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#### Generalized billiard paths and Morse theory for manifolds with corners

(1999)

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

(1998)

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 $-$ ...

#### Discrete Morse theory and the geometry of nonpositively curved simplicial complexes

(2001)

Understanding the conditions under which a simplicial complex collapses is a central issue in many problems in topology and combinatorics. Let K be a simplicial complex endowed with the piecewise Euclidean geometry given ...

#### Inverse spectral problems with incomplete knowledge of the spectrum

(1999)

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

(1993)

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 ...

#### Spaces with positive combinatorial curvature

(2005)

We present two results concerning spaces with "positive combinatorial curvature". The first is analogous to the Bonnet-Myers theorem and the second to the maximum-diameter sphere theorems of Toponogov [6] and Cheng [7]. ...

#### Morse-Bott functions and the Witten Laplacian

(1997)

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 ...

#### Subdivisions and secondary invariants

(2007)

Combinatorial transgressions are secondary invariants of a triangulable space analogous to transgressive forms such as those arising in Chern-Weil theory. Unlike combinatorial characteristic classes, combinatorial ...

#### Anomalies in finite dimensions

(1993)

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 ...