Abstract: In this talk I will show a sample of graphical user interfaces I have made which either illustrate concepts in geometry or else give geometrical slants to concepts a little bit outside of geometry. The topics will range from graph theory to number theory to computational geometry. This first talk will set the theme for the rest of the series, in which I use these kinds of interfaces to prove more serious results in mathematics.
Abstract: Outer billiards is a game played in the plane, on the outside of a convex shape, that is similar in spirit to ordinary billiards. When the shape is a polygon, the game often produces beautiful tilings of the plane. After presenting some general information about this topic, I will concentrate on my solution of the so-called Moser-Neumann problem, which asks whether one can have an unbounded orbit for an outer billiards system. (Yes.) I will illustrate many facets of my proof with demos from a graphical user interface I made in order to study this problem.
Abstract: (Part of this is joint work with my student Matei Coiculescu.) The 3-dimensional solvable Lie group SOL is probably the strangest of the 8 Thurston geometries. Up until recently, the metric spheres in Sol were quite mysterious, and it was not even known if they are topological spheres. Using a Hamiltonian systems approach introduced years ago by Matt Grayson, and some tricks of our own, we exactly characterize the cut locus of SOL in terms of the arithmetic-geometric mean of Gauss, and as a byproduct prove that the metric spheres in SOL are all topological spheres, smooth away from 4 planar arcs. I will illustrate some ideas in the proof using a graphical user interface I made for the purpose of studying this problem.