Georgia Institute of Technology College of Sciences

Robustness of several nonparametric multivariate "threshold type" outlier identification procedures is studied, employing a masking breakdown point criterion subject to a fixed false positive rate. The procedures are based on four different outlyingness functions: the widely-used "Mahalanobis distance" version, a new one based on a "Mahalanobis quantile" function that we introduce, one based on the well-known "halfspace" depth, and one based on the well-known "projection" depth. In this treatment, multivariate location outlyingness functions are formulated as extensions of univariate versions using either "substitution" or "projection pursuit," and an equivalence paradigm relating multivariate depth, outlyingness, quantile, and centered rank functions is applied. Of independent interest, the new "Mahalanobis quantile" outlyingness function is not restricted to have elliptical contours, has a transformation-retransformation representation in terms of the well-known spatial outlyingness function, and corrects to full affine invariance the orthogonal invariance of that function. Here two special tools, also of independent interest, are introduced and applied: a notion of weak covariance functional, and a very general and flexible formulation of affine equivariance for multivariate quantile functions. The new Mahalanobis quantile function inherits attractive features of the spatial version, such as computational ease and a Bahadur-Kiefer representation. For the particular outlyingness functions under consideration, masking breakdown points are evaluated and compared within a contamination model. It is seen that for threshold type outlier identification the Mahalanobis distance and projection procedures are superior to the others, although all four procedures are quite suitable for robust ranking of points with respect to outlyingness. Reasons behind these differences are discussed, and directions for further study are indicated.

A Turning point is where solutions to differential equations change behavior from exponential to oscillatory. In this region approximate solutions given by the powerful WKB method break down. In a series of paper in the 30's and 40's Langer developed a transformation (the Langer transformation) that allows the development of good approximate solutions (in terms of Airy functions) in the region of the Turning point I will discuss a discrete analog of this transformation and show how it leads to nice asymptotic formulas for various orthogonal polynomials.

Granular materials are important in a wide variety of contexts, such as avalanches and industrial processing of powders and grains. In this talk, I discuss some of the issues in understanding how granular materials flow, and especially how they tend to segregate by size. The segregation process, known scientifically as kinetic sieving, and more colorfully as The Brazil Nut Effect, involves the tendency of small particles to fall into spaces created by large particles. The small particles then force the large particles upwards, as in a shaken can of mixed nuts, in which the large Brazil nuts tend to end up near the lid. I'll describe ongoing physics experiments, mathematical modeling of kinetic sieving, and the results of analysis of the models (which are nonlinear partial differential equations). Movies of simulations and exact solutions illustrate the role of shock waves after layers of small and large particles have formed.

Based on work of Schwarz and Oh, information coming from a filtration in Hamiltonian Floer homology can be used to construct "spectral invariants" for paths of Hamiltonian diffeomorphisms of symplectic manifolds. I will show how these invariants can be used to provide a unified approach to proving various old and new results in symplectic topology, including the non-degeneracy of the Hofer metric and some of its variants; a sharp version of an inequality between the Hofer-Zehnder capacity and the displacement energy; and a generalization of Gromov's non-squeezing theorem.