Georgia Institute of Technology College of Sciences

The malaria parasite Plasmodium falciparum requires a vertebrate host, such as a human, and a vector host, the Anopheles mosquito, to complete a full life cycle. The portion of the life cycle in the mosquito harbors both the only time of sexual reproduction, expanding genetic complexity, and the most severe bottlenecks experienced, restricting genetic diversity, across the entire parasite life cycle. In previous work, we developed a two-stage stochastic model of parasite diversity within a mosquito, and demonstrated the importance of heterogeneity amongst parasite dynamics across a population of mosquitoes. Here, we focus on the parasite dynamics component to evaluate the first appearance of sporozoites, which is key for determining the time at which mosquitoes first become infectious. We use Bayesian inference techniques with simple models of within-mosquito parasite dynamics coupled with experimental data to estimate a posterior distribution of parameters. We determine that growth rate and the bursting function are key to the timing of first infectiousness, a key epidemiological parameter.

The dual of the cone of non-negative quadratics (on a variety) is included in the dual of the cone of sums of squares. Moreover, all (points which span) extreme rays of the dual cone of non-negative quadratics is point evaluations on real points of the variety. Therefore, we are interested in extreme rays of the dual cone of sums of squares which do not come from point evaluations. The dual cone of sums of squares on a variety is called the Hankel spectrahetron and the smallest rank of extreme rays which do not come from point evaluations is called Hankel index of the variety. In this talk, I will introduce some algebraic (or geometric) properties which control the Hankel index of varieties and compute the Hankel index of rational curves obtained by projecting rational normal curve away from a point (which has almost minimal degree).

Agol introduced veering triangulations of mapping tori, whose combinatorics are canonically associated to the pseudo-Anosov monodromy. In unpublished work, Guéritaud and Agol generalise an alternative construction to any closed manifold equipped with a pseudo-Anosov flow without perfect fits. Schleimer and I build the reverse map. As a first step, we construct the link space for a given veering triangulation. This is a copy of R2, equipped with transverse stable and unstable foliations, from which the Agol-Guéritaud's construction recovers the veering triangulation. The link space is analogous to Fenley's orbit space for a pseudo-Anosov flow. Along the way, we construct a canonical circular ordering of the cusps of the universal cover of a veering triangulation. I will also talk about work with Giannopolous and Schleimer building a census of transverse veering triangulations. The current census lists all transverse veering triangulations with up to 16 tetrahedra, of which there are 87,047.

The most popular model for Image Denoising is without any doubt the ROF (for Rudin-OsherFatemi) model. However, since the ROF approach has some drawbacks (the stair-case effect being one of them) practitioners have been looking for alternatives. One of them is the Elastica model, relying on the minimization in an appropriate functional space of the energy functional $J$ defined by

$$ J(v)=\varepsilon \int_{\Omega} \left[ a+b\left| \nabla\cdot \frac{\nabla v}{|\nabla v|}\right|^2 \right]|\nabla v| d\mathbf{x} + \frac{1}{2}\int_{\Omega} |f-v|^2d\mathbf{x} $$

where in $J(v)$: (i) $\Omega$ is typically a rectangular region of $R^2$ and $d\mathbf{x}=dx_1dx_2$. (ii) $\varepsilon, a$ and $b$ are positive parameters. (iii) function $f$ represents the image one intends to denoise.

Minimizing functional $J$ is a non-smooth, non-convex bi-harmonic problem from Calculus of  Variations. Its numerical solution is a relatively complicated issue. However, one can achieve this task rather easily by combining operator-splitting and finite element approximations. The main goal of this lecture is to describe such a methodology and to present the results of numerical experiments which validate it.