Georgia Institute of Technology College of Sciences

We will consider the inverse problem of determining the sound
speed or index of refraction of a medium by measuring the travel times of
waves going through the medium. This problem arises in global seismology
in an attempt to determine the inner structure of the Earth by measuring
travel times of earthquakes. It also has several applications in optics
and medical imaging among others.

The problem can be recast as a geometric problem: Can one determine
the Riemannian metric of a Riemannian manifold with boundary by
measuring the distance function between boundary points? This is the
boundary rigidity problem.

We will also describe some recent results, joint with Plamen Stefanov
and Andras Vasy, on the partial data case, where you are making
measurements on a subset of the boundary.

I'll talk about the structure of the collection of all n-ples of eigenvalues of elements of Zariski-dense subgroups D of SL(n,R). Subgroups like this appear, for instance, as the images of holonomy representations of geometric structures. Our focus is a deep and useful result of Benoist, which states that the natural cone one is led to consider here has strong convexity and non-degeneracy properties, and a succinct, qualitative characterization of the cones that so arise from Zariski-dense subgroups. The theorem comes out of a study of the dynamics of the actions of D on spaces of flags such as RP^n and the collection of open subsemigroups of SL(n,R). Everything in this talk is from Benoist’s paper Propriétés Asymptotiques des Groupes Linéaires (GAFA, 2002), and holds in far more generality than I'll state.

Transport equations arise in the modelling of several complex systems, including mean field games. Such equations often involve nonlinear dependence of the solution in the drift. These nonlinear transport equations can be understood by developing a theory for transport equations with irregular drifts. In this talk, I will outline the well-posedness theory for certain transport equations in which the drift has a one-sided bound on the divergence, yielding contractive or expansive behavior, depending on the direction in which the equation is posed. The analysis requires studying the relationship between the transport and continuity equations and the associated ODE flow. The theory is then used to discuss certain nonlinear transport equations arising in the study of finite state-space mean field games. This is joint work with P.-L. Lions.

In high dimensional contact and symplectic topology, finding interesting constructions for Legendrian submanifolds is an active area of research. Further, it is desirable that the constructions lend themselves nicely to computation of invariants. The doubling construction was defined by Ekholm, which uses Lagrangian fillings of a Legendrian knot in standard contact R^{2n-1} to produce a closed Legendrian submanifold in standard contact R^{2n+1}. Later Courte-Ekholm showed that symmetric doubles of embedded fillings are "uninteresting". In recent work the symmetric doubling construction was generalised to any contact manifold, giving two isotopic constructions related to open book decompositions of the ambient manifold. In a separate joint work with James Hughes, we explore the asymmetric doubling construction through Legendrian weaves.