# The geometry and dynamics of periodic equi-areal sequences. Tulane Colloquia in Mathematics. Tulane University Mathematics Department. February 25, 2016

Invited Talk

A sequence of functions $f = (f_i)$ ($-\infty < i < \infty$) on a surface $S$ is said to be \emph{equi-areal} (or sometimes, \emph{equi-Poisson}) if it satisfies the relations $$ df_{i-1}\wedge df_i = df_i\wedge df_{i+1}\ (\not=0) $$ for all $i$. In other words, the successive pairs $(f_i,f_{i+1})$ are local coordinates on $S$ that induce the same area form on $S$, independent of $i$. One says that $f$ is \emph{$n$-periodic} if $f_i = f_{i+n}$ for all $i$. The $n$-periodic equi-areal sequences for certain values of $n$ turn out to have close connections with interesting problems in both dynamical systems and in the theory of cluster algebras. In this talk, I will explain what is known about the classification (up to a natural notion of equivalence) of such sequences and their surprising relationships with differential geometry, cluster algebras, and the theory of overdetermined differential equations. I won’t assume that the audience knows much differential geometry, just basic multi-variable calculus, and the emphasis will be on describing the interesting results rather than on technical details.

### Service Performed By

- Bryant, Robert Phillip Griffiths Professor of Mathematics

### Date

- February 25, 2016

### Service or Event Name

- Tulane Colloquia in Mathematics

### Host Organization

- Tulane University Mathematics Department

### Location or Venue

- Tulane University Mathematics Department