Georgia Institute of Technology College of Sciences

Classically, there are two model category structures on coalgebras in the category of chain complexes over a field. In one, the weak equivalences are maps which induce an isomorphism on homology. In the other, the weak equivalences are maps which induce a weak equivalence of algebras under the cobar functor. We unify these two approaches, realizing them as the two extremes of a partially ordered set of model category structures on coalgebras over a cooperad satisfying mild conditions.

Mathematical models of physical phenomena often include parameters that are hard or impossible to measure directly or are subject to variability, and are thus considered uncertain. Different aspects of modeling under uncertainty include forward uncertainty propagation, statistical inver- sion of uncertain parameters, optimal design of experiments, and optimization under uncertainty. I will focus on recent advances in numerical methods for infinite-dimensional Bayesian inverse problems and optimal experimental de- sign. I will also discuss the problem of risk-averse optimization under uncertainty with applications to control of PDEs with uncertain parameters. The driving applications are systems governed by PDEs with uncertain parameter fields, such as ow in the subsurface with an uncertain permeability field, or the diffusive transport of a contaminant with an uncertain initial condition. Such problems are computationally challenging due to expensive forward PDE solves and infinite-dimensional (high-dimensional when discretized) parameter spaces.

We look at two combinatorial problems which can be solvedusing careful estimates for the distribution of smooth numbers. Thefirst is the Ramsey-theoretic problem to determine the maximal size ofa subset of of integers containing no 3-term geometric progressions.This problem was first considered by Rankin, who constructed such asubset with density about 0.719. By considering progressions among thesmooth numbers, we demonstrate a method to effectively compute thegreatest possible upper density of a geometric-progression-free set.Second, we consider the problem of determining which prime numberoccurs most frequently as the largest prime divisor on the interval[2,x], as well as the set prime numbers which ever have this propertyfor some value of x, a problem closely related to the analysis offactoring algorithms.

TBA

Sir Ronald Ross’ discovery of the transmission mechanism of malaria in 1897 inspired a suite of mathematical models for the transmission of vector-borne disease, known as Ross-Macdonald models. I introduce a common formulation of the Ross-Macdonald model and discuss its extension to address a current topic in malaria control: the introduction of malaria vaccines. Following over two decades of research, vaccine trials for the malaria vaccine RTS,S have been completed, demonstrating an efficacy of roughly 50% in young children. Regions with high malaria prevalence tend to have high levels of naturally acquired immunity (NAI) to severe malaria, leading to large asymptomatic populations. I introduce a malaria model developed to address concerns about how these vaccines will perform in regions with existing NAI, discuss some analytic results and their public health implications, and reframe our question as an optimal control problem.