Seminars and Colloquia Schedule

Ramified optimal transportation in geodesic metric spaces

Series
CDSNS Colloquium
Time
Monday, March 7, 2011 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Qinglan Xia – University of California Davis
An optimal transport path may be viewed as a geodesic in the space of probability measures under a suitable family of metrics. This geodesic may exhibit a tree-shaped branching structure in many applications such as trees, blood vessels, draining and irrigation systems. Here, we extend the study of ramified optimal transportation between probability measures from Euclidean spaces to a geodesic metric space. We investigate the existence as well as the behavior of optimal transport paths under various properties of the metric such as completeness, doubling, or curvature upper boundedness. We also introduce the transport dimension of a probability measure on a complete geodesic metric space, and show that the transport dimension of a probability measure is bounded above by the Minkowski dimension and below by the Hausdorff dimension of the measure. Moreover, we introduce a metric, called "the dimensional distance", on the space of probability measures. This metric gives a geometric meaning to the transport dimension: with respect to this metric, the transport dimension of a probability measure equals to the distance from it to any finite atomic probability measure.

From the "slicing problem" to "KLS Conjecture": The concentration of measure phenomenon in log-concave measures

Series
Joint ACO and ARC Colloquium
Time
Monday, March 7, 2011 - 13:30 for 1 hour (actually 50 minutes)
Location
Klaus 1116W
Speaker
Grigoris Paouris – Texas A &M University

Tea and light refreshments 2:30 p.m.  in Room 2222

We will discuss several open questions on the concentration of measure on log-concave measures and we will present the main ideas of some recent positive results.

Statistical Shape Analysis of Target Boundaries in 2D Sonar Imagery

Series
Applied and Computational Mathematics Seminar
Time
Monday, March 7, 2011 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Darshan Bryner – Naval Surface Warfare Center/FSU
There are several definitions of the word shape; of these, the most important to this research is “the external form or appearance of someone or something as produced by its outline.” Shape Analysis in this context focuses specifically on the mathematical study of explicit, parameterized curves in 2D obtained from the boundaries of targets of interest in Synthetic Aperture Sonar (SAS) imagery. We represent these curves with a special “square-root velocity function,” whereby the space of all such functions is a nonlinear Riemannian manifold under the standard L^2 metric. With this curve representation, we form the mathematical space called “shape space” where a shape is considered to be the orbit of an equivalence class under the group actions of scaling, translation, rotation, and re-parameterization. It is in this quotient space that we can quantify the distance between two shapes, cluster similar shapes into classes, and form means and covariances of shape classes for statistical inferences. In this particular research application, I use this shape analysis framework to form probability density functions on sonar target shape classes for use as a shape prior energy term in a Bayesian Active Contour model for boundary extraction in SAS images. Boundary detection algorithms generally perform poorly on sonar imagery due to their typically low signal to noise ratio, high speckle noise, and muddled or occluded target edges; thus, it is necessary that we use prior shape information in the evolution of an active contour to achieve convergence to a meaningful target boundary.

Lecture series on the disjoint paths algorithm

Series
Graph Theory Seminar
Time
Monday, March 7, 2011 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 168
Speaker
Paul Wollan – School of Mathematics, Georgia Tech and University of Rome
The k-disjoint paths problem takes as input a graph G and k pairs of vertices (s_1, t_1),..., (s_k, t_k) and determines if there exist internally disjoint paths P_1,..., P_k such that the endpoints of P_i are s_i and t_i for all i=1,2,...,k. While the problem is NP-complete when k is allowed to be part of the input, Robertson and Seymour showed that there exists a polynomial time algorithm for fixed values of k. The existence of such an algorithm is the major algorithmic result of the Graph Minors series. The original proof of Robertson and Seymour relies on the whole theory of graph minors, and consequently is both quite technical and involved. Recent results have dramatically simplified the proof to the point where it is now feasible to present the proof in its entirety. This seminar series will do just that, with the level of detail aimed at a graduate student level.

Arithmetic of the Legendre curve

Series
Algebra Seminar
Time
Monday, March 7, 2011 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Doug Ulmer – Georgia Tech
Let k be a field (not of characteristic 2) and let t be an indeterminate. Legendre's elliptic curve is the elliptic curve over k(t) defined by y^2=x(x-1)(x-t). I will discuss the arithmetic of this curve (group of solutions, heights, Tate-Shafarevich group) over the extension fields k(t^{1/d}). I will also mention several variants and open problems which would make good thesis topics.

