The Laplacian appears in not only in mathematical physics, but also in representation theory, geometry, and dynamical systems. There is a well-known but still somewhat mysterious relationship between the spectrum of the Laplcian and the geometry and topology of the underlying domain. For example, I am interested in whether the second Neumann eigenfunction on a convex plane domain has any critical points in the interior of the domain. In Euclidean triangles have no hot spots, Sugata Mondal and I show that acute triangles have no interior critical points.
Homeomorphisms of surfaces can be studied via their action on the orbifold universal cover of the moduli space of hyperbolic surfaces, the Teichmueller space. I am interested in the most prevalent type of homeomorphism---known as pseudo-Anosov maps---and the action of such a homeomorphism on spaces related to Teichmueller space. (See, for example, Ellipses in translation surfaces.)
The function theory of the quotient of the upper half plane H by the modular group SL(2,Z) is fundamental in number theory. For example, Selberg developed his well-known trace formula to show that H/SL(2,Z) has a plethora of Laplace eigenfunctions that are square-integrable. However, Phillips and Sarnak conjectured that the random hyperbolic surface has no square-integrable eigenfunctions. In Hyperbolic triangles without embedded eigenvalues , Luc Hillairet and I give the strongest evidence to date for this conjecture.