Georgia Institute of Technology College of Sciences

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.

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.

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).

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.