Seminars and Colloquia by Series

Soul Theorem and moduli spaces

Series
Research Horizons Seminar
Time
Wednesday, October 28, 2009 - 12:00 for 1 hour (actually 50 minutes)
Location
Skiles 171
Speaker
Igor Belegradek – School of Mathematics, Georgia Tech –
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

Series
Research Horizons Seminar
Time
Wednesday, October 21, 2009 - 12:00 for 1 hour (actually 50 minutes)
Location
Skiles 171
Speaker
Doron Lubinsky – School of Mathematics, Georgia Tech –
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.

Series
Research Horizons Seminar
Time
Wednesday, October 14, 2009 - 12:00 for 1 hour (actually 50 minutes)
Location
Skiles 171
Speaker
Sung Ha Kang – School of Mathematics, Georgia Tech –
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?

Series
Research Horizons Seminar
Time
Wednesday, October 7, 2009 - 12:00 for 1 hour (actually 50 minutes)
Location
Skiles 171
Speaker
Stavros Garoufalidis – School of Mathematics, Georgia Tech –
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.

Series
Research Horizons Seminar
Time
Wednesday, September 30, 2009 - 12:00 for 1 hour (actually 50 minutes)
Location
Skiles 171
Speaker
Brett Wick – School of Mathematics, Georgia Tech –
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.

Series
Research Horizons Seminar
Time
Wednesday, September 23, 2009 - 12:00 for 1 hour (actually 50 minutes)
Location
Skiles 171
Speaker
Stavros Garoufalidis – Georgia Tech School of Mathematics –
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

Series
Research Horizons Seminar
Time
Wednesday, September 16, 2009 - 12:00 for 1 hour (actually 50 minutes)
Location
Skiles 171
Speaker
William T. Trotter – School of Mathematics, Georgia Tech –

Please Note: (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)

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.

Some Problems and Results in Additive Combinatorics.

Series
Research Horizons Seminar
Time
Wednesday, September 9, 2009 - 12:00 for 1 hour (actually 50 minutes)
Location
Skiles 171
Speaker
Ernie Croot – School of Mathematics, Georgia Tech –
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?

Series
Research Horizons Seminar
Time
Wednesday, April 22, 2009 - 12:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Evans Harrell – School of Mathematics, Georgia Tech
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

Series
Research Horizons Seminar
Time
Wednesday, April 15, 2009 - 12:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Sung Ha Kang – School of Mathematics, Georgia Tech
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.

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