Mathematical models have become essential tools in the increasingly quantitative world of biology. In some cases, mathematics can reveal patterns in pre-existing static biological data. In other cases, mathematical models can interact dynamically with experimental biology by providing insight into observed phenomena as well as generating novel and non-intuitive hypotheses to motivate experimental design. In this talk, I will present my recent work in both of these realms of mathematical biology.

I developed a mathematical model to discover inversion structural variants in human populations from pre-existing SNP data. Inversion chromosomal variants have long been considered important in understanding speciation because large inversions create reproductive isolation by suppressing recombination between inverted and normal chromosomes. Recent studies have identified many polymorphic inversion variants in human populations. Many of these inversions appear to have functional consequences and have been associated with genetic disorders and complex diseases, such as asthma. In addition, there is evidence some inversions may be under positive selection. I created a probabilistic mixture to identify putative inversion polymorphisms from phased haplotype data. By examining characteristic differences in allele frequencies around candidate inversion breakpoints, I partition the population into "normal" and "inverted" chromosomes. Predictions from my model are supported by previously validated inversions and represent a rich new source of candidate inversion polymorphisms.

In collaboration with experimental yeast biologists, I developed and validated a new model of prion transmission. Prion proteins are responsible for severe neurodegenerative disorders, such as bovine spongiform encephalopathy in cattle ("mad cow" disease) and Creutzfeldt-Jakob disease in humans. These diseases arise when a protein adopts an abnormal folded state and persists when that form self-replicates. While prion diseases are progressive, evidence in yeast suggests that this process can be reversed and eliminated. To understand the mechanistic basis of this "curing", I developed a stochastic model of prion dynamics that suggested a new theory for prion transmission. Results from my model guided experimental design, leading to new and non-intuitive insights about propagation of the abnormal fold through a population.

This study is important in understanding the mechanism and dynamics of some biological problems such as tumor invasion and wound healing. Firstly, we describe microscopically the model and we derive the corresponding mesoscopic approximation, via the mean field assumption. In the following, we upscale our model providing a PDE which serves as
a macroscopic manifestation of the underlying cellular interactions. We focus on investigating the speed and the structure of the invasion
front, using the above mentioned approximations, as functions of the underling cell phenotypes and microenvironmental factors (i.e. nutrients).

The Cartan-Kahler theory for exterior differential systems (EDS) is a powerful tool in the study of over-determined systems and geometric systems with symmetry. I will present the machinery for linear Pfaffian systems, the setting in which the theory is most commonly applied. (The prolongation of any EDS is linear Pfaffian.) We will discuss tableau, torsion, involutivity, prolongation and Cartan's Test. I will illustrate the method with many examples, and hope to convince the audience of the power and elegance of the machinery.

If time permits, I will briefly discuss the Cartan-Kalher theory for general EDS: this is the machinery R. Bryant used to prove the existence (and construct examples) of metrics of exceptional holonomy in 1987.