Seminars and Colloquia by Series

The Point Mass Problem on the Real Line

Series
Analysis Seminar
Time
Wednesday, September 8, 2010 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Manwah WongGeorgia Tech
In this talk, I will talk about recent developments on the point mass problem on the real line. Starting from the point mass formula for orthogonal polynomials on the real line, I will present new methods employed to compute the asymptotic formulae for the orthogonal polynomials and how these formulae can be applied to solve the point mass problem when the recurrence coefficients are asymptotically identical. The technical difficulties involved in the computation will also be discussed.

The size of crossings in antichains

Series
Research Horizons Seminar
Time
Wednesday, September 8, 2010 - 12:00 for 1 hour (actually 50 minutes)
Location
Skiles 171 (NOTICE THE CHANGE OF ROOM)
Speaker
Tom TrotterSchool of Mathematics - Georgia Institute of Technology

Please Note: Hosted by: Yao Li and Ricardo Restrepo

Combinatorial mathematics exhibits a number of elegant, simply stated problems that turn out to be surprisingly challenging. In this talk, I report on a problem of this type on which I have been working with Noah Streib, Stephen Young and Ruidong Wang from Georgia Tech, as well as Piotr Micek, Bartek Walczak and Tomek Krawczyk, all computer scientists from Poland. Given positive integers $k$ and $w$, what is the largest integer $t = f(k,w)$ for which there exists a family $\mathcal{F}$ of $t$ vectors in $N^{w}$ so that: \begin{enumerate} \item Any two vectors in the family $\mathcal{F}$ are incomparable in the product ordering; and \item There do not exist two vectors $A$ and $B$ in the family for which there are distinct $i$ and $j$ so that $a_i\ge k +b_i$ and $b_j \ge k + a_j$. \end{enumerate} The Polish group posed the problem to us at the SIAM Discrete Mathematics held in Austin, Texas, this summer. They were able to establish the following bounds: \[ k^{w-1} \le t \le k^w \] We were able to show that the lower bound is essentially correct by showing that there is a constant $c_w$ so that $t \l c_w k^{w-1}$. But recent work suggests that the lower bound might actually be tight.

Character varieties of knots and tropical curves

Series
Tropical Geometry Seminar
Time
Wednesday, September 8, 2010 - 10:00 for 1 hour (actually 50 minutes)
Location
Skiles 114
Speaker
Stavros GaroufalidisGeorgia Tech
The moduli space of representations of a fundamental group of a knot in SL(2,C) is an affine algebraic variety, and generically a complex curve, with an explicit projection to C^2. The ideal that defines this curve has special type described by binomial and linear equations. I will motivate this curve using elementary hyperbolic geometry, and its Newton polygon in the plane using geometric topology. Finally, I will describe a heuristic method for computing the Newton polygon without computing the curve itself, using tropical implitization, work in progress with Josephine Yu. The talk will be concrete, with examples of concrete curves that come from knots. This talk involves classical mathematics. A sequel of it will discuss quantum character varieties of knots and tropical curves.

Synchronization of Cows

Series
Mathematical Biology Seminar
Time
Tuesday, September 7, 2010 - 17:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Mason PorterOxford University
The study of collective behavior---of animals, mechanical systems, or even abstract oscillators---has fascinated a large number of researchers from observational geologists to pure mathematicians. We consider the collective behavior of herds of cattle. We first consider some results from an agent-based model and then formulate a mathematical model for the daily activities of a cow (eating, lying down, and standing) in terms of a piecewise affine dynamical system. We analyze the properties of this bovine dynamical system representing the single animal and develop an exact integrative form as a discrete-time mapping. We then couple multiple cow "oscillators" together to study synchrony and cooperation in cattle herds, finding that it is possible for cows to synchronize less when the coupling is increased. [This research is in collaboration with Jie Sun, Erik Bollt, and Marian Dawkins.]

Curve complexes and mapping class groups

Series
Geometry Topology Working Seminar
Time
Friday, September 3, 2010 - 13:00 for 2 hours
Location
Skiles 114
Speaker
Dan MargalitGeorgia Tech
The mapping class group is the group of symmetries of a surface (modulo homotopy). One way to study the mapping class group of a surface S is to understand its action on the set of simple closed curves in S (up to homotopy). The set of homotopy classes of simple closed curves can be organized into a simplicial complex called the complex of curves. This complex has some amazing features, and we will use it to prove a variety of theorems about the mapping class group. We will also state some open questions. This talk will be accessible to second year graduate students.

The number of vertices in a 6-critical graph is linear in its genus

Series
Graph Theory Seminar
Time
Thursday, September 2, 2010 - 12:05 for 1 hour (actually 50 minutes)
Location
Skiles 114
Speaker
Luke PostleMath, GT
A deep theorem of Thomassen shows that for any surface there are only finitely many 6-critical graphs that embed on that surface. We give a shorter self-contained proof that for any 6-critical graph G that embeds on a surface of genus g, that |V(G)| is at most linear in g. Joint work with Robin Thomas.

The Aleksandrov problem and optimal transport on $S^n$

Series
School of Mathematics Colloquium
Time
Thursday, September 2, 2010 - 11:00 for 1 hour (actually 50 minutes)
Location
249 Skiles
Speaker
Vladimir OlikerEmory University
The purpose of this talk is to describe a variational approach to the problemof A.D. Aleksandrov concerning existence and uniqueness of a closed convexhypersurface in Euclidean space $R^{n+1}, ~n \geq 2$ with prescribed integral Gauss curvature. It is shown that this problem in variational formulation is closely connected with theproblem of optimal transport on $S^n$ with a geometrically motivated cost function.

A Variational Estimate for Paraproducts

Series
Analysis Seminar
Time
Wednesday, September 1, 2010 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 114
Speaker
Yen DoGeorgia Tech
We show variational estimates for paraproducts, which can be viewed as bilinear generalizations of L\'epingle’s variational estimates for martingale averages or scaled families of convolution operators. The heart of the matter is the case of low variation exponents. Joint work with Camil Muscalu and Christoph Thiele.

Universality Limits for Random Matrices and orthogonal Polynomials

Series
Research Horizons Seminar
Time
Wednesday, September 1, 2010 - 12:00 for 1 hour (actually 50 minutes)
Location
Skiles 114
Speaker
Doron LubinskySchool of Mathematics - Georgia Tech
Orthogonal Polynomials play a key role in analysis of random matrices. We discuss universality limits in the so-called unitary case, showing how the universality limit reduces to an asymptotic involving reproducing kernels associated with orthogonal polynomials. As a consequence, we show that universality holds in measure for any compactly supported measure.

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