Series: ACO Student Seminar
Constraint Programming is a powerful technique developed by the Computer Science community to solve combinatorial problems. I will present the model, explain constraint propagation and arc consistency, and give some basic search heuristics. I will also go through some illustrative examples to show the solution process works.
Series: Research Horizons Seminar
In this introduction to knot theory we will focus on a class of knots called rational knots. Here the word rational refers to a beautiful theorem by J. Conway that sets up a one to one correspondence between these knots and the rational numbers using continued fractions. We aim to give an elementary proof of Conway's theorem and discuss its application to the study of DNA recombination. No knowledge of topology is assumed.
Series: Mathematical Biology Seminar
Variational Principles and Partial Differential Equations for Some Model Problems in Polycrystal Plasticity
Series: PDE Seminar
The yield set of a polycrystal may be characterized using variational principles associated to suitable supremal functionals. I will describe some model problems for which these can be obtained via Gamma-convergence of a class of "power-law" functionals acting on fields satisfying appropriate differential constraints, and I will indicate some PDEs which play a role in the analysis of these problems.
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 especially welcome!
Series: Geometry Topology Seminar
This is an expository talk. A classical theorem of Mazur gives a simple criterion for two closed manifolds M, M' to become diffeomorphic after multiplying by the Euclidean n-space, where n large. In the talk I shall prove Mazur's theorem, and then discuss what happens when n is small and M, M' are 3-dimensional lens spaces. The talk shall be accessible to anybody with interest in geometry and topology.
Monday, September 29, 2008 - 13:00 , Location: Skiles 255 , Silas Alben , School of Mathematics, Georgia Tech , Organizer: Haomin Zhou
We discuss two problems. First: When a piece of paper is crumpled, sharp folds and creases form. These are distributed over the sheet in a complex yet fascinating pattern. We study experimentally a two-dimensional version of this problem using thin strips of paper confined within rings of shrinking radius. We find a distribution of curvatures which can be fit by a power law. We provide a physical argument for the power law using simple elasticity and geometry. The second problem considers confinement of charged polymers to the surface of a sphere. This is a generalization of the classical Thompson model of the atom and has applications in the confinement of RNA and DNA in viral shells. Using computational results and asymptotics we describe the sequence of configurations of a simple class of charged polymers.
Series: Probability Working Seminar
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.