- You are here:
- GT Home
- Home
- News & Events

Series: Research Horizons Seminar

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.

Series: Research Horizons Seminar

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.

Series: Research Horizons Seminar

(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.

Series: Research Horizons Seminar

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.

Series: Research Horizons Seminar

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.

Series: Research Horizons Seminar

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.

Series: Research Horizons Seminar

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.

Series: Research Horizons Seminar

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.

Series: Research Horizons Seminar

I'll give a brief introduction to the to Quantum Statistical Mechanics in the case of systems of Fermions (e.g. electrons) and try to show that a lot of the mathematical problems can be framed in term of counting (Feynman) graphs or estimating large determinants.

Series: Research Horizons Seminar

I will give a modern bijective proof of Kirchhoff's classical theorem relating the number of spanning trees in a graph to the Laplacian matrix of the graph. The proof will highlight some analogies between graph theory and algebraic geometry.