Applications of representation theory to dynamics and spectral geometry.

We address the areas of dynamics and spectral geometry through the use of representation theory. In Chapter III we study dynamics. In particular, we establish that measures on the unit tangent bundle SX of a positively curved Riemannian manifold (X, m) which are even and invariant under the geodesic flow are not determined by their projection to X. This problem is motivated by a result of Flaminio which shows that in the case of S2, with a positively curved metric, a measure on S(S2 ) that is even and invariant under the geodesic flow is determined by its projection to S2. In Chapter IV we take up the study of spectral geometry. Specifically, we generalize Sunada's method for producing pairs of isospectral manifolds and use this generalization to produce pairs of compact, connected and simply connected normal homogeneous spaces which are isospectral and non-isometric. These are the first examples of isospcetral normal homogeneous spaces and also the first examples of isospectral homogeneous spaces G/ H1 and G/H2, where H1, H2 ≤ G are non-trivial connected subgroups. These homogeneous examples will also provide us with examples of group actions which have discrete spectra and are isospectral yet not measurably conjugate. We also show that if SU(3) is endowed with the bi-invariant metric, then the Aloff-Wallach spaces SU(3)/Tp.q can be distinguished from one another by the smallest positive eigenvalue of the Laplacian on functions, taking multiplicities into account.PhDMathematicsPure SciencesUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/126589/2/3016965.pd