Departmental Colloquium

Swimming, or self locomotion through a fluid, is done by algae, bacteria, birds, and whales. It even occurs inside of cells. Swimming becomes especially fascinating when it involves collectives that interact through the fluid. I'll talk about a few examples. One involves experiments and models that explore the interactions of many flapping flyers. Surprising effects occur due to the ability of the fluid to store information on the history of the flow. At a very different scale I'll discuss how biological motor proteins can collectively drive flow and transport in the cell, such as the "swimming" and positioning of the pronuclear complex prior to cell division.

For hundreds of years, mathematics has had a close connection with the biological sciences, and with physiology in particular. Indeed, some of the most famous equations in applied mathematics were originally motivated by physiological problems, such as the flow of blood in blood vessels. In this lecture I shall look briefly at some famous examples of how mathematics has been used to study physiological problems, including Bernoulli's smallpox model from 1765, mathematical models of the cochlea from Helmholtz to Hudspeth, and the propagation of action potentials in neurons. I'll end with a brief look at some modern applications of mathematics to physiology, and show how physiology continues to inspire and direct new mathematical investigations.

A holomorphic one-form defines a flat metric such that the underlying complex curve can be realized as a plane polygon. Varying the shape of the polygon induces an SL(2,R)-action on the Hodge bundle, which is called the Teichmueller dynamics. Among all orbit closures, those of minimal dimension correspond to Teichmueller curves. In this talk I will give an introduction to this beautiful subject, with a focus on the interplay between billiards in polygon and intersection theory on moduli space. As an application, we prove a conjecture of Kontsevich-Zorich about a non-varying property of Teichmueller curves in low genus (joint work with M. Moeller). I promise this talk will be accessible to everyone!