Living cells exhibit many forms of spatial-temporal dynamics, including recently-discovered traveling waves. Cells use these traveling waves to organizer their insides, to improve cell-cell communication, and tune their ability to move around the body. Some of these traveling waves arise from excitability (positive feedback) and non-local coupling (dynamics that spread spatially on timescales much faster than the timescale of wave motion). In collaboration with graduate students at UC Irvine, our research has studied two traveling waves involving the mechanics of the cytoskeletal protein actin: one that is approximately equivalent to a reaction-diffusion system [Barnhart et al, 2017, Current Biology], and one that is not [Manakova et al, 2016, Biophys J]. For the non-reaction-diffusion wave, we demonstrate conditions for wave travel analogous to ones previously derived for reaction-diffusion waves. We also demonstrate the existence of a "pinned" regime of parameter space absent in the equivalent reaction-diffusion system.
Many classical problems in dynamical systems have been studied for more than a century. Despite enormous progress, several have resisted solution. Descriptive Set Theory provides the tools to prove impossibility results, both in the quantitative and the qualitative behavior of dynamical systems.
I will talk about some challenging problems and the role of Fourier concentration in an emerging field called learning theory. Suppose we want to learn a certain data. The basic question we ask is what is the minimal amount of information needed to learn (predict) the data with high precision and probability, and what is the role of Fourier analysis in these questions.
Given a property P of graphs, it is natural to wonder how likely a given finite graph satisfies property P, that is, given some fixed natural number n, what proportion of the n vertex graphs satisfies property P. If P is a "first-order" property, then there is a 0-1 law that says the proportion of graphs of size n that satisfy P approaches either 0 or 1 as n tends to infinity. A simple proof of this fact uses the model theory of the so-called random (or Rado) graph. We will present the proof of this result and then discuss recent work, joint with Bradd Hart and Alex Kruckman, which show these ideas can be used to prove an approximate 0-1 law for finite metric spaces. The metric space version of the random graph that is relevant turns out to be somewhat surprising!
