Abstract. It has been shown in [DPSU] that, under some additional assumptions, two simple domains with the same scattering data are equivalent. We show that the simplicity of a region can be read from the metric in the boundary and the scattering data. This lets us extend the results in [DPSU] to regions with the same scattering data, where only one is known apriori to be simple. We will then use this results to resolve a local version of a question by Robert Bryant. That is, we show that a surface of constant curvature can not be modified in a small region while keeping all the curves of some fixed constant geodesic curvatures closed.

Abstract. We consider the scattering and lens rigidity of compact surfaces with boundary that have a trapped geodesic. In particular we show that the flat cylinder and the flat Möbius strip are determined by their lens data. We also see by example that the flat Möbius strip is not determined by it's scattering data. We then consider the case of negatively curved cylinders with convex boundary and show that they are lens rigid.

Consider a compact manifold M with boundary ∂M endowed with a Riemannian metric g and a magnetic field . Given a point and direction of entry at the boundary, the scattering relation determines the point and direction of exit of a particle of unit charge, mass, and energy. In this paper we show that a magnetic system (M, ∂M, g, ) that is known to be real-analytic and that satisfies some mild restrictions on conjugate points is uniquely determined up to a natural equivalence by . In the case that the magnetic field is taken to be zero, this gives a new rigidity result in Riemannian geometry that is more general than related results in the literature.

Abstract. We say that a pair of points x and y is secure if there exist a finite set of blocking points such that any geodesic between x and y passes through one of the blocking points. The main point of this paper is to exhibit new examples of blocking phenomena both in the manifold and the billiard table setting. As an approach to this, we study if the product of secure configurations (or manifolds) is also secure.We introduce the concept of midpoint security that imposes that the geodesic reaches a blocking point exactly at its midpoint. We prove that products of midpoint secure configurations are midpoint secure. On the other hand, we give an example of a compact C 1 surface that contains secure configurations that are not midpoint secure. This surface provides the first example of an insecure product of secure configurations, as well as billiard table examples.

Abstract. In this paper we give conditions on f so that problemhas at least two radial bound state solutions with any prescribed number of zeros, and such that u(0) belongs to a specific subinterval of (0, ∞). This property will allow us to give conditions on f so that this problem has at least any given number of radial solutions having a prescribed number of zeros.