Series: Research Horizons Seminar
I will give a modern bijective proof of Kirchhoff's classical theorem relating the number of spanning trees in a graph to the Laplacian matrix of the graph. The proof will highlight some analogies between graph theory and algebraic geometry.
Series: PDE Seminar
We study generalized traveling front solutions of reaction-diffusion equations modeling flame propagation in combustible media. Although the case of periodic media has been studied extensively, until very recently little has been known for general disordered media. In this talk we will address questions of existence, uniqueness, and stability of traveling fronts in this framework.
Series: CDSNS Colloquium
Permutation entropy was introduced as a complexity measure of time series. Formally, it replaces the symbol blocks in the definition of Shannon entropy by the so-called ordinal patterns –a digest of the ups-and-downs along a finite orbit in a totally ordered state space. Later, this concept was extended to self maps of n-dimensional intervals, in metric and topological versions. It can be proven that, under some assumptions, the metric and topological permutation entropy coincide with their corresponding conventional counterparts. Besides its use as an entropy estimator, permutation entropy has found some interesting applications. We will talk about the detection of determinism in noisy time series, and the recovery of the control parameter from the symbolic sequences of a unimodal map (which allows to cryptanalize some chaotic ciphers).
Series: Geometry Topology Seminar
I will discuss a couple of applications of transverse knot theory to the classification of contact structures and braid theory. In particular I will make the statement "transverse knots classify contact structures" precise and then prove it (if we have time). I will also discuss how progress on two of Orevkov's questions concerning quasi-positive knots that have implications for Hilbert's 16th problem.
Series: Probability Working Seminar
This term, the main topic for the Probability Working Seminar will be the coupling method, broadly understood. In the first talk, some basics on coupling will be discussed along with classical examples such as the ergodic theorem for Markov chains.
Friday, February 13, 2009 - 15:00 , Location: Skiles 269 , Igor Belegradek , School of Mathematics, Georgia Tech , Organizer: John Etnyre
Comparison geometry studies Riemannian manifolds with a given curvature bound. This minicourse is an introduction to volume comparison (as developed by Bishop and Gromov), which is fundamental in understanding manifolds with a lower bound on Ricci curvature. Prerequisites are very modest: we only need basics of Riemannian geometry, and fluency with fundamental groups and metric spaces. In the first (2 hour) lecture I shall explain what volume comparison is and derive several applications.
Series: SIAM Student Seminar
Let V be a vector space over the field C of complex numbers and let GL(V) be the group of isomorphisms of onto itself. Suppose G is a finite group. A linear representation of G in V is a homomorphism from the group G into the group GL(V). In this talk, I will give a brief introduction to some basic theorems about linear representations of finite groups with concentration on the decomposition of a representation into irreducible sub-representations, and the definition and some nice properties of the character. At the end of the talk, I will re-prove the Burnside lemma in the group theory from the representation theory approach. Since I began learning the topic only very recently, hence an absolute novice myself, I invite all of you to the talk to help me learn the knowledge through presenting it to others. If you are familiar with the topic and want to learn something new, my talk can easily be a disappointment.
On creating a model assessment tool independent of data size and estimating the U statistic variance
Series: Stochastics Seminar
If viewed realistically, models under consideration are always false. A consequence of model falseness is that for every data generating mechanism, there exists a sample size at which the model failure will become obvious. There are occasions when one will still want to use a false model, provided that it gives a parsimonious and powerful description of the generating mechanism. We introduced a model credibility index, from the point of view that the model is false. The model credibility index is defined as the maximum sample size at which samples from the model and those from the true data generating mechanism are nearly indistinguishable. Estimating the model credibility index is under the framework of subsampling, where a large data set is treated as our population, subsamples are generated from the population and compared with the model using various sample sizes. Exploring the asymptotic properties of the model credibility index is associated with the problem of estimating variance of U statistics. An unbiased estimator and a simple fix-up are proposed to estimate the U statistic variance.
Series: Job Candidate Talk
In late 1980's Manin et al put forward a precise conjecture about the density of rational points on Fano varieties. Over the last two decades some progress has been made towards proving this conjecture. But the conjecture is far from being proved even for the case of two dimensional Fano varieties or del Pezzo surfaces. These surfaces are geometrically classified according to `degree', and the geometric, as well as, the arithmetic complexity increases as the degree drops. The most interesting cases of Manin's conjecture for surfaces are degrees four and lower. In this talk I will mainly focus on the arithmetic of these del Pezzo surfaces, and report some of my own results (partly joint with Henryk Iwaniec). I will also talk about some other problems which apparently have a different flavor but, nonetheless, are directly related with the problem of rational points on surfaces.