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.