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Series: Research Horizons Seminar

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.

Series: Research Horizons Seminar

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.

Series: Research Horizons Seminar

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.

Series: Research Horizons Seminar

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.

Series: Research Horizons Seminar

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.

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.