Seminars and Colloquia Schedule

Feature Based Fusion of Multimodal Data for Object Classification

Series
Applied and Computational Mathematics Seminar
Time
Monday, October 4, 2010 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 002
Speaker
Michael Burkhart – Gatech, Math
The over-abundance of remotely sensed data has resulted inthe realization that we do not have nor could ever acquire asufficient number of highly trained image analysts to parse theavailable data.  Automated techniques are needed to perform low levelfunctions, identifying scenarios of importance from the availabledata, so that analysts may be reserved for higher level interpretativeroles. Data fusion has been an important topic in intelligence sincethe mid-1980s and continues to be a necessary concept in thedevelopment of these automated low-level functions. We propose anapproach to multimodal data fusion to combine images of varyingspatial and spectral resolutions with digital elevation models.Furthermore, our objective is to perform this fusion at the imagefeature level, specifically utilizing Gabor filters because of theirresemblance to the human visual system.

Legendrian contact homology for Seifert fibered spaces

Series
Geometry Topology Seminar
Time
Monday, October 4, 2010 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Joan Licata – Stanford University
In this talk, I'll focus on Seifert fibered spaces whose fiber structure is realized by the Reeb orbits of an appropriate contact form. I'll describe a rigorous combinatorial formulation of Legendrian contact homology for Legendrian knots in these manifolds. This work is joint with J. Sabloff.

Higher-Order Three-Term Recurrences and Asymptotics of Multiple Orthogonal Polynomials

Series
Analysis Seminar
Time
Tuesday, October 5, 2010 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Sasha Aptekarev – Keldish Institute for Applied Mathematics
The asymptotic theory is developed for polynomial sequences that are generated by the three-term higher-order recurrence Q_{n+1} = zQ_n - a_{n-p+1}Q_{n-p}, p \in \mathbb{N}, n\geq p, where z is a complex variable and the coefficients a_k are positive and satisfy the perturbation condition \sum_{n=1}^\infty |a_n-a|<\infty . Our results generalize known results for p = 1, that is, for orthogonal polynomial sequences on the real line that belong to the Blumenthal-Nevai class. As is known, for p\geq 2, the role of the interval is replaced by a starlike set S of p+1 rays emanating from the origin on which the Q_n satisfy a multiple orthogonality condition involving p measures. Here we obtain strong asymptotics for the Q_n in the complex plane outside the common support of these measures as well as on the (finite) open rays of their support. In so doing, we obtain an extension of Weyl's famous theorem dealing with compact perturbations of bounded self-adjoint operators. Furthermore, we derive generalizations of the classical Szeg\"o functions, and we show that there is an underlying Nikishin system hierarchy for the orthogonality measures that is related to the Weyl functions. Our results also have application to Hermite-Pad\'e approximants as well as to vector continued fractions.

Nonlinear Schroedinger equation with a Magnetic Potential

Series
PDE Seminar
Time
Tuesday, October 5, 2010 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Prof. Shijun Zheng – Georgia Southern University –
The dissipative mechanism of Schroedinger equation is mathematically described by the decay estimate of solutions. In this talk I mainly focused on the use of harmonic analysis techniques to obtain suitable time decay estimates and then prove the local wellposedness for semilinear Schroedinger equation in certain external magnetic field. It turns out that the scattering with a potential may lead to understanding of the wellposedness of NLS in the presence of nonsmooth or large initial data. Part of this talk is a joint work with Zhenqiu Zhang.

Master's Thesis. Limit theorems for a one dimensional system with random switchings.

Series
Dissertation Defense
Time
Tuesday, October 5, 2010 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 114
Speaker
Tobias Hurth – School of Mathematics, Georgia Tech
We consider a simple one-dimensional random dynamical system with two driving vector fields and random switchings between them. We show that this system satisfies a one force - one solution principle and compute the unique invariant density explicitly. We study the limiting behavior of the invariant density as the switching rate approaches zero or infinity and derive analogues of classical probability theory results such as central limit theorem and large deviation principle.

Lifting Tropical Curves and Linear Systems on Graphs

Series
Tropical Geometry Seminar
Time
Wednesday, October 6, 2010 - 10:05 for 1 hour (actually 50 minutes)
Location
Skiles 114
Speaker
Eric Katz – UT Austin
Tropicalization is a procedure for associating a polyhedral complex to a subvariety of an algebraic torus. We study the question on which graphs arise from tropicalizing algebraic curves. By using Baker's technique of specialization of linear systems from curves to graphs, we are able to give a necessary condition for a balanced weighted graph to be the tropicalization of a curve. Our condition reproduces the known necessary conditions and also gives new conditions.

Panel discussion with students about the job market for mathematicians.

Series
Research Horizons Seminar
Time
Wednesday, October 6, 2010 - 12:00 for 1 hour (actually 50 minutes)
Location
Skiles 171
Speaker
Doug Ulmer - Luca Dieci – School of Mathematics - Georgia Institute of Technology

Hosts: Yao Li and Ricardo Restrepo

The Research Horizons seminar this week will be a panel discussion on the job market for mathematicians. Professor Doug Ulmer and Luca Dieci will give a presentation with general information on the academic job market and the experience of our recent students, in and out of academia. The panel will then take questions from the audience. 

