# The geometry and dynamics of uniformly periodic recursion relations. Vanderbilt University Mathematics Colloquium Series. Vanderbilt University Mathematics Department. April 21, 2016

Invited Talk

When H(x) is a real-valued function of a real variable, the 2-step recursion relation x_{i+1} = H(x_i) - x_{i-1} is said to be n-periodic if it is periodic with period n for all initial conditions (x_0,x_1). These n-periodic recursion relations and their generalizations 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 recursion relations and their surprising relationship with differential geometry, cluster algebras, and the theory of overdetermined differential equations. I won’t assume that the audience knows any 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

- April 21, 2016

### Service or Event Name

- Vanderbilt University Mathematics Colloquium Series

### Host Organization

- Vanderbilt University Mathematics Department

### Location or Venue

- Vanderbilt University