Geometry in the Presence of Symmetries

I enjoy working on problems in Riemannian geometry which involve Lie groups and group actions. In particular, I like to consider questions in spectral geometry. That is, I am interested in the extent to which one can “hear” the geometry of a Riemannian manifold. Recently, I have also started to think about problems concerning Riemannian submersions and manifolds of nonnegative curvature. If you are an undergraduate or graduate student with an interest in conducting research in geometry I would be more than happy to discuss projects with you and/or to suggest some background reading.

Papers

**Detecting the moments of inertia of a molecule via its rotational spectrum**, in preparation.with E. Makover & H. Parlier, Manuscripta Math. 139 (2012), no. 3-4, 515-534.**Constructing metrics with a partially prescribed stable norm,**with B. de Smit & R. Gornet, Geom. Dedicata 153 (2012), no. 1, 343-352.*Isospectral surfaces with distinct covering spectra via Cayley graphs,***Two remarks on the length spectrum of a Riemannian manifold**, with B. Schmidt, Proc. Amer. Math. Soc. 139 (2011), 4113-4119.**Sunada’s method and the covering spectrum**, with B. de Smit & R. Gornet, J. Differential Geom., 86 (2010), no. 3, 501-537.**Spectral isolation of bi-invariant metrics on compact Lie groups**, with C. Gordon & D. Schueth, Ann. Inst. Fourier (Grenoble) 60 (2010), 1617-1628.**Spectral isolation of naturally reductive metrics on simple Lie groups**, with C. Gordon, Math. Z. 266 (2010), 979-995.**Equivariant isospectrality and Sunada's method**, Arch. Math. 95 (2010), 75-85.**Measures invariant under the geodesic flow and their projections**, Proc. Amer. Math. Soc. 131 (2003), 2933-2936.**Isospectral simply-connected homogeneous spaces and the spectral rigidity of group actions**, Comment. Math. Helv. 77 (2002), 701-717.

Thesis

**Applications of representation theory to dynamics and spectral geometry**, Ph. D. Thesis, University of Michigan, Ann Arbor, MI USA 48104, 2001. (Advisor: Ralf Spatzier )