In this thesis, we study the dynamics of magnetic flows on compact nilmanifolds. Magnetic flows are generalizations of geodesic flows. They model the motion of a particle of unit mass and unit charge in a smooth manifold M in the presence of a...
In this thesis we study 1=k-geodesics, those closed geodesics that minimize on any subinterval of length L=k, where L is the length of the geodesic. These curves arise as critical points of the uniform energy, a function introduced in Morse theory...
Spectral theory is the subfield of differential geometry which provided the solution to Kac''s famous question, "Can you hear the shape of a drum?" That is, can we use the Laplace spectrum of a manifold to draw conclusions about its geometry or...
We associate, to each positive integer n , a Cayley graph to the group PSL(2.Ζ[subscript n]). We then consider the isoperimetric numbers of these graphs. In chapter three we determine upper bounds for the isoperimetric number by a detailed...
A metric structure on a set gives a concept of distance between any two elements of that set, and it induces a topology. In this thesis, we provide several ways to put a metric structure on the collection of CW complexes. We accomplish this by...
This thesis investigates the embedding theory of orders in central simple algebras, placing a particular emphasis on the role that the phenomenon known as selectivity plays in the theory. Although the notion of selectivity is completely algebraic,...
Medical imaging methods have become increasingly important in diagnosing diseases and assisting therapeutic treatment. In particular, early detection of breast cancer is considered as a critical factor in reducing the mortality rate of women....
We prove the existence of nontrivial multiparameter isospectral deformations of metrics on the classical compact simple Lie groups SO (n) (n = 9, n ≥11), Spin(n) (n = 9, n ≥11), SU (n) (n ≥7), and Sp (n) (n ≥5). The proof breaks into three...
Motivated by quantum mechanics and geometric optics, it is a long-standing problem whether the length spectrum of a compact Riemannian manifold can be recovered from its Laplace spectrum. One route to proving that the length spectrum depends on the...
Given graded C *-algebras A and B , we define the notion of an admissible pair ([straight phi], D ) for A and B . Associated to an admissible pair ([straight phi], D ) is an equivalence class of asymptotic morphisms from A to B . Under certain...
Riemannian manifolds. Singularities (Mathematics). Laplacian operator. Spectral theory (Mathematics). Riemann surfaces. Curves on surfaces. Geometry
Historically, inverse spectral theory has been concerned with the relationship between the geometry and the spectrum of compact Riemannian manifolds, where spectrum means the eigenvalue spectrum of the Laplace operator as it acts on smooth...
Manifolds (Mathematics) Geodesics (Mathematics) Space and time.
We investigate weak and strong refocusing of light rays in a space-time and related concepts. A strongly causal space-time ( X^ n +1 , g ) is emphstrongly refocusing at x ∈ X if there is a point y ≠ x such that all null-geodesics through y pass...
Goldman and Turaev constructed a Lie bialgebra structure on the free Z-module generated by free homotopy classes of loops on an oriented surface. Turaev conjectured that the cobracket of A is zero if and only if A is a power of a simple class. Chas...