Georgia Institute of Technology College of Sciences

Suppose you want to stir a pot of soup with several spoons. What is the most efficient way to do this? Thurston's theory of surface homeomorphisms gives us a concrete way to analyze this question. That is, to each mixing pattern we can associate a real number called the entropy. We'll start from scratch with a simple example, state the Nielsen-Thurston classification of surface homeomorphisms, and give some open questions about entropies of surface homeomorphisms.

In this talk we will discuss the vanishing viscosity limit of the Navier-Stokes equations to the isentropic Euler equations for one-dimensional compressible fluid flow. We will follow the approach of R.DiPerna (1983) and reduce the problem to the study of a measure-valued solution of the Euler equations, obtained as a limit of a sequence of the vanishing viscosity solutions. For a fixed pair (x,t), the (Young) measure representing the solution encodes the oscillations of the vanishing viscosity solutions near (x,t). The Tartar-Murat commutator relation with respect to two pairs of weak entropy-entropy flux kernels is used to show that the solution takes only Dirac mass values and thus it is a weak solution of the Euler equations in the usual sense. In DiPerna's paper and the follow-up papers by other authors this approach was implemented for the system of the Euler equations with the artificial viscosity. The extension of this technique to the system of the Navier-Stokes equations is complicated because of the lack of uniform (with respect to the vanishing viscosity), pointwise estimates for the solutions. We will discuss how to obtain the Tartar-Murat commutator relation and to work out the reduction argument using only the standard energy estimates. This is a joint work with Gui-Qiang Chen (Oxford University and Northwestern University).

It has been suggested by Y. Meyer and numerically confirmed by many othersthat dual spaces are good for texture recovery. Among the dual spaces, ourwork focuses on Sobolev spaces of negative differentiability to recovertexture from noisy blurred images. Such Sobolev spaces are good to modeloscillatory component, on the other hand, the spaces themselves hardlydistinguishes texture component from noise component because noise is alsoconsidered to be a highly oscillatory component. In this talk, in additionto oscillatory component recovery, we will further investigate aone-parameter family of Sobolev norms to achieve such a separation task.

I will talk about how tropical geometry can be used for implicitization and elimination problems. Implicitization is the problem of finding the defining equations (implicit equations) of an algebraic variety from a given parameterization. Elimination is the problem of finding the defining equations of a projection of an algebraic variety. In some instances such as the case when the polynomials involved have generic coefficients, we give a combinatorial construction of the tropical varieties without actually computing the defining polynomials. Tropical varieties can then be used to compute invariants of the original varieties.