Isospectral Graph Reductions, Estimates of Matrices' Spectra, and Eventually Negative Schwarzian Systems

Series
Dissertation Defense
Time
Tuesday, March 8, 2011 - 09:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Benjamin Webb – School of Mathematics, Georgia Tech
Real world networks typically consist of a large number of dynamical units with a complicated structure of interactions. Until recently such networks were most often studied independently as either graphs or as coupled dynamical systems. To integrate these two approaches we introduce the concept of an isospectral graph transformation which allows one to modify the network at the level of a graph while maintaining the eigenvalues of its adjacency matrix. This theory can then be used to rewire dynamical networks, considered as dynamical systems, in order to gain improved estimates for whether the network has a unique global attractor. Moreover, this theory leads to improved eigenvalue estimates of Gershgorin-type. Lastly, we will discuss the use of Schwarzian derivatives in the theory of 1-d dynamical systems.

Math Modeling of Biological Memory

Series
Mathematical Biology Seminar
Time
Tuesday, March 8, 2011 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Vadim L. Stefanuk – Russian Academy of Sciences
Some properties of biological memory are briefly described. The examples of short term memory and extra long term memory are drawn from psychological literature and from the personal experience. The short term memory is modeled here with the two types of mathematical models, both models being special cases of the Locally Organized Systems (LOS). The first model belongs to Prof. Mikhail Tsetlin of Moscow State University. His original ?pile of books? model was independently rediscovered a new by a number of scientists throughout the World. Tsetlin?s model demonstrates some very important properties of a natural memory organization. However mathematical study of his model turned out to be rather complicated. The second model belongs to the present author and has somewhat similar properties. However, it is organized in a completely different manner. In particular it contains some parameters, which makes the model rather interesting mathematically and pragmatically. The Stefanuk?s model has many interpretations and will be illustrated here with some biologically inspired examples. Both models founded a number of practical applications. These models demonstrate that the short term memory, which is heavily used by humans and by many biological subsystems is arranged reasonably. For humans it helps to keep the knowledge in the way facilitating its fast extraction. For biological systems the models explain the arrangement of storage of various micro organisms in a cell in an optimal manner to provide for the living.

Efficiently Learning Gaussian Mixtures

Series
ACO Seminar
Time
Tuesday, March 8, 2011 - 16:30 for 1 hour (actually 50 minutes)
Location
KACB 1116
Speaker
Greg Valiant – University of California, Berkeley
Given data drawn from a mixture of multivariate Gaussians, a basic problem is to accurately estimate the mixture parameters. This problem has a rich history of study in both statistics and, more recently, in CS Theory and Machine Learning. We present a polynomial time algorithm for this problem (running time, and data requirement polynomial in the dimension and the inverse of the desired accuracy), with provably minimal assumptions on the Gaussians. Prior to this work, it was unresolved whether such an algorithm was even information theoretically possible (ie, whether a polynomial amount of data, and unbounded computational power sufficed). One component of the proof is showing that noisy estimates of the low-order moments of a 1-dimensional mixture suffice to recover accurate estimates of the mixture parameters, as conjectured by Pearson (1894), and in fact these estimates converge at an inverse polynomial rate. The second component of the proof is a dimension-reduction argument for how one can piece together information from different 1-dimensional projections to yield accurate parameters.

2-dimensional TQFTs and Frobenius Algebras

Series
Geometry Topology Student Seminar
Time
Wednesday, March 9, 2011 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Alan Diaz – Georgia Tech
An n-dimensional topological quantum field theory is a functor from the category of closed, oriented (n-1)-manifolds and n-dimensional cobordisms to the category of vector spaces and linear maps. Three and four dimensional TQFTs can be difficult to describe, but provide interesting invariants of n-manifolds and are the subjects of ongoing research. This talk focuses on the simpler case n=2, where TQFTs turn out to be equivalent, as categories, to Frobenius algebras. I'll introduce the two structures -- one topological, one algebraic -- explicitly describe the correspondence, and give some examples.

The mathematics of service processes

Series
Research Horizons Seminar
Time
Wednesday, March 9, 2011 - 12:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Ton Dieker – ISYE - Georgia Institute of Technology

Hosts: Amey Kaloti and Ricardo Restrepo

This talk gives an overview of the mathematics of service processes, with a focus on several problems I have been involved in. In many service environments, resources are shared and delays arise as a result; examples include bank tellers, data centers, hospitals, the visa/mortgage application process.I will discuss some frequently employed mathematical tools in this area. Since randomness is inherent to many service environments, I will focus on stochastic processes and stochastic networks.

