Seminars and Colloquia by Series

Some Results on Linear Discrepancy for Partially Ordered Sets

Series
Dissertation Defense
Time
Thursday, October 1, 2009 - 14:00 for 2 hours
Location
Skiles 255
Speaker
Mitch Keller – School of Mathematics, Georgia Tech
Tanenbaum, Trenk, and Fishburn introduced the concept of linear discrepancy in 2001, proposing it as a way to measure a partially ordered set's distance from being a linear order. In addition to proving a number of results about linear discrepancy, they posed eight challenges and questions for future work. This dissertation completely resolves one of those challenges and makes contributions on two others. This dissertation has three principal components: 3-discrepancy irreducible posets of width 3, degree bounds, and online algorithms for linear discrepancy. The first principal component of this dissertation provides a forbidden subposet characterization of the posets with linear discrepancy equal to 2 by completing the determination of the posets that are 3-irreducible with respect to linear discrepancy. The second principal component concerns degree bounds for linear discrepancy and weak discrepancy, a parameter similar to linear discrepancy. Specifically, if every point of a poset is incomparable to at most \Delta other points of the poset, we prove three bounds: the linear discrepancy of an interval order is at most \Delta, with equality if and only if it contains an antichain of size \Delta+1; the linear discrepancy of a disconnected poset is at most \lfloor(3\Delta-1)/2\rfloor; and the weak discrepancy of a poset is at most \Delta-1. The third principal component of this dissertation incorporates another large area of research, that of online algorithms. We show that no online algorithm for linear discrepancy can be better than 3-competitive, even for the class of interval orders. We also give a 2-competitive online algorithm for linear discrepancy on semiorders and show that this algorithm is optimal.

Trigonometric Grassmannian and a difference W-algebra

Series
Analysis Seminar
Time
Wednesday, September 30, 2009 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Plamen Illiev – Georgia Tech
The trigonometric Grassmannian parametrizes specific solutions of the KP hierarchy which correspond to rank one solutions of a differential-difference bispectral problem. It can be considered as a completion of the phase spaces of the trigonometric Calogero-Moser particle system or the rational Ruijsenaars-Schneider system. I will describe the characterization of this Grassmannian in terms of representation theory of a suitable difference W-algebra. Based on joint work with L. Haine and E. Horozov.

Introduction to Sheaf Cohomology

Series
Other Talks
Time
Wednesday, September 30, 2009 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Matt Baker – School of Mathematics, Georgia Tech
After a few remarks to tie up some loose ends from last week's talk on locally ringed spaces, I will discuss exact sequences of sheaves and give some natural examples coming from real, complex, and algebraic geometry. In the context of these examples, we'll see that a surjective map of sheaves (meaning a morphism of sheaves which is surjective on the level of stalks) need not be surjective on global sections. This observation will be used to motivate the need for "sheaf cohomology" (which will be discussed in detail in subsequent talks).

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.

Optimizing influenza vaccine distribution

Series
Mathematical Biology Seminar
Time
Wednesday, September 30, 2009 - 11:00 for 1.5 hours (actually 80 minutes)
Location
Skiles 269
Speaker
Jan Medlock – Clemson University –
The recent emergence of the influenza strain (the "swine flu") and delays in production of vaccine against it illustrate the importance of optimizing vaccine allocation.  Using an age-dependent model parametrized with data from the 1957 and 1918 influenza pandemics, which had dramatically different mortality patterns, we determined optimal vaccination strategies with regard to five outcome measures: deaths, infections, years of life lost, contingent valuation and economic costs.  In general, there is a balance between vaccinating children who transmit most and older individuals at greatest risk of mortality, however, we found that when at least a moderate amount of an effective vaccine is available supply, all outcome measures prioritized vaccinating schoolchildren.  This is vaccinating those most responsible for transmission to indirectly protect those most at risk of mortality and other disease complications.  When vaccine availability or effectiveness is reduced, the balance is shifted toward prioritizing those at greatest risk for some outcome measures. The amount of vaccine needed for vaccinating schoolchildren to be optimal depends on the general transmissibility of the influenza strain (R_0).  We also compared the previous and new recommendations of the CDC and its Advisory Committee on Immunization Practices are below optimum for all outcome measures. In addition, I will discuss some recent results using mortality and hospitalization data from the novel H1N1 "swine flu" and implications of the delay in vaccine availability.

