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Friday, September 26, 2008 - 14:00 ,
Location: Skiles 269 ,
Jim Krysiak ,
School of Mathematics, Georgia Tech ,
Organizer: John Etnyre

This will be a presentation of the classical result on the existence of three closed nonselfintersecting geodesics on surfaces diffeomorphic to the sphere. It will be accessible to anyone interested in topology and geometry.

Series: Stochastics Seminar

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.

Series: Combinatorics Seminar

Motivated by a question raised by P\'or and Wood in connection with compact embeddings of graphs into the grid {\mathbb Z}^d, we consider generalizations of the no-three-in-line-problem. For several pairs (d,k,l) we give algorithmic or probabilistic, combinatorial lower, and upper bounds on the largest sizes of subsets S of grid-points in the d-dimensional T \times ... \times T-grid, where T is large and no l distinct grid-points of S are contained in a k-dimensional affine or linear subspace.

Series: Research Horizons Seminar

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.

Wednesday, September 24, 2008 - 11:00 ,
Location: Skiles 255 ,
Reinhard Laubenbacher ,
Virginia Bioinformatics Institute and Department of Mathematics, Virginia Tech ,
Organizer: Christine Heitsch

Since John von Neumann introduced cellular automata in the 1950s to study self-replicating systems, algebraic models of different kinds have increased in popularity in network modeling in systems biology. Their common features are that the interactions between network nodes are described by "rules" and that the nodes themselves typically take on only finitely many states, resulting in a time-discrete dynamical system with a finite state space. Some advantages of such qualitative models are that they are typically intuitive, can accommodate noisy data, and require less information about a variety of kinetic and other parameters than differential equations models. Yet they can capture essential network features in many cases. This talk will discuss examples of different types of algebraic models of molecular networks and a common conceptual framework for their analysis.

Series: PDE Seminar

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.

Series: ACO Student Seminar

The notion of a correlation decay, originating in statistical physics, has recently played an important role in yielding deterministic approximation algorithms for various counting problems. I will try to illustrate this technique with two examples: counting matchings in bounded degree graphs, and counting independent sets in certain subclasses of claw-free graphs.

Series: Geometry Topology Seminar

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.

Series: Geometry Topology Seminar

I will discuss a relation between the HOMFLY polynomial of a knot, its extension for a closed 3-manifold, a special function, the trilogarithm, and zeta(3). Technically, this means that we consider perturbative U(N) Chern-Simons theory around the trivial flat connection, for all N, in an ambient 3-manifold. This is rigorous, and joint with Marcos Marino and Thang Le.

Series: Analysis Seminar

The Horn inequalities give a characterization of eigenvalues of self-adjoint n by n matrices A, B, C with A+B+C=0. The proof requires powerful tools from algebraic geometry. In this talk I will talk about our recent result of these inequalities that are indeed valid for self-adjoint operators of an arbitrary finite factors. Since in this setting there is no readily available machinery from algebraic geometry, we are forced to look for an analysts friendly proof. A (complete) matricial form of our result is known to imply an affirmative answer to the Connes' embedding problem. Geometers in town especially welcome!