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Robert Bryant : The geometry of periodic equi-areal sequences

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 low 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.

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