Seminars and Colloquia by Series

Two weight Inequality for Hilbert transform

Series
Analysis Working Seminar
Time
Monday, February 22, 2010 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Michael Lacey – Georgia Tech
We will start a discussion of arXiv:1001.4043, which characterizes the two weight inequality for the Hilbert transform, including the statement of the theorem, and some examples of how this question arises. Joint work with Ignacio Uriate-Tuero, and Eric Sawyer.

Georgia Scientific Computing Symposium

Series
Other Talks
Time
Saturday, February 20, 2010 - 09:00 for 8 hours (full day)
Location
Skiles 249
Speaker
Georgia Scientific Computing Symposium – School of Mathematics, Georgia Tech
The purpose of the Georgia Scientific Computing Symposium (GSC 2010) is to provide an opportunity for professors, postdocs and graduate students in the Atlanta area to meet in an informal setting, to exchange ideas, and to highlight local scientific computing research. The one-day symposium is open to the whole research community. The event is free but registration is required.

Harris' ergodic theorem for Markov chains revisited

Series
Probability Working Seminar
Time
Friday, February 19, 2010 - 15:00 for 1.5 hours (actually 80 minutes)
Location
Skiles 169
Speaker
Tobias Hurth – Georgia Tech
In my talk, I will present the main results of a recent article by Martin Hairer and Jonathan Mattingly on an ergodic theorem for Markov chains. I will consider Markov chains evolving in discrete time on an abstract, possibly uncountable, state space. Under certain regularity assumptions on the chain's transition kernel, such as the existence of a Foster-Lyapunov function with small level sets (what exactly is meant by that will be thoroughly explained in the talk), one can establish the existence and uniqueness of a stationary distribution. I will focus on a new proof technique for that theorem which relies on a family of metrics on the set of probability measures living on the state space. The main result of my talk will be a strict contraction estimate involving these metrics.

Introduction to the AJ Conjecture

Series
Geometry Topology Working Seminar
Time
Friday, February 19, 2010 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Anh Tran – Georgia Tech

Please Note: This is part 1 of a two part talk. The second part will continue next week.

I will introduce the AJ conjecture (by Garoufalidis) which relates the A-polynomial and the colored Jones polynomial of a knot in the 3-sphere. Then I will verify it for the trefoil and the figure 8 knots (due to Garoufalidis) and torus knots (due to Hikami) by explicit calculations.

A Survey of Hardy Inequalities and their Optimization

Series
SIAM Student Seminar
Time
Friday, February 19, 2010 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Craig A. Sloane – School of Mathematics, Georgia Tech
This will be an introductory talk about Hardy inequalities. These inequalities are solutions to optimization problems, and their results are well-known. I will survey these results, and discuss some of the techniques used to solve these problems. The applications of Hardy inequalities are broad, from PDE's and mathematical physics to brownian motion. This talk will also serve as a lead-in to my talk at the Analysis seminar next Wednesday in which I discuss some current results that Michael Loss and I have obtained.

A Hasse principle for homogeneous spaces over function fields of p-adic curves

Series
School of Mathematics Colloquium
Time
Thursday, February 18, 2010 - 16:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Raman Parimala – Department of Mathematics and Computer Science, Emory University
Let k be a p-adic field and K/k function field in one variable over k. We discuss Hasse principle for existence of rational points on homogeneous spaces under connected linear algebraic groups. We illustrate how a positive answer to Hasse principle leads for instance to the result: every quadratic form in nine variables over K has a nontrivial zero.

Asymptotic enumeration of surface maps and its connection with other mathematical objects

Series
Graph Theory Seminar
Time
Thursday, February 18, 2010 - 12:05 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Professor Jason Gao – School of Mathematics and Statistics Carleton University –
A map is a connected graph G embedded in a surface S (a closed 2-manifold) such that all components of S -- G are simply connected regions. A map is rooted if an edge is distinguished together with a direction on the edge and a side of the edge. Maps have been enumerated by both mathematicians and physicists as they appear naturally in the study of representation theory, algebraic geometry, and quantum gravity. In 1986 Bender and Canfield showed that the number of n-edge rooted maps on an orientable surface of genus g is asymptotic to t_g n^{5(g-1)/2}12n^n, (n approaches infinity), where t_g is a positive constant depending only on g. Later it was shown that many families of maps satisfy similar asymptotic formulas in which tg appear as \universal constants". In 1993 Bender et al. derived an asymptotic formula for the num- ber of rooted maps on an orientable surface of genus g with i faces and j vertices. The formula involves a constant tg(r) (which plays the same role as tg), where r is determined by j=i.In this talk, we will review how these asymptotic formulas are obtained using Tutte's recursive approach. Connections with random trees, representation theory, integrable systems, Painleve I, and matrix integrals will also be mentioned. In particular, we will talk aboutour recent results about a simple relation between tg(r) and tg, and asymptotic formulas for the numbers of labeled graphs (of various connectivity)of a given genus. Similar results for non-orientable surfaces will also be discussed.

