Georgia Institute of Technology College of Sciences

The traditional epidemiological approach to characterize transmission of infectious disease consists of compartmentalizing hosts into susceptible, exposed, infected, recovered (SEIR), and vectors into susceptible, exposed and infected (SEI), and variations of this paradigm (e.g. SIR, SIR/SI, etc.). Compartmentalized models are based on a series of simplifying assumptions and have been successfully used to study a broad range of disease transmission dynamics. These paradigm is challenged when the within-host dynamics of disease is taken into account with aspects such as: (i) Simultaneous Infection: An infection can include the simultaneous presence of several distinct pathogen genomes, from the same or multiple species, thus an individual might belong to multiple compartments simultaneously. This precludes the traditional calculation of the basic reproductive number. (ii) Antigenic diversity and variation: Antigenic diversity, defined as antigenic differences between pathogens in a population, and antigenic variation, defined as the ability of a pathogen to change antigens presented to the immune system during an infection, are central to the pathogen's ability to 1) infect previously exposed hosts, and 2) maintain a long-term infection in the face of the host immune response. Immune evasion facilitated by this variability is a critical factor in the dynamics of pathogen growth, and therefore, transmission.This talk explores an alternate mechanistic formulation of epidemiological dynamics based upon studying the influence of within-host dynamics in environmental transmission. A basic propagation number is calculated that could guide public health policy.

In Computer Vision and multi-view geometry one considers several cameras in general position as a collection of projection maps. One would like to understand how to reconstruct the 3-dimensional image from the 2-dimensional projections. [Hartley-Zisserman] (and others such as Alzati-Tortora and Papadopoulo-Faugeras) described several natural multi-linear (or tensorial) constraints which record certain relations between the cameras such as the epipolar, trifocal, and quadrifocal tensors. (Don't worry, the story stops at quadrifocal tensors!) A greater understanding of these tensors is needed for Computer Vision, and Algebraic Geometry and Representation Theory provide some answers.I will describe a uniform construction of the epipolar, trifocal and quadrifocal tensors via equivariant projections of a Grassmannian. Then I will use the beautiful Algebraic Geometry and Representation Theory, which naturally arrises in the construction, to recover some known information (such as symmetry and dimensions) and some new information (such as defining equations). Part of this work is joint with Chris Aholt (Microsoft).

In this talk, the two approaches for computing the long time behavior of highly oscillatory dynamical systems will be introduced. Firstly, a generalization of the backward-forward HMM (BF HMM) will be discussed. It is intended to deal with the multiple time scale (>2) behavior of certain nonlinear systems where the non-linearity is introduced as a perturbation to a primarily linear problem. Focusing on the Fermi-Pasta-Ulam problem, I propose a three-scale version of the BF HMM. Secondly, I will consider a multiscale method using a signal processingidea. The dynamics on the slow time scale can be approximated by an averaged system gained by fltering out the fast oscillations. An Adaptive Local Iterative Filtering (ALIF) algorithm is used to do such averaging with respect to fast oscillations.