Georgia Institute of Technology

Panel Discussion with Students About the Hiring Process.

Wed, 11/18/2009 - 12:00pm, Skiles 269

Drs. Ulmer, Harrell, and Wick, School of Mathematics, Georgia Tech        Organizer: Craig Sloane

The Research Horizons seminar this week will be a panel discussion on the academic job market for mathematicians.  The discussion will begin with an overview by Doug Ulmer of the hiring process, with a focus on the case of research-oriented universities.  The panel will then take questions from the audience.  Professor Wick was hired last year at Tech, so has recently been on the students' side of the process.  Professor Harrell has been involved with hiring at Tech for many years and can provide a perspective on the university side of the process.

Topological aspects in the theory of aperiodic solids and tiling spaces

Wed, 11/11/2009 - 12:00pm, Skiles 171

Jean Bellissard, School of Mathematics, Georgia Tech, email        Organizer: Craig Sloane

An assembly of atoms in a solid phase will be described through the notion of Delone sets and related to tilings. The Hull and the tiling space wiill be defined. It will be shown that the tiling space and the Hull can be constructed through an inverse limit of CW-complexes built out of the tiles and of the local patches. From then various cohomologies can be defined and allow to distinguish between these atomic distributions. The question of whether these topological invariant can be seen in experiments will be addressed.

Dynamical Systems, Graphs, Entropies, Dynamical Networks, and Statistical Mechanics

Wed, 11/04/2009 - 12:00pm, Skiles 171

Leonid Bunimovich, School of Mathematics, Georgia Tech, email        Organizer: Craig Sloane

Dynamical systems theory is concerned with systems that change in time (where time can be any semigroup). However, it is quite rare that one can find the solutions for such systems or even a "sizable" subset of such solutions. An approach motivated by this fact, that goes back to Poincaré, is to study instead partitions of the (phase) space M of all states of a dynamical system and consider the evolution of the elements of this partition (instead of the evolution of points of M). I'll explain how the objects in the title appear, some relations between them, and formulate a few general as well as more specific open problems suitable for a PhD thesis in dynamical systems, mathematical biology, graph theory and applied and computational mathematics. This talk will also serve to motivate and introduce to the topics to be given in tomorrow's colloquium.

Soul Theorem and moduli spaces

Wed, 10/28/2009 - 12:00pm, Skiles 171

Igor Belegradek, School of Mathematics, Georgia Tech, email        Organizer: Craig Sloane

The Soul Theorem, proved by Cheeger and Gromoll forty year ago, reveals a beautiful structure of noncompact complete manifolds of nonnegative curvature. In the talk I will sketch a proof of the Soul Theorem, and relate it to my current work on moduli spaces of nonnegatively curved metrics.

Title: Orthogonal and Biorthogonal Polyonmials

Wed, 10/21/2009 - 12:00pm, Skiles 171

Doron Lubinsky, School of Mathematics, Georgia Tech, email        Organizer: Craig Sloane

Orthogonal polynomials are an important tool in many areas of pure and applied mathematics. We outline one application in random matrix theory. We discuss generalizations of orthogonal polynomials such as the Muntz orthogonal polynomials investigated by Ulfar Stefansson. Finally, we present some conjectures about biorthogonal polynomials, which would be a great Ph.D. project for any interested student.

Introduction of variational approaches to image segmentation.

Wed, 10/14/2009 - 12:00pm, Skiles 171

Sung Ha Kang, School of Mathematics, Georgia Tech, email        Organizer: Craig Sloane

Image segmentation has been widely studied, specially since Mumford-Shah functional was been proposed. Many theoretical works as well as numerous extensions have been studied rough out the years. This talk will focus on introduction to these image segmentation functionals.  I will start with the review of Mumford-Shah functional and discuss Chan-Vese model.  Some new extensions will be presented at the end.

What is a totally positive matrix?

Wed, 10/07/2009 - 12:00pm, Skiles 171

Stavros Garoufalidis, School of Mathematics, Georgia Tech, email        Organizer: Craig Sloane

In linear algebra classes we learn that a symmetic matrix with real entries has real eigenvalues. But many times we deal with nonsymmetric matrices that we want them to have real eigenvalues and be stable under a small perturbation. In the 1930's totally positive matrices were discovered in mechanical problems of vibtrations, then lost for over 50 years. They were rediscovered in the 1990's as esoteric objects in quantum groups and crystal bases. In the 2000's these matrices appeared in relation to Teichmuller space and its quantization. I plan to give a high school introduction to totally positive matrices.

A Primer on Analytic Function Theory.

Wed, 09/30/2009 - 12:00pm, Skiles 171

Brett Wick, School of Mathematics, Georgia Tech, email        Organizer: Craig Sloane

 In the last 10 years there has been a resurgence of interest in questions about certain spaces of analytic functions. In this talk we will discuss various advances in the study of these spaces of functions and highlight questions of current interest in analytic function theory. We will give an overview of recent advances in the Corona Problem, bilinear forms on spaces of analytic functions, and highlight some methods to studying these questions that use more discrete techniques. 

