Seminars and Colloquia by Series

Self-intersection of random paths

Series
Combinatorics Seminar
Time
Friday, October 17, 2008 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Ravi MontenegroUniversity of Massachussetts
The Birthday Paradox says that if there are N days in a year, and 1.2*sqrt(N) days are chose uniformly at random with replacement, then there is a 50% probability that some day was chosen twice. This can be interpreted as a statement about self-intersection of random paths of length 1.2*sqrt(N) on the complete graph K_N with loops. We prove an extension which shows that for many graphs random paths with length of order sqrt(N) will have the same self-intersection property. We finish by discussing an application to the Pollard Rho Algorithm for Discrete Logarithm. (joint work with Jeong-Han Kim, Yuval Peres and Prasad Tetali).

Three closed, nonselfintersecting geodesics on the sphere

Series
Geometry Topology Working Seminar
Time
Friday, October 17, 2008 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Jim KrysiakSchool of Mathematics, Georgia Tech
This will be a continuation of the previous talk by this title. Specifically, 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.

A Geometric Description of Adaptation in Nonparametric Functional Estimation

Series
Stochastics Seminar
Time
Thursday, October 16, 2008 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Tony CaiDepartment of Statistics, The Wharton School, University of Pennsylvania
Adaptive estimation of linear functionals occupies an important position in the theory of nonparametric function estimation. In this talk I will discuss an adaptation theory for estimation as well as for the construction of confidence intervals for linear functionals. A between class modulus of continuity, a geometric quantity, is shown to be instrumental in characterizing the degree of adaptability and in the construction of adaptive procedures in the same way that the usual modulus of continuity captures the minimax difficulty of estimation over a single parameter space. Our results thus "geometrize" the degree of adaptability.

Topology of Riemannian submanifolds with prescribed boundary

Series
School of Mathematics Colloquium
Time
Thursday, October 16, 2008 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Mohammad GhomiSchool of Mathematics, Georgia Tech
We prove that a smooth compact submanifold of codimension $2$ immersed in $R^n$, $n>2$, bounds at most finitely many topologically distinct compact nonnegatively curved hypersurfaces. This settles a question of Guan and Spruck related to a problem of Yau. Analogous results for complete fillings of arbitrary Riemannian submanifolds are obtained as well. On the other hand, we show that these finiteness theorems may not hold if the codimension is too high, or the prescribed boundary is not sufficiently regular. Our proofs employ, among other methods, a relative version of Nash's isometric embedding theorem, and the theory of Alexandrov spaces with curvature bounded below, including the compactness and stability theorems of Gromov and Perelman. These results consist of joint works with Stephanie Alexander and Jeremy Wong, and Robert Greene.

Dynamics of Functions with an Eventual Negative Schwarzian Derivaitve

Series
Research Horizons Seminar
Time
Wednesday, October 15, 2008 - 12:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Ben WebbSchool of Mathematics, Georgia Tech
In the study of one dimensional dynamical systems it is often assumed that the functions involved have a negative Schwarzian derivative. However, as not all one dimensional systems of interest have this property it is natural to consider a generalization of this condition. Specifically, we consider the interval functions of a real variable having some iterate with a negative Schwarzian derivative and show that many known results generalize to this larger class, that is to functions with an eventual negative Schwarzian derivative. The property of having an eventual negative Schwarzian derivative is nonasymptotic therefore verification of whether a function has such an iterate can often be done by direct computation. The introduction of this class was motivated by some maps arising in neuroscience.

Dynamics and implications of some models of hepatitis B virus infection

Series
Mathematical Biology Seminar
Time
Wednesday, October 15, 2008 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Yang KuangArizona State University
Chronic HBV infection affects 350 million people and can lead to death through cirrhosis-induced liver failure or hepatocellular carcinoma. We present the rich dynamics of two recent models of HBV infection with logistic hepatocyte growth and a standard incidence function governing viral infection. One of these models also incorporates an explicit time delay in virus production. All model parameters can be estimated from biological data. We simulate a course of lamivudine therapy and find that the models give good agreement with clinical data. Previous models considering constant hepatocyte growth have permitted only two dynamical possibilities: convergence to a virus free or an endemic steady state. Our models admit periodic solutions. Minimum hepatocyte populations are very small in the periodic orbit, and such a state likely represents acute liver failure. Therefore, the often sudden onset of liver failure in chronic HBV patients can be explained as a switch in stability caused by the gradual evolution of parameters representing the disease state.

Knots in contact 3-manifolds

Series
Geometry Topology Seminar
Time
Friday, October 10, 2008 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Vera VertesiSchool of Mathematics, Georgia Tech
In this talk I will give a purely combinatorial description of Knot Floer Homology for knots in the three-sphere (Manolescu-Ozsvath-Szabo- Thurston). In this homology there is a naturally associated invariant for transverse knots. This invariant gives a combinatorial but still an effective way to distinguish transverse knots (Ng-Ozsvath-Thurston). Moreover it leads to the construction of an infinite family of non-transversely simple knot-types (Vertesi).

The giant component in a random subgraph of a given graph

Series
Combinatorics Seminar
Time
Thursday, October 9, 2008 - 16:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Lincoln LuUniversity of South Carolina
We consider a random subgraph G_p of a host graph G formed by retaining each edge of G with probability p. We address the question of determining the critical value p (as a function of G) for which a giant component emerges. Suppose G satisfies some (mild) conditions depending on its spectral gap and higher moments of its degree sequence. We define the second order average degree \tilde{d} to be \tilde{d}=\sum_v d_v^2/(\sum_v d_v) where d_v denotes the degree of v. We prove that for any \epsilon > 0, if p > (1+ \epsilon)/\tilde{d} then almost surely the percolated subgraph G_p has a giant component. In the other direction, if p < (1-\epsilon)/\tilde{d} then almost surely the percolated subgraph G_p contains no giant component. (Joint work with Fan Chung Graham and Paul Horn)

Hyperbolic geometry and Jones polynomial of embedded graphs

Series
Graph Theory Seminar
Time
Thursday, October 9, 2008 - 12:05 for 1.5 hours (actually 80 minutes)
Location
Skiles 255
Speaker
Roland van der VeenUniversity of Amsterdam
The aim of this talk is to introduce techniques from knot theory into the study of graphs embedded in 3-space. The main characters are hyperbolic geometry and the Jones polynomial. Both have proven to be very successful in studying knots and conjecturally they are intimately related. We show how to extend these techniques to graphs and discuss possible applications. No prior knowledge of knot theory or geometry will be assumed.

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