Georgia Institute of Technology College of Sciences

Rob Kirby and I have been thinking for a while now about stable maps to 2-manifolds, which we call "Morse 2-functions", to stress the analogy with standard Morse theory, which studies stable maps to 1-manifolds. In this talk I will focus on the extent to which we can extend that analogy to the way in which handle decompositions combinatorialize Morse functions, especially in low dimensions. By drawing the images of attaching maps and some extra data, one describes the total space of a Morse function and the Morse function, up to diffeomorphism. I will discuss how much of that works in the context of Morse 2-functions. This is important because Rob Kirby and I have spent most of our time thinking about stable homotopies between Morse 2-functions, which should be thought of as giving "moves" between Morse 2-functions, but to honestly call them "moves" we need to make sure we have a reasonable way to combinatorialize Morse 2-functions to begin with.

We consider particle dynamics in the (unfolded) Ehrenfest Windtree Model and theflow along straight lines on a certain folded complex plane. Fixing some parameters,it turns out that both doubly periodic models cover one and the same L-shaped surface.We look at the case for which that L-shaped surface has a (certain kind of) structure preservingpseudo-Anosov. The dynamics in the eigendirection(s) of the pseudo-Anosovon both periodic covers is very different:The orbit diverges on the Ehrenfest model, but is dense on the folded complex plane.We show relations between the two models and present constructions of folded complex planes.If there is time we sketch some of the arguments needed to show escaping & density of orbits.There will be some figures showing the trajectories in different settings.

I will discuss a computation of the lower central series of the Torelli group as a symplectic module, which depends on some conjectures and was performed 15 years ago in unpublished joint work with Ezra Getzler. Renewed interest in this computation comes from recent work of Benson Farb on representation stability.

In this talk a curve complex HC(S) closely related to the "Cyclic Cycle Complex" (Bestvina-Bux-Margalit) and the "Complex of Cycles" (Hatcher) is defined for an orientable surface of genus g at least 2. The main result is a simple algorithm for calculating distances and constructing quasi-geodesics in HC(S). Distances between two vertices in HC(S) are related to the "Seifert genus" of the corresponding link in S x R, and behave quite differently from distances in other curve complexes with regards to subsurface projections.