Research

My primary research interests include geometric analysis, differential geometry, low dimensional topology, and PDE. A secondary research interest is graph theory, specifically interpretations of geometric and analytic phenomena on graphs.

Overview

The Cheeger Constant and Laplace Spectrum of Riemannian and Hyperbolic Manifolds

How does the geometry of a manifold (its shape, size, and curvature) affect analysis on the manifold and, in particular, the spectral theory of physically natural operators? To help better understand this relationship, I study the spectral theory of and the isoperimetric problem on Riemannian manifolds.

For a fixed t∈(0, Voln (M)), the isoperimetric problem on a Riemannian n-manifold M asks one to find n-dimensional subset A of M so that Voln(A)=t and so that Voln-1(∂A) is minimal among all subsets of M having n-volume equal to t. The ratio Voln-1(∂A)/Voln(A) is called the isoperimetric ratio of A. A quantity related to the isoperimetric problem on M is the Cheeger constant of M, denoted h(M), which can be though of as the infimum of isoperimetric ratios over all solutions to isoperimetric problems with t∈ (0, Voln(M)/2]. Cheeger showed that h(M) can be used to give a non-trivial lower bound on the smallest positive eigenvalue of the Laplacian on M, denoted λ1(M). When M is closed, Buser's inequality gives an upper bound on λ1(M) in terms of the dimension of M, a lower bound on its Ricci curvature, and h(M). With the help and inspiration of work of Buser and unpublished work of Ian Agol, I found an analog of Buser's inequality which gives an upper bound on all of the higher eigenvalues of M in terms of the Cheeger constant. This result is a comparison between the eigenvalues of the Laplacian on M and the eigenvalues of a Sturm-Liouville (ODE) problem which depends on all of the same invariants of the manifold as Buser's inequality, namely the dimension, lower bound on Ricci curvature, and h(M).

My initial motivation for studying h(M) and λ1(M) came from the work of Lackenby, who showed how these quantities are related several problems in the theory of 3-manifolds such as the virtually Haken conjecture (later resolved by Ian Agol) and the rank versus Heegard genus question for hyperbolic 3-manifolds (later resolved by Tao Li). Despite the resolution of many of these problems, Lackenby's work provides more than adequate motivation to classify sets achieving the Cheeger constant (Cheeger minimizers) and improve our ability to compute the Cheeger constant for hyperbolic 2- and 3-manifolds. Using a classification of isoperimetric regions of hyperbolic surfaces given by Adams and Morgan, I have given such a classification for hyperbolic surfaces with finite area in a recent preprint.

In forthcoming work with Grant Lakeland, we consider an application of the Cheeger constant to study hyperbolic reflection groups. More details on this work will be available soon!

G-Parking Functions

For a finite graph G, Deeparnab Chakrabarty, Prasad Tetali, and I considered combinatorial descriptions of G-parking functions. Specifically, their relationship to the spanning trees of G through acyclic orientations of the graph with a unique source and a unique sink. In addition, while there were several known combinatorial bijections between the spanning trees of G and G-parking functions, we gave a new bijection and discussed how our bijection differs from other bijections in the literature.

Manuscripts

Published

1. Mean Value Theorems for Riemannian Manifolds via the Obstacle Problem. Joint with Ivan Blank and Jeremy LeCrone. Journal of Geometric Analysis, 24 pages, DOI:10.1007/s12220-018-0093-4, arXiv:1704.07518.

2. 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.

3. 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.

4. Sturm-Liouville Estimates for the Spectrum and Cheeger Constant. Int Math Res Not, Vol. 2015, No. 16, pp. 7510–7551. rnu175. arXiv:1308.5936 MR3428972 MSC 58J50

5. 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 MR2592488 MSC 05C57

Preprints and Submitted

1. Volume growth, curvature, and Buser-type inequalities in graphs. Joint with Peter Ralli and Prasad Tetali. arXiv:1802.01952 Submitted

2. The Cheeger Constant, Isoperimetric Problems, and Hyperbolic Surfaces. arXiv:1509.08993 Revising

In Preparation

1. Spectra and Cheeger Constants of Arithmetic Hyperbolic Reflection Groups. Joint with Grant S. Lakeland and Holger Then. A video describing a portion of this work.

Miscellaneous Information

MathSciNet MR Author ID: 892840

Erdös Number: 2 (through Prasad Tetali)