The Vlasov-Poisson System with Steady Spatial Asymptotics

Series
PDE Seminar
Time
Tuesday, September 29, 2009 - 15:05 for 1.5 hours (actually 80 minutes)
Location
Skiles 255
Speaker
Stephen Pankavich – University of Texas, Arlington
We formulate a plasma model in which negative ions tend to a fixed, spatially-homogeneous background of positive charge. Instead of solutions with compact spatial support, we must consider those that tend to the background as x tends to infinity. As opposed to the traditional Vlasov-Poisson system, the total charge and energy are thus infinite, and energy conservation (which is an essential component of global existence for the traditional problem) cannot provide bounds for a priori estimates. Instead, a conserved quantity related to the energy is used to bound particle velocities and prove the existence of a unique, global-in-time, classical solution. The proof combines these energy estimates with a crucial argument which establishes spatial decay of the charge density and electric field.

Classification of Legendrian twist knots

Series
Geometry Topology Seminar
Time
Monday, September 28, 2009 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Vera Vertesi – MSRI
Legendrian knots are knots that can be described only by their projections(without having to separately keep track of the over-under crossinginformation): The third coordinate is given as the slope of theprojections. Every knot can be put in Legendrian position in many ways. Inthis talk we present an ongoing project (with Etnyre and Ng) of thecomplete classification of Legendrian representations of twist knots.

Biological aggregation patterns and the role of social interactions

Series
Applied and Computational Mathematics Seminar
Time
Monday, September 28, 2009 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Chad Topaz – Macalester College
Biological aggregations such as insect swarms, bird flocks, and fish schools are arguably some of the most common and least understood patterns in nature. In this talk, I will discuss recent work on swarming models, focusing on the connection between inter-organism social interactions and properties of macroscopic swarm patterns. The first model is a conservation-type partial integrodifferential equation (PIDE). Social interactions of incompressible form lead to vortex-like swarms. The second model is a high-dimensional ODE description of locust groups. The statistical-mechanical properties of the attractive-repulsive social interaction potential control whether or not individuals form a rolling migratory swarm pattern similar to those observed in nature. For the third model, we again return to a conservation-type PIDE and, via long- and short-wave analysis, determine general conditions that social interactions must satisfy for the population to asymptotically spread, contract, or reach steady state.

Fourier's Law, a brief mathematical review - Continued

Series
CDSNS Colloquium
Time
Monday, September 28, 2009 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Federico Bonetto – School of Mathematics, Georgia Tech

Please Note: This talk continues from last week's colloquium.

Fourier's Law assert that the heat flow through a point in a solid is proportional to the temperature gradient at that point. Fourier himself thought that this law could not be derived from the mechanical properties of the elementary constituents (atoms and electrons, in modern language) of the solid. On the contrary, we now believe that such a derivation is possible and necessary. At the core of this change of opinion is the introduction of probability in the description. We now see the microscopic state of a system as a probability measure on phase space so that evolution becomes a stochastic process. Macroscopic properties are then obtained through averages. I will introduce some of the models used in this research and discuss their relevance for the physical problem and the mathematical results one can obtain.

Cramer's Theorem

Series
Probability Working Seminar
Time
Friday, September 25, 2009 - 16:00 for 1 hour (actually 50 minutes)
Location
Skiles 154
Speaker
Linwei Xin – Georgia Tech
In this talk, we will introduce the classical Cramer's Theorem. The pattern of proof is one of the two most powerful tools in the theory of large deviations. Namely, the upper bound comes from optimizing over a family of Chebychef inequalities; while the lower bound comes from introducing a Radon-Dikodym factor in order to make what was originally "deviant" behavior look like typical behavior. If time permits, we will extend the Cramer's Theorem to a more general setting and discuss the Sanov Theorem. This talk is based on Deuschel and Stroock's .

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