My broad research interests include topics in geometric analysis, differential geometry, PDE, low dimensional topology, and graph theory. Thank you for your interest!
A research colloquium talk I gave to a general mathematics audience at California State University-San Bernardino.
Cheeger constants of hyperbolic reflection groups and Maass cusp forms of small eigenvalues. Joint with Grant Lakeland and Holger Then. arXiv:1908.00199. Accepted by Proc. Am. Math. Soc.
Volume growth, curvature, and Buser-type inequalities in graphs. Joint with Peter Ralli and Prasad Tetali. arXiv:1802.01952.To appear in Int Math Res Not.
Mean Value Theorems for Riemannian Manifolds via the Obstacle Problem. Joint with Ivan Blank and Jeremy LeCrone. Journal of Geometric Analysis, Vol 29, Issue 3, 2752-2775, DOI:10.1007/s12220-018-0093-4. arXiv:1704.07518.
A note on a Newtonian Approximation in a Schwarzchild background. Joint with Marcelo M. Disconzi. The African Review of Physics, Vol 13, 2018, 12 pages.
Torsion and ground state maxima: close but not the same. Joint with Richard S. Laugesen, Michael L. Minion, and Bartlomiej A. Siudeja. Irish Mathematical Society Bulletin 78 (Winter 2016), 81-88, arXiv:1507.01565.
Sturm-Liouville Estimates for the Spectrum and Cheeger Constant. Int Math Res Not, Vol. 2015, No. 16, pp. 7510-7551. rnu175. arXiv:1308.5936
G-parking functions, acyclic orientations and spanning trees. Joint with Deeparnab Chakrabarty and Prasad Tetali. Discrete Math. 310 (2010), no. 8, 1340-1353. arXiv:0801.1114
The Cheeger Constant, Isoperimetric Problems, and Hyperbolic Surfaces. arXiv:1509.08993 (October 12, 2020 - This manuscript is currently undergoing a substantial revision. While I hope to finish this soon, I am happy to answer emails about the specifics of this revision.)
Cheeger Constants of Principal Congruence Arithmetic Surfaces. Joint with Grant Lakeland and Jeffrey Meyer.
The Spanning Tree Modulus and the Laplace Spectrum of a Graph. Joint with Nathan Albin and Pietro Poggi-Corradini.
The Three Hat Problem. Joint with Yang Wang. arXiv:0710.2685 (A fun project that introduced me to the research process in mathematics!)
See Grant Lakeland present preliminary results on our joint work. This work eventually led to the paper Cheeger constants of hyperbolic reflection groups and Maass cusp forms of small eigenvalues completed with Grant Lakeland and Holger Then. This presentation was given shortly before Holger joined the project.
Various additional projects are in earlier stages of development or have not yet been announced. For additional information or potential collaboration, please contact me via email.