Thea Pignataro will carry out research in two areas. The first concerns the spectral theory of three dimensional manifolds and the second focusses on estimation of the topological entropy in dynamical systems. The spectral theory of manifolds centers around the derivation of geometric information from knowledge of the spectrum of the Laplacian on the manifold. This has its origins in questions such as "Can you hear the shape of a drum?", since the spectrum of the Laplacian is closely related to the harmonics of the surface. Her work on entropy is part of a wide ranging program of many researchers to measure the chaos in a system. Pignataro has already been involved in important work on the eigenvalues of the Laplacian on hyperbolic surfaces. The results there show that the small eigenvalues vary like the lengths of short separating geodesics. The more complicated the topology the more eigenvalues that can be estimated. She will now try to extend some of these ideas and results to three dimensional manifolds. This will involve overcoming quite difficult topological problems. In particular the ends of these manifolds are more complicated than those of surfaces and have only recently been understood. Her work on dynamical systems involves estimation of the topological entropy. This essentially measures the number of different orbit types of a system. She will investigate the connections between entropy and the volume growth of certain embedded submanifolds.
