Mercator Workshop 2022

Abstract: We will fix some topological data, a pants decomposition, of a closed surface of genus g and build hyperbolic structures by gluing hyperbolic pairs of pants along their boundary. The set of all hyperbolic metrics with a pants decomposition having a given set of lengths defines a (3g-3)-dimensional immersed torus in the (6g-6)-dimensional moduli space of hyperbolic metrics, a twist torus. Mirzakhani conjectured that as the lengths of the pants curves tend to infinity, that the corresponding twist torus equidistributes in the moduli space. In joint work-in-progress with Aaron Calderon, we confirm Mirzakhani`s conjecture. In the talk, we explain how to import tools in Teichmüller dynamics on the moduli space of flat surfaces with cone points to dynamics on the moduli space of hyperbolic surfaces with geodesic laminations.

Abstract: Despite the fact that the 3-body problem is an ancient conundrum that goes back to Newton, it is remarkably poorly understood, and is still a benchmark for modern developments. In this talk, I will give a (very) biased account of this classical problem, both from a modern theoretical perspective, i.e. outlining possible lines of attack coming from Symplectic Geometry, holomorphic curves and Floer theory; as well as comment on practical and numerical aspects, within the context of finding orbits for space mission design and space exploration.

Abstract: I will explain a rigidity result that intertwines the deep diagonal pentagram maps and Poncelet polygons, the first very nontrivial case of a general conjecture. The result is that the full orbit of a centrally symmetric octagon O under the 3-diagonal map is convex if and only if O is a Poncelet polygon. The proof involves a lucky guess concerning the symplectic nature and the invariants of the map, and then a careful analysis of the geometry of the associated Lagrangian level-set foliation in R4. I'll illustrate the result with a lot of pictures and computer demos.

Abstract: In a joint work with P. Sarkar, we showed local mixing for certain one parameter diagonal flows on higher rank Borel Anosov homogeneous spaces Γ \ G with respect to Bowen-Margulis-Sullivan measure. A natural follow up question is whether one can give a rate for this mixing, in particular, an exponential rate. Due to the higher rank nature of the space, this seems unlikely to be the case. However, the space turns out to be a vector bundle over a "rank one like" space homeomorphic to the Gromov hyperbolic space associated to Γ. In a work in preparation with P. Sarkar, we prove the Gromov geodesic flow is exponentially mixing with respect to the induced Bowen-Margulis-Sullivan measures. I will talk about this work.