Traveling waves arise in partial differential equations in a broad range of applications. The notion of stability of a traveling wave solution concerns its resilience to small perturbations and can frequently be inferred from an eigenvalue problem obtained by linearizing the PDE about the solution. I will discuss these ideas in the context of the FitzHugh--Nagumo system, a simplified model of nerve impulse propagation. I will present existence and stability results for (multi)pulse solutions, and I will describe a phenomenon whereby unstable eigenvalues accumulate as a single pulse is continuously deformed into a double pulse.

Suppose that at each vertex of the Cayley graph of a finitely generated group G is a person holding a dollar. Everybody is told to pass their dollar bill to a neighbor. This can be done so that each person’s net worth increases if and only if the group G is non-amenable. Thus, one can think of non-amenable groups as those where Ponzi schemes can benefit everyone. The Cayley graph of the free group with two generators is an infinite 4-valent tree. If everyone passes their dollar towards the origin then everyone’s net worth increases! Because we live in a world where Ponzi schemes don't work, we restrict our attention to amenable groups such as the integer lattice. For dynamically-defined operator families, the Hausdorff distance of the spectra is estimated by the distance of the underlying dynamical systems while the group is amenable. We prove that if the group has strict polynomial growth and both the group action and the coefficients are Lipschitz continuous, then the spectral estimate has a square root behavior or, equivalently, the spectrum map is $ \frac{1}{2} $-Hölder continuous.

Numerical simulation has played a key role in the study of dynamical systems, from modeling ecosystems to weather simulations. Archetypical examples of complex fractals generated by simple non-linear dynamical systems are Julia sets of quadratic polynomials. Computer-generated Julia sets are among the most familiar mathematical images, enjoyed both for their beauty and for the deep theory behind them. In a series of works with M. Braverman and others we have put to the test the modern paradigm of numerical simulation of chaotic dynamics, and asked whether images of Julia sets can always be computed if the parameters are known. My talk will describe some of the surprising results we have obtained, and several intriguing open problems.

I will describe recent advances in promoting renormalization techniques from one dimensional holomirphic setting to two dimensions, together with the set of questions that these techniques are aiming to address, such as rigidity and geometry of attractors.

We develop the basic spectral theory of ergodic Schrodinger operators when the underlying dynamics are given by a conservative ergodic transformation of a \sigma-finite measure space. Some fundamental results, such as the Ishii--Pastur theorem carry over to the infinite-measure setting. We also discuss some examples in which straightforward analogs of results from the probability-measure case do not hold. We will discuss some examples and some interesting open problems. 

The talk is based on a joint work with M. Boshernitzan, D. Damanik, and M. Lukic.