A satellite operator on knots is called an L-space satellite operator if the 2-component link, consisting of the pattern and the meridian of the solid torus, is an L-space link. Examples of such operators include cabling, Whitehead double and Mazur patterns. We present an algorithm to compute the knot Floer homology of the resulting knot in terms of the knot Floer homology of the companion knot, as well as the Alexander polynomials of the 2-component link and the pattern. This algorithm is based on the link surgery formula of Manolescu and Ozsváth, together with a reinterpretation and a connected sum formula of it by Zemke. A key ingredient in this algorithm is a formality result for 2-component L-space links, which says we can determine the knot Floer chain complex of such a link from the Alexander polynomial of it and its sublinks. This is joint work with Ian Zemke and Hugo Zhou.

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.