reflections: the Euclidean plane and beyond.
(Bradley University Mathematics Colloquium, 10/24/13)

Systoles and Dehn surgery for hyperbolic 3-manifolds.
(AMS Sectional Meeting, St. Louis, Special Session on Geometric Aspects of 3-Manifold Invariants, 10/20/13)

Testing congruence for arithmetic reflection groups.
(Université de Fribourg, Oberseminar Geometrie, 12/21/11)

Arithmetic reflection groups and congruence.
(University of Illinois at Chicago, Geometry, Topology and Dynamics Seminar, 11/14/11)