"Seismic tomography" is the term geophysicists use for a collection of methods to use seismic waves to image the interior of the Earth, much like in a CAT scan. Tomographic imaging has led to important discoveries, such as the observation that ocean floor subducts to the bottom of the Earth's mantle and - more recently - that plumes of hot material rise up from the lower mantle.

In its simplest form, the approximations of geometrical optics are applied to high frequency seismic waves. These waves then follow raypaths and the most useful observable is a travel time along the ray: T = \int ds / v(r). In a typical interpretation, \mathcal O (10^6) data with a signal-to-noise ratio of order 1 are inverted for \mathcal O (10^4-10^5) parameters. The mathematical challenge is mostly that of an adequate regularization of the problem that minimizes artifacts. More accurate travel time measurements can be obtained using cross-correlation on digital seismograms with sensitivity to lower frequency. For such waves a first order perturbation theory is needed to include the effects of wave diffraction around small anomalies. The travel time becomes then frequency dependent, and T is given by a volume integral, with an increase by several orders of magnitude in the numerical effort. Finally, for the lowest frequency waves we use the whole waveform as data. These waveforms can be modeled by summation of normal modes, but the problem is inherently nonlinear and again a ray approximation is needed to render the inverse problem feasible. The challenge is to relax this constraint and take effects of diffraction into account. We shall speculate about the possible role of wavelets in meeting these challenges.

Thermal or stochastic effects are prevalent in physical, chemical, and biological systems. Particularly in small systems, noise can overpower the deterministic dynamics and lead to "rare events", events which would never be seen in the absence of noise. One example is the thermally-driven switching of the magnetization in small memory elements. Wentzell-Freidlin large deviation theory is a mathematical tool for studying rare events. It estimates their probability and also the "most likely switching pathway," which is the pathway in phase space by which rare events are most likely to occur. We explain how large deviation theory and concepts from stochastic resonance may be applied to analyze thermally-activated magnetization reversal in the context of the spatially uniform Landau-Lifschitz-Gilbert equations. The time-scales of the experiment are critical. One surprising and physically relevant result is that in multiple-pulse experiments, nonconvential "short-time switching pathways" can dominate. The effect is dramatic: the usual pathway (connected with the Arrhenius-law) underestimates the probability of switching by an exponential factor.

An advantage of the method via large deviation theory is that it generalizes to systems with spatial variation. To discuss the complications and richness that emerge when spatial variation is taken into account, we consider the (simpler) Allen-Cahn equation. In this context, the rare event of interest is phase transformation from u = -1 to u = +1, and the most likely switching pathway is a pathway through function space. A natural reduced problem emerges in the "sharp-interface limit." We give a brief overview of some results (rigorous in d = 1, heuristic in d > 1.)

The first part of the talk is joint work with Bob Kohn and Eric Vanden-Eijnden. The second part includes work that is also joint with Felix Otto and Yoshihiro Tonegawa.