Alice in Wonderland learns how to compute determinants.

Wed, 09/23/2009 - 12:00pm, Skiles 171

Stavros Garoufalidis, Georgia Tech School of Mathematics, email        Organizer: Craig Sloane

Dodgson (the author of Alice in Wonderland) was an amateur mathematician who wrote a book about determinants in 1866 and gave a copy to the queen. The queen dismissed the book and so did the math community for over a century. The Hodgson Condensation method resurfaced in the 80's as the fastest method to compute determinants (almost always, and almost surely). Interested about Lie groups, and their representations? In crystal bases? In cluster algebras? In alternating sign matrices? OK, how about square ice? Are you nuts? If so, come and listen.

Partitions of the Subset Lattice into Intervals

Wed, 09/16/2009 - 12:00pm, Skiles 171

William T. Trotter, School of Mathematics, Georgia Tech, email        Organizer: Craig Sloane

Motivated by questions in algebra involving what is called "Stanley" depth, the following combinatorial question was posed to us by Herzog: Given a positive integer n, can you partition the family of all non-empty subsets of {1, 2, ..., n} into intervals, all of the form [A, B] where |B| is at least n/2. We answered this question in the affirmative by first embedding it in a stronger result and then finding two elegant proofs. In this talk, which will be entirely self-contained, I will give both proofs. The paper resulting from this research will appear in the Journal of Combinatorial Theory, Series A.

(joint work with Csaba Biro, Dave Howard, Mitch Keller and Stephen Young. Biro and Young finished their Ph.D.'s at Georgia Tech in 2008. Howard and Keller will graduate in spring 2010)

Some Problems and Results in Additive Combinatorics.

Wed, 09/09/2009 - 12:00pm, Skiles 171

Ernie Croot, School of Mathematics, Georgia Tech, email        Organizer: Craig Sloane

Additive combinatorics is a relatively new field, with many diverse and exciting research programmes.  In this talk I will discuss two of these programmes -- the continuing development of sum-product inequalities, and the unfolding progress on arithmetic progressions -- along with some new results proved by me and my collaborators.  Hopefully I will have time to mention some nice research problems as well.

Did you hear what's going round?

Wed, 04/22/2009 - 12:00pm, Skiles 255

Evans Harrell, School of Mathematics, Georgia Tech        Organizer: Craig Sloane

The eigenvalues of the Laplacian are the squares of the frequencies of the normal modes of vibration, according to the wave equation. For this reason, Bers and Kac referred to the problem of determining the shape of a domain from the eigenvalue spectrum of the Laplacian as the question of whether one can "hear" the shape. It turns out that in general the answer is "no." Sometimes, however, one can, for instance in extremal cases where a domain, or a manifold, is round. There are many "isoperimetric" theorems that allow us to conclude that a domain, curve, or a manifold, is round, when enough information about the spectrum of the Laplacian or a similar operator is known. I'll describe a few of these theorems and show how to prove them by linking geometry with functional analysis.

Mathematical approaches to image processing

Wed, 04/15/2009 - 12:00pm, Skiles 255

Sung Ha Kang, School of Mathematics, Georgia Tech        Organizer: Craig Sloane

This talk will focus on mathematical approaches using PDE and variational models for image processing. I will discuss general problems arising from image reconstructions and segmentation, starting from Total Variation minimization (TV) model and Mumford-Shah segmentation model, and present new models from various developments. Two main topics will be on variational approaches to image reconstruction and multi-phase segmentation. Many challenges and various problems will be presented with some numerical results.

PDE Techniques in Wavelet Transforms and Applications Image Processing, Part II

Wed, 04/08/2009 - 12:00pm, Skiles 255

Hao Min Zhou, School of Mathematics, Georgia Tech        Organizer: Craig Sloane

This talk will be a continuation of the one I gave in this Seminar on March~11. I will present a brief introduction to use partial differential equations (PDE) and variational techniques (including techniques developed in computational fluid dynamics (CFD)) into wavelet transforms and Applications in Image Processing. Two different approaches are used as examples. One is PDE and variational frameworks for image reconstruction. The other one is an adaptive ENO wavelet transform designed by using ideas from Essentially Non-Oscillatory (ENO) schemes for numerical shock capturing.

PDE Techniques in Wavelet Transforms and Applications Image Processing

Wed, 03/11/2009 - 12:00pm, Skiles 255

Hao Min Zhou, School of Mathematics, Georgia Tech        Organizer: Craig Sloane

In this talk, I will present an brief introdution to use partial differential equation (PDE) and variational techniques (including techniques developed in computational fluid dynamics (CFD)) into wavelet transforms and Applications in Image Processing. Two different approaches are used as examples. One is PDE and variational frameworks for image reconstruction. The other one is an adaptive ENO wavelet transform designed by using ideas from Essentially Non-Oscillatory (ENO) schemes for numerical shock capturing.