I will describe the statistical mechanical derivation of mesoscopic free energy functionals for systems interacting via both short range and long range (Kac) potentials.  These functionals are useful for describing the spatial structure of components in various types of phase transitions.  I will then consider the phenomenological Cahn Hilliard free energy functional and derive criteria for stability of droplets of the minority phase just inside the coexistence region.

The ability to effectively control a fluid would enable many exciting technological advances, including the design of quieter, more efficient aircraft. Most of the flow control strategies tried so far have been largely ad hoc, and have not used many of the available tools from control theory and dynamical systems, which can guide controller design as well as placement of sensors and actuators. These tools require knowledge of a model of the system in terms of a system of differential equations, and the equations governing a fluid, though known, are too complex for these tools to apply. This talk addresses model reduction techniques, which are used to simplify existing models, to obtain low-order models tractable enough to be used for analysis and control, while retaining the essential physics. These techniques provide a bridge between complex problems and the mathematical tools useful for their analysis.

Specifically, the talk will focus on recent developments of two techniques, Proper Orthogonal Decomposition (POD) and balanced truncation. Each of these techniques has strengths and weaknesses, and we show how ideas from both techniques may be combined, to exploit their strengths. We illustrate the methods by obtaining reduced-order models for a compressible flow past a cavity, and an incompressible channel flow.

Natural images taken with a digital camera can be viewed as vectors in a high-dimensional vector space whose dimension is the number of pixels. To understand the set of natural images within this vector space is a very interesting problem, but as stated it is very difficult and likely intractable. A. Lee, D. Mumford, and K. Pedersen have created a data set consisting of small (3 by 3) patches, and one can then ask questions about this set. We (V. de Silva, T. Ishkanov, and myself) have used algebraic topological techniques to obtain information about this set, and I will discuss this application of topological methods in this talk. I will also discuss potential applications in compression and in the neuroscience of vision.

Sphere packings in high dimensions are of great interest to mathematicians and physicists, and have direct applications in communications theory. Remarkably, no one has been able to provide exponential improvement on a 100-year-old lower bound on the maximal packing density due to Minkowski in d-dimensional Euclidean space \Re^d. The asymptotic behavior of this bound is controlled by 2^{-d} in high dimensions. Using an optimization procedure that we introduced earlier [1] and a conjecture concerning the existence of disordered sphere packings in \Re^d, we obtain a provisional lower bound on the density whose asymptotic behavior is controlled by 2^{-0.7786}, thus providing the putative exponential improvement of Minkowski's bound [2]. The conjecture states that a hard-core nonnegative tempered distribution is a pair correlation function of a translationally invariant disordered sphere packing in \Re^d for asymptotically large d if and only if the Fourier transform of the autocovariance function is nonnegative. The conjecture is supported by two explicit analytically characterized disordered packings, numerical simulations in low dimensions, and known necessary conditions that only have relevance in very low dimensions. A byproduct of our approach is an asymptotic lower bound on the average kissing number whose behavior is controlled by 2^{0.2213}, which is to be compared to the best known asymptotic lower bound on the individual kissing number of 2^{0.2075}. Interestingly, our optimization procedure is precisely the dual of a primal linear program devised by Cohn and Elkies [3] to obtain upper bounds on the density, and hence has implications for linear programming bounds.

• S. Torquato and F. H. Stillinger, "Controlling the Short-Range Order and Packing Densities of Many-Particle Systems," Journal of Physical Chemistry B, 106, 8354 (2002); ibid, 106, 11406 (2002).
• S. Torquato and F. H. Stillinger, "New Provisional Lower Bounds on the Optimal Density of Sphere Packings," http://arxiv.org/abs/math.MG/0508381.
• H. Cohn and N. Elkies, "New upper bounds on sphere packings I," Annals of Mathematics, 157, 689 (2003); H. Cohn, "New upper bounds on sphere packings II," Geometry and Topology, 6, 329 (2002).

"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.