Ph.D. Recipients and their Thesis Abstracts

Mathematical Physics

Algebra, Analysis, Combinatorics, Differential Equations / Dynamical Systems, Differential Geometry, Geometry, Interdisciplinary, Lie Groups, Logic, Mathematical Physics, PDE / Numerical Analysis, Probability, Statistics, Topology

The Heat Kernel Weighted Hodge Laplacian on Noncompact Manifolds

Abstract: On a compact orientable Riemannian manifold, the Hodge Laplacian \Delta has compact resolvent, therefore a spectral gap, and the dimension of the space {\Cal H}^p = ker \Delta^p of harmonic p-forms is a topological invariant. By contrast, on complete noncompact Riemannian manifolds \Delta is known to have various pathologies, among them the absence of a spectral gap and either a "too large" or "too small" space {\Cal H}^p.

We use a heat kernel measure d\mu to determine the size of the space of square-integrable forms and in constructing the appropriate Laplacian \Delta_\mu. We recover in the noncompact case certain results of Hodge's theory of \Delta in the compact case. If the Ricci curvature of a noncompact connected Riemannian manifold M is bounded below, then this "heat kernel weighted Laplacian" \Delta_\mu acts on functions on M in precisely the manner we would wish, that is, it has a spectral gap and a one-dimensional kernel. We prove that the kernel of \Delta_\mu on n-forms is zero-dimensional on M, as we expect from topology, if the Ricci curvature is nonnegative. On Euclidean space, there is a complete Hodge theory for \Delta_\mu. Weighted Laplacians have a duality analogous to Poincaré duality on noncompact manifolds, and a Künneth formula. We show that heat kernel-like measures give desirable spectral properties (compact resolvent) in certain general cases. In particular, we use measures with Gaussian decay to justify the statement: "every topologically tame manifold has a strong Hodge decomposition."

On the Essential Self-Adjointness of Dirichlet Operators on Nonlinear Path Space

Abstract: There are two points of view in studying loop groups depending on the domain one chooses for the differential operators. If G is a compact Lie group and g is its Lie algebra, one can consider functions defined on

W_g= {b: [0, 1] \rightarrow g; b continuous, b(0) = 0}

or functions defined on

W_G= {g: [0, 1] \rightarrow G; g continuous, g(0) = e.}

Getzler and Malliavin have formulated their work on analysis over Loop Spaces in terms of functions on W_g. We choose to work with spaces of functions on W_G since they reflect topological properties of G. We call W_G nonlinear path space. We make substantial progress towards the solution of the problem of essential self-adjointness of the number operator over the nonlinear path space in the natural presumed core. Consider the Cameron-Martin space H over g and for h in H let \partial _h be the directional derivative on W_G. The adjoint operator \partial*_h with respect to the "Wiener measure" P on W_G is given by –\partial_h+ j_h. For each partition 0 < T_1 < ... < T_m of [0,1] define the subspace {\cal F}^m_h of L^2(W_G, P) by pulling back C^{\infty}_c(R x G^m)with respect to the map \theta_m(\gamma) = (j_h, \gamma(T_1), ..., \gamma(T_m)) on W_G. The completion \bar{{\cal F}}^m_h of {\cal F}^m_h in L^2(W_G, P) is isomorphic to L^2(R x G^m, \rho_m) where \rho_m is the density of the distribution law of \theta_m. We appeal to Malliavin calculus to prove smoothness of \rho_m and then use Wielen's technique to prove that the Dirichlet operator \partial*_h \partial _h|_{{\cal F}^m_h} as an operator on L^2(R x G^m, \rho_m) is essentially self-adjoint in {\cal F}^m_h. The essential self-adjointness of \partial*_h \partial _h in U {\cal F}^m_h is deduced easily due to the invariance of {\cal F}^m_h with respect to \partial*_h \partial _h.