Stevel Trettel, Assistant Professor of Mathematics at the University of San Francisco, will give a colloquium.

Title: Classical Physics in Curved Space

Abstract:

Classical physics, as formalized by Newton, traditionally operates within the confines of a flat ambient space. However this is not a requirement: the Lagrangian and Hamiltonian approaches offer a more flexible framework, allowing us to contemplate classical worlds on curved background spaces. In this talk, we explore some of the surprising differences of such worlds, focusing on hyperbolic space when convenient for exposition.

We will look at the way curvature affects different aspects of the world: from the geometry of optics, to geometric forces, and finally dynamics; culminating with some current work of Brian Day, Sabetta Matsumoto and myself on the control theory of deformable bodies.  As a taste, here are two surprising facts: (1) there is no Galilean relativity: inside a sealed box in hyperbolic geometry it is possible to perform an experiment which detects your precise velocity.  And (2): it's possible to ‘swim’ in the vacuum in hyperbolic space - to move your arms and legs in a specific pattern that causes you to translate along a geodesic (compare to Euclidean space, where you cannot move your center of mass by internal deformations only).

Bio:  Steve Trettel is an assistant professor of mathematics at the University of San Francisco.  He works in geometry, topology and mathematical illustration, and is always looking for new mathematical ideas to draw beautiful pictures of.  Outside of math, Steve is interested in languages, and enjoys cycling around the Bay Area.