My passion for mathematical research evolves from questions in spectral geometry, spectral graph theory, and theoretical physics as well as their connections to representation theory, Hamiltonian dynamics, and combinatorics. I particularly aim to transfer and unify ideas between these areas.

Broadly speaking, spectral geometry and spectral graph theory are concerned with relationships between geometric objects, e.g., Riemannian manifolds or graphs, and the spectra of associated self- adjoint operators, e.g., Laplace-Beltrami operators or graph Laplacians. The majority of my research (2008-present) has been concerned with negative results in inverse spectral geometry and spectral graph theory, i.e., with the construction of geometric objects that cannot be spectrally distinguished. Since spectral theory is the core element of quantum mechanics, its relations to classical mechanics have been investigated for nearly a century. In line with this tradition, I studied the classical and quantum aspects of periodic magnetic fields on manifolds as part of my Ph.D. (2009-2012). Using my background in functional analysis, I also collaborated with a former colleague at Dartmouth College on a side project in combinatorial representation theory (2013).

  1. Spectral geometry (Laplace-Beltrami and Steklov isospectral manifolds)
    • On inaudible properties of broken drums – Isospectrality with mixed Dirichlet-Neumann boundary conditions [submitted, pdf]
      I express the transplantation method in graph- and representation-theoretic terms, which allows for the generating of new transplantable pairs from given ones as well as a computer-aided search for isospectral pairs. In particular, I show that the Dirichlet spectrum of a manifold does not determine whether it is connected and that an orbifold can be Dirichlet isospectral to a manifold.
    • Line graphs and the transplantation method [submitted, pdf]
      I extend the graph-theoretic formulation of the transplantation method. Led by the theory of Brownian motion, I introduce vertex-colored and edge-colored line graphs that give rise to block diagonal transplantation matrices. In particular, I rephrase the transplantation method in terms of representations of free semigroups, and provide a method for generating adjacency cospectral weighted directed graphs.
    • Robin and Steklov isospectral manifolds, with Carolyn Gordon and David Webb [preprint, pdf]
      We adapt known methods for constructing isospectral manifolds to the context of mixed Dirichlet-Neumann-Robin boundary conditions. In so doing, we construct strongly Steklov isospectral manifolds whose Dirichlet-to- Neumann operators are isospectral at all frequencies.
  2. Spectral geometry (Schrödinger operators and Hamiltonian dynamics)
    • Magnetic Schrödinger operators and Mañé’s critical value [preprint, pdf]
      I study periodic magnetic Schrödinger operators on covers of closed manifolds in relation to Mañé’s critical energy values of the corresponding classical Hamiltonian systems. In particular, I show that the bottom of the spectrum is bounded from above by Mañé’s critical energy value provided that the covering transformation group is amenable. I moreover determine the spectra for various homogeneous spaces with left-invariant magnetic fields.
  3. Spectral graph theory (Isospectral quantum graphs, cospectral discrete graphs, graph zeta functions)
    • Zeta-equivalent digraphs: Simultaneous cospectrality [submitted, pdf]
      I introduce a zeta function of digraphs that determines, and is determined by, the spectra of all linear combinations of the adjacency matrix, its transpose, the out-degree matrix, and the in-degree matrix. I derive further characterizations of this zeta function and present a method for constructing zeta-equivalent digraphs.
    • Changing gears: Isospectrality via eigenderivative transplantation, with Peter Doyle [submitted, pdf]
      We introduce a new method for constructing isospectral quantum graphs that is based on transplanting derivatives of eigenfunctions. We also present simple digraphs with the same reversing zeta function, which generalizes the Bartholdi zeta function to digraphs.
  4. Combinatorial representation theory
    • Centralizers of the infinite symmetric group, with Zajj Daugherty, DMTCS Proceedings, 2014. [pdf]
      We introduce new approaches to the study of centralizer algebras of the infinite symmetric group S∞. Led by the double commutant relationship between finite symmetric groups and partition algebras, as well as the theory of symmetric functions in non-commuting variables, we study faithful non-unitary representations of S∞ on sequence spaces that have invariant elements and lead to centralizer algebras contained in partition algebras.