Georgia Institute of Technology College of Sciences

In the world of moderate Reynolds number, everyday turbulence of fluids flowing across planes and down pipes a velvet revolution is taking place. Experiments are almost as detailed as the numerical simulations, DNS is yielding exact numerical solutions that one dared not dream about a decade ago, and dynamical systems visualization of turbulent fluid's state space geometry is unexpectedly elegant. We shall take you on a tour of this newly breached, hitherto inaccessible territory. Mastery of fluid mechanics is no prerequisite, and perhaps a hindrance: the talk is aimed at anyone who had ever wondered why - if no cloud is ever seen twice - we know a cloud when we see one? And how do we turn that into mathematics? (Joint work with J. F. Gibson)

Let A and B be attractors of two point-fibred iterated function systems with coding maps f and g. A transformations from A into B can be constructed by composing a branch of the inverse of f with g. I will outline the shape of the theory of such transformations, which are termed "fractal" because their graphs are typically of non-integer dimension. I will also describe the remarkable geometry of these transformations when the generating iterated functions systems are projective. Finally, I will show how they can be used to provide new insights into dynamical systems and also how they can be used to manipulate, filter, process and efficiently store digital images, and how they can be used in image synthesis, leading to applications in the visual arts.

The first order small-time approximation of the marginal distribution of a L\'evy process has been known for long-time. In this talk, I present higher order expansions polynomial in time for the distributions of a L\'evy process. As a secondary objective, I illustrate the application of our expansions in the estimation of financial models with jumps as well as in the study of the small-term asymptotic behavior of the implied volatility for this class of financial models. This talk presents joint work with C. Houdr\'e and M. Forde. Associated reading (available in the web site of the speaker): (1) Small-time expansions for the transition distribution of Levy processes. J.E. Figueroa-L\'opez and C. Houdré. Stochastic Processes and their Applications 119 pp. 3862-3889, 2009. (2) Nonparametric estimation of time-changed Levy models under high-frequency data. J.E. Figueroa-L\'opez. Advances in Applied Probability vol. 41, number 4, pp. 1161-1188, 2009. (3) The small-maturity smile for exponential Levy model. J.E. Figueroa-L\'opez and M. Forde. Preprint.