Georgia Institute of Technology College of Sciences

For dimensions n greater than or equal to 3, and integers N greater than 1, there is a distribution of points P in a unit cube [0,1]^{n}, of cardinality N, for which the discrepancy function D_N associated with P has an optimal Exponential Orlicz norm. In particular the same distribution will have optimal L^p norms, for 1 < p < \infty. The collection P is a random digit shift of the examples of W.L. Chen and M. Skriganov.

In this talk we will first present several breakdown mechanisms of Uniformly Hyperbolic Invariant Tori (FHIT) in area-preserving skew product systems by means of numerical examples. Among these breakdowns we will see that there are three types: Hyperbolic to elliptic (smooth bifurcation), the Non-smooth breakdown and the Folding breakdown. Later, we will give a theoretical explanation of the folding breakdown. Joint work with Alex Haro.

In synthetic aperture radar (SAR) imaging, two important applications are formation of high resolution images and motion estimation of moving targets on the ground. In scenes with both stationary targets and moving targets, two problems arise. Moving targets appear in the computed image as a blurred extended target in the wrong location. Also, the presence of many stationary targets in the vicinity of the moving targets prevents existing algorithms for monostatic SAR from estimating the motion of the moving targets. In this talk I will discuss a data pre-processing strategy I developed to address the challenge of motion estimation in complex scenes. The approach involves decomposing the SAR data into components that correspond to the stationary targets and the moving targets, respectively. Once the decomposition is computed, existing algorithms can be applied to compute images of the stationary targets alone. Similarly, the velocity estimation and imaging of the moving targets can then be carried out separately.The approach for data decomposition adapts a recent development from compressed sensing and convex optimization ideas, namely robust principle component analysis (robust PCA), to the SAR problem. Classicalresults of Szego on the distribution of eigenvalues for Toeplitz matrices and more recent results on g-Toeplitz and g-Hankel matrices are key for the analysis. Numerical simulations will be presented.

A tropical complex is a Delta-complex together with some additional numerical data, which come from a semistable degeneration of a variety. Tropical complexes generalize to higher dimensions some of the analogies between curves and graphs. I will introduce tropical complexes and explain how they relate to classical algebraic geometry.