Energy estimates for the random displacement model

Series
Analysis Seminar
Time
Wednesday, March 9, 2011 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Michael Loss – School of Mathematics, Georgia Tech
This talk is about a random Schroedinger operator describing the dynamics of an electron in a randomly deformed lattice. The periodic displacement configurations which minimize the bottom of the spectrum are characterized. This leads to an amusing problem about minimizing eigenvalues of a Neumann Schroedinger operator with respect to the position of the potential. While this configuration is essentially unique for dimension greater than one, there are infinitely many different minimizing configurations in the one-dimensional case. This is joint work with Jeff Baker, Frederic Klopp, Shu Nakamura and Guenter Stolz.

Exact asymptotic behavior of the Pekar-Thomasevich functional

Series
Math Physics Seminar
Time
Wednesday, March 9, 2011 - 16:30 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Rafael D. Benguria – Physics Department, Catholic University of Chile
An explicit asymptotic expression for the ground-state energy of the Pekar-Tomasevich functional for the N-polaron is found, when the positive repulsion parameter U of the electrons is less than twice the coupling constant of the polaron. This is joint workwith Gonzalo Bley.

First-Fit is Linear on (r+s)-free Posets

Series
Graph Theory Seminar
Time
Thursday, March 10, 2011 - 12:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Kevin Milans – University of South Carolina
First-Fit is an online algorithm that partitions the elements of a poset into chains. When presented with a new element x, First-Fit adds x to the first chain whose elements are all comparable to x. In 2004, Pemmaraju, Raman, and Varadarajan introduced the Column Construction Method to prove that when P is an interval order of width w, First-Fit partitions P into at most 10w chains. This bound was subsequently improved to 8w by Brightwell, Kierstead, and Trotter, and independently by Narayanaswamy and Babu. The poset r+s is the disjoint union of a chain of size r and a chain of size s. A poset is an interval order if and only if it does not contain 2+2 as an induced subposet. Bosek, Krawczyk, and Szczypka proved that if P is an (r+r)-free poset of width w, then First-Fit partitions P into at most 3rw^2 chains and asked whether the bound can be improved from O(w^2) to O(w). We answer this question in the affirmative. By generalizing the Column Construction Method, we show that if P is an (r+s)-free poset of width w, then First-Fit partitions P into at most 8(r-1)(s-1)w chains. This is joint work with Gwena\"el Joret.

Cantor Boundary Behavior of Analytic Functions

Series
Analysis Seminar
Time
Thursday, March 10, 2011 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Ka-Sing Lau – Hong Kong Chinese University
There is a large literature to study the behavior of the image curves f(\partial {\mathbb D}) of analytic functions f on the unit disc {\mathbb D}. Our interest is on the class of analytic functions f for which the image curves f(\partial {\mathbb D}) form infinitely many (fractal) loops. We formulated this as the Cantor boundary behavior (CBB). We develop a general theory of this property in connection with the analytic topology, the distribution of the zeros of f'(z) and the mean growth rate of f'(z) near the boundary. Among the many examples, we showed that the lacunary series such as the complex Weierstrass functions have the CBB, also the Cauchy transform F(z) of the canonical Hausdorff measure on the Sierspinski gasket, which is the original motivation of this investigation raised by Strichartz.

Coupling at infinity

Series
Stochastics Seminar
Time
Thursday, March 10, 2011 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Jonathan Mattingly – Duke University, Mathematics Department
I will discuss how the idea of coupling at time infinity is equivalent to unique ergodicity of a markov process. In general, the coupling will be a kind of "asymptotic Wasserstein" coupling. I will draw examples from SDEs with memory and SPDEs. The fact that both are infinite dimensional markov processes is no coincidence.

Lorenz flow and random effect

Series
CDSNS Colloquium
Time
Friday, March 11, 2011 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Weiping Li – Oklahoma State University
In this talk, I will explain the correspondence between the Lorenz periodic solution and the topological knot in 3-space.The effect of small random perturbation on the Lorenz flow will lead to a certain nature order developed previously by Chow-Li-Liu-Zhou. This work provides an answer to an puzzle why the Lorenz periodics are only geometrically simple knots.

Contact geometry and Heegaard Floer invariants for noncompact 3-manifolds

Series
Geometry Topology Seminar
Time
Friday, March 11, 2011 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Shea Vela-Vick – Columbia University
I plan to discuss a method for defining Heegaard Floer invariants for 3-manifolds. The construction is inspired by contact geometry and has several interesting immediate applications to the study of tight contact structures on noncompact 3-manifolds. In this talk, I'll focus on one basic examples and indicate how one defines a contact invariant which can be used to give an alternate proof of James Tripp's classification of tight, minimally twisting contact structures on the open solid torus. This is joint work with John B. Etnyre and Rumen Zarev.