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Series: SIAM Student Seminar

I will describe some interesting properties of frames and Gabor frames in particular. Then we will examine how frames may lead to interesting decompositions of integral operators. In particular, I will share some theorems for pseudodifferential operators and Fourier integral operators arising from Gabor frames.

Series: Stochastics Seminar

The Black‐Scholes model for stock price as geometric Brownian motion, and the
corresponding European option pricing formula, are standard tools in mathematical
finance. In the late seventies, Cox and Ross developed a model for stock price based
on a stochastic differential equation with fractional diffusion coefficient. Unlike the
Black‐Scholes model, the model of Cox and Ross is not solvable in closed form, hence
there is no analogue of the Black‐Scholes formula in this context. In this talk, we
discuss a new method, based on Stratonovich integration, which yields explicitly
solvable arbitrage‐free models analogous to that of Cox and Ross. This method gives
rise to a generalized version of the Black‐Scholes partial differential equation. We
study solutions of this equation and a related ordinary differential equation.

Series: Dissertation Defense

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.

Series: Analysis Seminar

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.

Series: Other Talks

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).

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.

Wednesday, September 30, 2009 - 11:00 ,
Location: Skiles 269 ,
Jan Medlock ,
Clemson University ,
medlock@clemson.edu ,
Organizer:

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.

Series: PDE Seminar

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.

Series: Geometry Topology Seminar

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.

Monday, September 28, 2009 - 13:00 ,
Location: Skiles 255 ,
Chad Topaz ,
Macalester College ,
Organizer:

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.