Sobolev orthogonal polynomials in two variables and partial differential equations

Series
Analysis Seminar
Time
Wednesday, October 6, 2010 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Miguel Pinar – Dpto. Matematica Aplicada, Universidad de Granada
Sobolev orthogonal polynomials in two variables are defined via inner products involving gradients. Such a kind of inner product appears in connection with several physical and technical problems. Matrix second-order partial differential equations satisfied by Sobolev orthogonal polynomials are studied. In particular, we explore the connection between the coefficients of the second-order partial differential operator and the moment functionals defining the Sobolev inner product. Finally, some old and new examples are given.

Generalized Borcherds Products and Two number theoretic applications

Series
School of Mathematics Colloquium
Time
Thursday, October 7, 2010 - 11:05 for 1 hour (actually 50 minutes)
Location
Skiles 249
Speaker
Ken Ono – University of Wisconsin at Madison and Emory University –
n his 1994 ICM lecture, Borcherds famously introduced an entirely new conceptin the theory of modular forms. He established that modular forms with very specialdivisors can be explicitly constructed as infinite products. Motivated by problemsin geometry, number theorists recognized a need for an extension of this theory toinclude a richer class of automorphic form. In joint work with Bruinier, the speakerhas generalized Borcherds's construction to include modular forms whose divisors arethe twisted Heegner divisors introduced in the 1980s by Gross and Zagier in theircelebrated work on the Birch and Swinnerton-Dyer Conjecture. This generalization,which depends on the new theory of harmonic Maass forms, has many applications.The speaker will illustrate the utility of these products by resolving open problemson the following topics: 1) Parity of the partition function 2) Birch and Swinnerton-Dyer Conjecture and ranks of elliptic curves.

Homogenization of the G-equation in random media

Series
Stochastics Seminar
Time
Thursday, October 7, 2010 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 002
Speaker
Alexei Novikov – Penn State
The G-equation is a Hamilton-Jacobi level-set equation, that is used in turbulent combustion theory. Level sets of the solution represent a flame surface which moves with normal velocity that is the sum of the laminar flame velocity and the fluid velocity. In this work I will discuss the large-scale long-time asymptotics of these solutions when the fluid velocity is modeled as a stationary incompressible random field. The main challenge of this work comes from the fact that our Hamiltonian is noncoercive. This is a joint work with J.Nolen.

The entropy production problem and Villani's conjecture

Series
SIAM Student Seminar
Time
Friday, October 8, 2010 - 13:05 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Amit Einav – School of Mathematics, Georgia Tech
In 1956 Mark Kac published his paper about the Foundation of Kinetic Theory in which he gave a mathematical, probabilistic description of a system of N particles colliding randomly. An interesting result that was found, though not causing any surprise, was the convergence to the stable equilibrium state. The question of the rate of the L2 convergence interested Kac and he conjectured that the spectral gap governing the convergence is uniformly bounded form below as N goes to infinity. While this was proved to be true, and even computed exactly, many situations show that the time scale of the convergence for very natural cases is proportional to N, while we would hope for an exponential decay. A different approach was considered, dealing with a more natural quantity, the entropy. In recent paper some advancement were made about evaluating the rate of change, and in 2003 Villani conjectured that the corresponding 'spectral gap', called the entropy production, is of order of 1/N. In our lecture we'll review the above topics and briefly discuss recently found results showing that the conjecture is essentially true.

Non-commutative Geometry I

Series
Geometry Topology Working Seminar
Time
Friday, October 8, 2010 - 14:00 for 2 hours
Location
Skiles 171
Speaker
Jean Bellissard – Ga Tech

Note this is a 2 hour talk (with a short break in the middle).

This series of lecture will try to give some basic facts about Noncommutative Geometry for the members of the School of Mathematics who want to learn about it. In the first lecture, the basics tools will be presented, (i) the philosophy and the notion of space, and (ii) the notion of C*-algebra, (iii) groupoids. As many examples as possible will be described to illustrate the purpose. In the following lectures, in addition to describing these tools more thoroughly, two aspects can be developed depending upon the wishes of the audience: A- Topology, K-theory, cyclic cohomology B- Noncommutative metric spaces and Riemannian Geometry.

Long cycles in 3-connected graphs with bounded degrees

Series
Combinatorics Seminar
Time
Friday, October 8, 2010 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Guantao Chen – Department of Mathematics and Statistics, Georgia State University
In 1993 Jackson and Wormald conjectured that if G is a 3-connected n-vertex graph with maximum degree d \ge 4 then G has a cycle of length \Omega(n^{\log_{d-1} 2}). In this talk, I will report progresses on this conjecture and related problems.

Concentration inequalities for matrix martingales

Series
Probability Working Seminar
Time
Friday, October 8, 2010 - 15:05 for 1.5 hours (actually 80 minutes)
Location
Skiles 249
Speaker
Stas Minsker – School of Math, Georgia Tech
We will present probability inequalities for the sums of independent, selfadjoint random matrices. The focus is made on noncommutative generalizations of the classical bounds of Azuma, Bernstein, Cherno ff, Hoeffding, among others. These inequalities imply concentration results for the empirical covariance matrices. No preliminary knowledge of probability theory will be assumed. (The talk is based on a paper by J. Tropp).