We study the inverse problem of determining both the source of a wave and its speed inside a medium from measurements of the solution of the wave equation on the boundary. This problem arises in photoacoustic and thermoacoustic tomography, and has important applications in medical imaging. We prove that if $c^{-2}$ is harmonic in $\omega \subset \R^3$ and identically 1 on $\omega^c$, where $\omega$ is a simply connected region, then a non-trapping wave speed $c$ can be uniquely determined from the solution of the wave equation on boundary of $\Omega \supset \supset \omega$ without the knowledge of the source. We also show that if the wave speed $c$ is known and only assumed to be bounded then, under mild assumptions on the set of discontinuous points of $c$, the source of the wave can be uniquely determined from boundary measurements.

In  Ramsey  Theory, ultrafilters often play an instrumental role.
By using nonstandard models of the integers, one can replace those
third-order objects (ultrafilters are families of subsets) by simple
points.

In this talk we present a nonstandard technique that is grounded
on the above observation, and show its use in proving some new results
in Ramsey Theory of Diophantine equations.
 

A classical result of Goldman states that character variety of an oriented surface is a symplectic algebraic variety, and that the Goldman Lie algebra of free loops on the surface acts by Hamiltonian vector fields on the character variety. I will describe a vast generalization of these results, including to higher dimensional manifolds where the role of the Goldman Lie algebra is played by the Chas-Sullivan string bracket in the string topology of the manifold. These results follow from a general statement in noncommutative geometry. In addition to generalizing Goldman's result to string topology, we obtain a number of other interesting consequences including the universal Hitchin system on a Riemann surface. This is joint work with Chris Brav.

A classical result of Goldman states that character variety of an oriented surface is a symplectic algebraic variety, and that the Goldman Lie algebra of free loops on the surface acts by Hamiltonian vector fields on the character variety. I will describe a vast generalization of these results, including to higher dimensional manifolds where the role of the Goldman Lie algebra is played by the Chas-Sullivan string bracket in the string topology of the manifold. These results follow from a general statement in noncommutative geometry. In addition to generalizing Goldman's result to string topology, we obtain a number of other interesting consequences including the universal Hitchin system on a Riemann surface. This is joint work with Chris Brav.