Electrolyte and cell volume regulation is essential in physiological systems. After a brief introduction to cell volume control and electrophysiology, I will discuss the classical pump-leak model of electrolyte and cell volume control. I will then generalize this to a PDE model that allows for the modeling of tissue-level electrodiffusive, convective and osmotic phenomena. This model will then be applied to the study of cortical spreading depression, a wave of ionic homeostasis breakdown, that is the basis for migraine aura and other brain pathologies.

We describe a short, direct, alternative to the DeTurck trick to prove the
uniqueness of solutions to a large class of curvature flows of all orders,
including the Ricci flow, the L^2 curvature flow, and other flows related
to the ambient obstruction tensor. Our approach is based on the analysis
of simple energy quantities defined in terms of the actual solutions to the
equations, and allows one to avoid the step -- itself potentially
nontrivial in the noncompact setting -- of solving an auxiliary parabolic
equation (e.g., a k-harmonic-map heat-type flow) in order to overcome the
gauge-invariance-based degeneracy of the original flow. We also
demonstrate that, by the consideration of a certain energy
quotient/frequency-type quantity, one can give a short and quantitative
proof (avoiding Carleman inequalities) of the global backward uniqueness of
solutions to a large class of these equations.

The ABC Conjecture, roughly stated says that the equation A+B+C=0 has no solutions for relatively prime, highly divisible integers A, B, and C. If the divisibility criteria are relaxed, then solutions exist and a conjecture of Mazur predicts the density of such solutions. We discuss techniques for proving this conjecture for certain ranges of parameters.

In 2010, Zagier defined the notion of a "quantum modular form,'' and offered several diverse examples, including Kontsevich's 'strange' function. Here, we construct infinite families of quantum modular forms, and prove one of Ramanujan's remaining claims about mock theta functions in his last letter to Hardy as a special case of our work. We will show how quantum modular forms underlie new relationships between combinatorial mock modular and modular forms due to Dyson and Andrews-Garvan. This is joint work with Ken Ono (Emory U.) and Rob Rhoades (CCR-Princeton).

We study the result of repeatedly differentiating a random entire function whose zeros are the points of a Poisson process of intensity 1 on $\mathbb{R}$. Based on joint work with Robin Pemantle.