Activities This Week

In this talk, we explain how the classical Toda lattice model, one of the earliest examples of nonlinear completely integrable systems, can be used to demonstrate that certain configurations in the classical phase space are symplectic balls in disguise. No background in symplectic geometry is needed. The talk is based on joint work with Vinicius Ramos and Daniele Sepe.

Part of the talk is expository: I will explain how supersingular isogeny graphs can be used to construct cryptographic hash functions and survey some of the rich mathematics involved. Then, with this motivation in mind, I will discuss two recent theorems by Jonathan Love and myself. The first concerns the generation of maximal orders by elements of particular norms. The second states that maximal orders of elliptic curves are determined by their theta functions.

In studying higher dimensional Schrödinger operators of quasicrystals, one is lead to find suitable periodic approximations. This means in particular that the spectrum converges as a set to the limiting spectrum. It turns out that for this to hold, the convergence of the underlying dynamical systems is exactly what is needed. This is the starting point of the present talk.

We focus on aperiodic subshifts defined through symbolic substitutions. These substitution subshifts provide models of aperiodic ordered systems. We find natural sequence candidate of subshifts to approximate the aforementioned substitution subshift. We characterize when these sequences converge, and if so at what asymptotic rate. Some well-known examples of substitution subshifts are discussed during the talk. We will also discuss the motivation for this characterization, arising from an attempt to study higher dimensional quasi-crystals. This is based on a Joint work with Ram Band, Siegfried Beckus and Felix Pogorzelski.