Seminars and Colloquia by Series

Introduction to metric and comparison geometry

Series
Geometry Topology Working Seminar
Time
Friday, February 13, 2009 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Igor BelegradekSchool of Mathematics, Georgia Tech
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.

Baby Representation Theory of Finite Groups

Series
SIAM Student Seminar
Time
Friday, February 13, 2009 - 12:30 for 2 hours
Location
Skiles 269
Speaker
Yi HuangSchool of Mathematics, Georgia Tech
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
Time
Thursday, February 12, 2009 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Jiawei LiuDepartment of Mathematics & Statistics, Georgia State University
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.

Rational Points on Surfaces

Series
Job Candidate Talk
Time
Thursday, February 12, 2009 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Ritabrata MunshiRutgers University
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.

Intersections of Schubert varieties and eigenvalue inequalities

Series
Research Horizons Seminar
Time
Wednesday, February 11, 2009 - 12:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Wing Suet LiSchool of Mathematics, Georgia Tech
The Horn inequalities give a characterization of eigenvalues of self-adjoint n by n matrices A, B, C with A+B+C=0. The original proof by Klyachko and Knutson-Tao, requires tools from algebraic geometry, among other things. Our recent work provides a proof using only elementary tools that made it possible to generalize the Horn inequalities to finite von Neumann factors. No knowledge of von Neumann algebra is required.

Ramified optimal transport and their applications

Series
PDE Seminar
Time
Tuesday, February 10, 2009 - 15:05 for 1.5 hours (actually 80 minutes)
Location
Skiles 255
Speaker
Quinlan XiaUniversity of California, Davis
The transportation problem can be formulated as the problem of finding the optimal way to transport a given measure into another with the same mass. In mathematics, there are at least two different but very important types of optimal transportation: Monge-Kantorovich problem and ramified transportation. In this talk, I will give a brief introduction to the theory of ramified optimal transportation. In terms of applied mathematics, optimal transport paths are used to model many "tree shaped" branching structures, which are commonly found in many living and nonliving systems. Trees, river channel networks, blood vessels, lungs, electrical power supply systems, draining and irrigation systems are just some examples. After briefly describing some basic properties (e.g. existence, regularity) as well as numerical simulation of optimal transport paths, I will use this theory to explain the dynamic formation of tree leaves. On the other hand, optimal transport paths provide excellent examples for studying geodesic problems in quasi-metric spaces, where the distance functions satisfied a relaxed triangle inequality: d(x,y) <= K(d(x,z)+d(z,y)). Then, I will introduce a new concept "dimensional distance" on the space of probability measures. With respect to this new metric, the dimension of a probability measure is just the distance of the measure to any atomic measure. In particular, measures concentrated on self-similar fractals (e.g. Cantor set, fat Cantor sets) will be of great interest to us.

A New Nonlinear Long Memory Volatility Process

Series
Mathematical Finance/Financial Engineering Seminar
Time
Tuesday, February 10, 2009 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Rehim KilicSchool of Economics, Georgia Tech
This paper introduces a new nonlinear long memory volatility process, denoted by Smooth Transition FIGARCH, or ST-FIGARCH, which is designed to account for both long memory and nonlinear dynamics in the conditional variance process. The nonlinearity is introduced via a logistic transition function which is characterized by a transition parameter and a variable. The model can capture smooth jumps in the altitude of the volatility clusters as well as asymmetric response to negative and positive shocks. A Monte Carlo study finds that the ST-FIGARCH model outperforms the standard FIGARCH model when nonlinearity is present, and performs at least as well without nonlinearity. Applications reported in the paper show both nonlinearity and long memory characterize the conditional volatility in exchange rate and stock returns and therefore presence of nonlinearity may not be the source of long memory found in the data.

Influence of anti-viral drug treatments on evolution of HIV-1 pathogen

Series
CDSNS Colloquium
Time
Monday, February 9, 2009 - 16:30 for 2 hours
Location
Skiles 255
Speaker
Zhilan FengDepartment of Mathematics, Purdue University
Mathematical models are used to study possible impact of drug treatment of infections with the human immunodeficiency virus type 1 (HIV-1) on the evolution of the pathogen. Treating HIV-infected patients with a combination of several antiretroviral drugs usually contributes to a substantial decline in viral load and an increase in CD4+ T cells. However, continuing viral replication in the presence of drug therapy can lead to the emergence of drug-resistant virus variants, which subsequently results in incomplete viral suppression and a greater risk of disease progression. As different types of drugs (e.g., reverse transcriptase inhibitors,protease inhibitors and entry inhibitors) help to reduce the HIV replication at different stages of the cell infection, infection-age-structured models are useful to more realistically model the effect of these drugs. The model analysis will be presented and the results are linked to the biological questions under investigation. By demonstrating how drug therapy may influence the within host viral fitness we show that while a higher treatment efficacy reduces the fitness of the drug-sensitive virus, it may provide a stronger selection force for drug-resistant viruses which allows for a wider range of resistant strains to invade.

Projection and Nyström methods for FIE on bounded and unbounded intervals

Series
Applied and Computational Mathematics Seminar
Time
Monday, February 9, 2009 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Giuseppe MastroianniDept. of Mathematics and Informatics, Univ. of Basilicata, Italy)
In this talk I will show a simple projection method for Fredholm integral equation (FIE) defined on finite intervals and a Nyström method for FIE defined on the real semiaxis. The first method is based the polynomial interpolation of functions in weighted uniform norm. The second one is based on a Gauss truncated quadrature rule. The stability and the convergence of the methods are proved and the error estimates are given.

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