April 1: Timothy Susse (The Graduate Center, CUNY)
Title: SCL and Knot Complements
Abstract: We present a topological algorithm that can be used to compute SCL in amalgams of free Abelian groups over cyclic subgroups. In particular, we will show that such groups are PQL, implying that SCL is rational for any element of the group. The most interesting examples of such groups are fundamental groups of torus knot complements. Our approach will allow us to classify and study all surfaces with boundary in these spaces and provide combinatorial proofs of some old theorems of Waldhausen.
April 22: Guillaume Dreyer (Notre Dame)
Title: Parametrizing Hitchin Components
Abstract: Attached.
April 29: Rodrigo Trevino (Cornell)
Title: Recent developments on ergodic properties of translation flows
Abstract: The geodesic flow on a flat 2-dimensional torus is an example of a translation flow and all ergodic properties of this system are well-understood. For translation flows on flat surfaces of higher genus some things are known and some are not. I will talk about some recent results concerning the ergodic properties of translation flows for flat surfaces of finite area. These include surfaces of finite area but of infinite topological type. I will also discuss their relationship with orbits with optimal escape rate for the Teichmuller (geodesic) flow on the moduli space of flat surfaces.
May 1: Denis Sullivan (CUNY Graduate Center and Stony Brook) joint CAMP/PDE
Title: The algebra of Poincare duality and statistics for Navier-Stokes
Abstract: If the pieces of differential graded algebra hold together as they should, one obtains a tower of finite dimensional ODE's related by semiconjugacies [i.e., maps that commute with time evolution] that are derived from the Navier Stokes PDE in 3D -- which is itself used to model incompressible Newtonian fluid motion. Using these ODEs one can show the existence of measures at the top of the tower which project to consistent stationary measures for the ODE of each model in the tower. One can calculate algebraically the ODEs by algorithms that depend only on the combinatorics of cellular decompositions. These ODEs can be treated numerically to compute stationary measures.
May 6: Marian Gidea (Northeastern Illinois/Institute of Advanced Studies)
Title: Hamiltonian Instability Driven by Recurrent Dynamics
Abstract: We present some novel approaches to the instability problem of Hamiltonian systems (in particular, to the Arnold Diffusion problem). We show that, under generic conditions, perturbations of geodesic flows by recurrent dynamics yield trajectories whose energy grows to infinity in time (at a linear rate, which is optimal). We also show that small, generic perturbations of integrable Hamiltonian systems yield trajectories that travel large distances in the phase space. The systems that we consider are very general. The methodology relies on the theory of normal hyperbolicity and on the recurrent properties of the dynamics. The moral is: a little recurrence goes a long way.
May 13: Saar Hersonsky
Title: From tiling and packing to uniformization, I & II.
Abstract: The celebrated Riemann mapping theorem asserts that a non-empty simply connected open subset of the complex plane (which is not the whole of it) is conformally equivalent to the open unit disk in the complex plane. Conformally equivalent means that there exists an angle-preserving map from this subset to the open unit disk which is one to one and onto, and that map has an inverse with the same properties. Such a map preserves the shape of any sufficiently small figure, while possibly rotating and scaling (but not reflecting) it. Can one visualize this map? We will start with a remarkable conjecture made by Thurston which was first proved by Rodin-Sullivan, continue with related work of Dehn, Schramm and Cannon-Floyd-Parry that involves tiling by squares, and end with current research directions aim at addressing more general questions such as: Given a surface with some combinatorics defined on it, such as a cellular decomposition, can one use the combinatorics to obtain an effective version of uniformization theorems?
May 20: Kathryn Lindsey (Cornell)
Title: Invariant components of hyperelliptic translation surfaces
Abstract: The flow in a fixed direction on a translation surface determines a decomposition of the surface into closed invariant sets, each of which is either periodic or minimal. A natural question is "how many periodic and minimal components can a surface of a given type have?" In the case of translation surfaces which are elements of the hyperelliptic connected components of the moduli space of translation surfaces, the answer is surprising. I will present a characterization of the pairs of nonnegative integers (p,m) for which there exists a translation surface in the hyperelliptic connected components $\mathcal{H}^{hyp}(2g-2)$ or $\mathcal{H}^{hyp}(g-1,g-1)$ with precisely p periodic components and m minimal components.