Geometry Labs United 2017

Plenary Abstracts

Non-Euclidean Virtual Reality

Speaker: Sabetta Matsumoto, Georgia Tech
 (joint work with Vi Hart, Andrea Hawksley, and Henry Segerman)
Abstract:
The properties of euclidean space seem natural and obvious to us, to the point that it took mathematicians over two thousand years to see an alternative to Euclid’s parallel postulate. The eventual discovery of hyperbolic geometry in the 19th century shook our assumptions, revealing just how strongly our native experience of the world blinded us from consistent alternatives, even in a field that many see as purely theoretical. Non-euclidean spaces are still seen as unintuitive and exotic, but with direct immersive experiences we can get a better intuitive feel for them. The latest wave of virtual reality hardware, in particular the HTC Vive, tracks both the orientation and the position of the headset within a room-sized volume, allowing for such an experience. We use this nacent technology to explore the three-dimensional geometries of the Thurston/Perelman geometrization theorem. This talk focuses on our simulations of \(\mathbb{H}^3\) and \(\mathbb{H}^2 \times E\).

On the Possible Values of \(\pi\), and Other Adventures in Normed Linear Spaces

Speaker: Kelly Delp, Cornell University

Abstract:

Let \(\Omega\) be a closed, bounded, convex set in the plane that contains and is symmetric about the origin. For example, \(\Omega\) could be a regular \(2n\)-gon centered at the origin. Such a set can be used to define a linear norm on the plane, which induces a so called Minkowski metric. 

In this talk, I’ll survey some results about this family of metric spaces, including the work of former undergraduates Sam Reed and Michael Filipski. Reed computed the values of \(\pi\) in the regular \(2n\)-gon metrics, and Filipski gave an explicit description of a \(2\)-parameter sine function in the taxicab metric. As time allows, I’ll share my motivation for learning more about these metric spaces, which lies in their connection to convex projective structures on surfaces. I’ll also have plenty of animations, and will discuss how they were created. 

Sensual Mathematics

Speaker: Kirsi Peltonen, Aalto University

Abstract:

I will describe some ideas and concrete openings at Aalto University to enhance interaction between science and arts. This is a talk for a broad audience.

Photo: Henri Vogt