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Generalized billiard paths and Morse theory for manifolds with corners
A billiard path on a manifold M embedded in Euclidean space is a series of line segments connecting reflection points on M. In a generalized billiard path we also allow the path to pass through M. The two segments at a ...
Can you hear the size of the vertices? An inverse spectral problem of Laplacians on weighted graphs
Let G be a simple graph with n vertices. We define a Laplacian $\Delta$ on G which depends on an assignment of a weight to each vertex of G. One of the eigenvalues of $\Delta$ will always be 0. We fix the remaining (n $-$ ...
Inverse spectral problems with incomplete knowledge of the spectrum
To solve an inverse spectral problem, we try to discover an operator of a certain form that has a prescribed spectrum. In this thesis we proceed in two different settings, both times considering the potential function as ...
Critical points of the determinant of the Laplace operator
The determinant of the Laplace operator, det $\Delta$, is a function on the set of metrics on a compact manifold. Generalizing the work of Osgood, Phillips, and Sarnak on surfaces, we consider one-parameter variations of ...
Morse-Bott functions and the Witten Laplacian
Given a compact Riemannian manifold (N, g), a flat vector bundle V over N, and a Morse-Bott function h, Witten considered the following one-parameter deformation of the differential d in the de Rham complex of V-valued ...
Anomalies in finite dimensions
A problem in quantum mechanics is finding a determinant function on linear maps between two vector spaces. In this paper we consider the question in the context of finite dimensional vector spaces. Given two finite dimensional ...