A combinatorial approach to the interpolation method and scaling limits in sparse random graphs

Series
ACO Colloquium
Time
Wednesday, February 17, 2010 - 16:30 for 1 hour (actually 50 minutes)
Location
Skiles 255 (Refreshments at 4pm in Skiles 236)
Speaker
David Gamarnik – Professor, M.I.T. –
We establish the existence of scaling limits for several combinatorial optimization models on Erdos-Renyi and sparse random regular graphs. For a variety of models, including maximum independent sets, MAX-CUT, coloring and K-SAT, we prove that the optimal value appropriately rescaled, converges to a limit with high probability (w.h.p.), as the size of the underlying graph divergesto infinity. For example, as a special case we prove that the size of a largest independent set in these graphs, normalized by the number of nodes converges to a limit w.h.p. thus resolving an open problem. Our approach is based on developing a simple combinatorial approach to an interpolation method developed recently in the statistical physics literature. Among other things, theinterpolation method was used to prove the existence of the so-called free energy limits for several spin glass models including Viana-Bray and random K-SAT models. Our simpler combinatorial approach allows us to work with the zero temperature case (optimization) directly and extend the approach to many other models. Additionally, using our approach, we establish the large deviationsprinciple for the satisfiability property for constraint satisfaction problems such as coloring, K-SAT and NAE(Not-All-Equal)-K-SAT. The talk will be completely self-contained. No background on random graph theory/statistical physics is necessary. Joint work with Mohsen Bayati and Prasad Tetali

On-Line Graph Coloring

Series
ACO Student Seminar
Time
Wednesday, February 17, 2010 - 13:30 for 1 hour (actually 50 minutes)
Location
ISyE Executive Classroom
Speaker
William T. Trotter – School of Mathematics, Georgia Tech
On-line graph coloring has a rich history, with a very large number of elegant results together with a near equal number of unsolved problems. In this talk, we will briefly survey some of the classic results including: performance on k-colorable graphs and \chi-bounded classes. We will conclude with a sketch of some recent and on-going work, focusing on the analysis of First Fit on particular classes of graphs.

Irregular activity and propagation of synchrony in complex, spiking neural networks

Series
Mathematical Biology Seminar
Time
Wednesday, February 17, 2010 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Raoul-Martin Memmesheimer – Center for Brain Science, Faculty of Arts and Sciences Harvard University
Mean field theory for infinite sparse networks of spiking neurons shows that a balanced state of highly irregular activity arises under a variety of conditions. The state is considered to be a model for the ground state of cortical activity. In the first part, we analytically investigate its irregular dynamics in finite networks keeping track of all individual spike times and the identities of individual neurons. For delayed, purely inhibitory interactions, we show that the dynamics is not chaotic but in fact stable. Moreover, we demonstrate that after long transients the dynamics converges towards periodic orbits and that every generic periodic orbit of these dynamical systems is stable. These results indicate that chaotic and stable dynamics are equally capable of generating the irregular neuronal activity. More generally, chaos apparently is not essential for generating high irregularity of balanced activity, and we suggest that a mechanism different from chaos and stochasticity significantly contributes to irregular activity in cortical circuits. In the second part, we study the propagation of synchrony in front of a background of irregular spiking activity. We show numerically and analytically that supra-additive dendritic interactions, as recently discovered in single neuron experiments, enable the propagation of synchronous activity even in random networks. This can lead to intermittent events, characterized by strong increases of activity with high-frequency oscillations; our model predicts the shape of these events and the oscillation frequency. As an example, for the hippocampal region CA1, events with 200Hz oscillations are predicted. We argue that these dynamics provide a plausible explanation for experimentally observed sharp-wave/ripple events.

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