Date: Tuesday, February 10, 2015

Time: 12:30PM

Location: Rockefeller Hall 310

**Title:** Continued fractions: dynamics and geometry

**Abstract:**

A regular continued fraction is an expression for a real number in a particular, some might even say peculiar, manner. Finite length continued fractions are rational numbers, and in general the regular continued fraction expansion of a real number reveals the sequence of rational numbers "best" approximating the real number. Continued fraction expansions can be calculated by way of a certain function sending the unit interval to itself. Gauss found the correct way to calculate probabilities of entries in continued fractions, but we do not know how he discovered the underlying "measure."

In 1977, a group of Japanese mathematicians studied a particular function sending the unit square to itself that Gauss could have employed. One can interpret that 2-dimensional function in terms of hyperbolic geometry. This then relates flow through a certain 3-dimensional space and "those peculiar" continued fraction expansions. In this expository talk, we explain these various notions and hint at ongoing research.

Pizza lunch at noon in Rockefeller Hall 305