Tuesday, January 28, 2020
01/28/2020 - 4:15pm
In this talk I will discuss about the geometry of constant Gaussian curvature (CGC) surfaces inside hyperbolic ends, and how they relate to the structures of their pleated boundary and their boundary at infinity. In particular, I will describe how the “classical” Thurston’s and Schwarzian parametrizations of the space of hyperbolic ends can be recovered as the limit of two families of parametrizations, introduced by Labourie in terms of the data of immersions of the CGC-surfaces. In addition, I will mention a series of consequences of this phenomenon in relation to the notions of dual and W-volume, such as a new characterization of the renormalized volume in terms of the CGC-foliation, and a generalization of McMullen's Kleinian reciprocity theorem.
01/28/2020 - 6:00pm
Why is 22/7 such a good approximation to pi? What does the space of invertible 2x2 matrices look like? What about the space of lattices in the plane? We will answer all these questions and connect the answers by introducing an elegant geometric way to think about continued fractions. As a bonus, along the way we’ll happen to meet a short, intuitive, geometric proof of the polar decomposition from linear algebra.