Quantized higher Teichmüller theory, as described by Fock and Goncharov, assigns an algebra and its representation to a surface and a Lie group. This assignment is equivariant with respect to the action of the mapping class group of the surface, and is conjectured to give an analog of a modular functor, that is it should respect the operation…

The cellular cytoskeleton is essential in proper cell function as well as in organism development. These filaments represent the roads along which most protein transport occurs inside cells. I will discuss examples where questions about filament-motor protein interactions require the development of novel mathematical modeling, analysis, and simulation. In the development of egg cells into…

Fusion categories are rich mathematical structures generalizing the representation categories of finite groups. They arise in many areas of mathematics and physics. Most strikingly, they have emerged as models for particle-like excitations with exotic exchange statistics in low dimensional quantum field theories. We will provide an introduction to these ideas and discuss recent results on…

A hypersurface in a Riemannian manifold is called minimal if its mean curvature vanishes identically.  Minimal surfaces have fascinated mathematicians since the time of Euler, and tremendous progress has been made in understanding the structure of the space of embedded minimal surfaces in various ambient manifolds in the last 50 years.  After surveying the subject, I will…

This talk will cover two approaches for modeling blood flow in the human body.  The first approach describes blood transport in elastic vessels and requires the numerical solution of a nonlinear hyperbolic system on branching vessel networks.  I will discuss some mathematical properties of these equations that seem to be useful for analysis of numerical schemes,…

What can we tell about the interior structure of our planet, if we observe the travel time of a large number of earthquakes? This is the time it takes for a seismic wave to travel from the epicenter of the  earthquake to the seismic sensor. In the geometric literature, the boundary rigidity problem on a compact Riemannian manifold…

Many materials are built from a grid of flexible but nearly inextensible rods that behaves as a shell-like structure. Everyday examples range from fabrics made of 1000s of interwoven yarns; to kitchen strainers made of 100s of plastically deforming wires; to architectural gridshells or medical stents made of 10s of elastically deforming rods. In this…

Pertussis, commonly known as whooping cough, is caused by the bacterial pathogen Bordetella pertussis. Completely susceptible individuals experience severe disease, with the hallmark whooping cough, but those with partial immunity have milder, if any symptoms. Immunity following natural infection (or immunization) may wane, increasing susceptibility with time since exposure. In this talk, we begin by examining…

Students often take precalculus or college algebra as a terminal math course, leaving them with the impression that mathematics lacks real meaning. Due to the increasingly interdisciplinary nature of the mathematical sciences, we are well-poised to intervene and design inspiring general education courses that reveal the utility of mathematics. In this talk, I will share…

A circle packing is the mathematical name for a collection of circles. I am interested in circle packings with a fixed pattern of tangencies between the circles. Given a tangency pattern, one might ask questions like, "Can I find a circle packing with that tangency pattern?" and "How many such circle packings can I find?"…