When transitioning from studying Euclidean space to more Riemannian manifolds, one must first unlearn many special properties of the flat world. The same is true in physics: while one can make sense of classical physics on an arbitrary curved background space, many seemingly foundational concepts (like the center of mass) turn out to have no place in the general theory.  Freed from the constraints such properties induce, classical physics on a curved background space has many surprises in store.  In this talk I will share some stories related to joint work with Brian Day and Sabetta Matsumoto on understanding and simulating such situations, focusing on hyperbolic space when convenient.  To give a taste, here are two such surprises:
(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 with no external forces.  The arguments for the former are readily accessible to beginning graduate students in geometry, and the latter illustrates a use of gauge theory in classical mechanics, following work of Wilczek and Montgomery.
 

The study of real planar curves dates back to antiquity, where the ancient Greeks studied curves defined on the plane cut out by polynomials of two variables. We’ll provide a friendly overview to beautiful formulas of Plücker which govern the “shape” of planar curves. We will discuss the Shapiro—Shapiro conjecture and connections to the real Schubert calculus, and end by presenting some new conjectures and computational evidence joint with Frank Sottile.

In a work on the geometry of minimal submanifolds written in 1968, Shiing-Shen Chern invited more efforts and reflections to identify relationships between intrinsic and extrinsic curvature invariants of submanifolds in various ambient spaces. After 1993, when Bang-Yen Chen introduced the first of his curvature invariants, namely scal - inf(sec), a lot of work has been done to explore this avenue, which represents an active research area. We will survey some of these results obtained in the last three decades, and conclude our talk with new relationships between intrinsic and extrinsic curvature invariants.

A manifold is called formal if it has the same rational homotopy type
as the cohomology ring. We first consider the formality of a sphere bundle
over a formal manifold. In this case the formality is entirely determined by
the Bianchi-Massey tensor, which is a 4-tensor on a subspace of the
cohomology ring, introduced by Crowley and Nordstrom. As a special case, we
will see that if a manifold and its unit tangent bundle are both formal,
then the manifold has either Euler characteristic zero or rational
cohomology ring generated by one element. Finally we discuss the case of
a general base manifold, and give an obstruction to formality.