November 5, 2022 at Brown University

GATSBY is a conference organized by the geometry/topology groups at Brown and Yale. All are welcome!

Speakers

Schedule (talks in the Foxboro auditorium in Brown's math department)

10-10:30 Coffee/breakfast

10:30-11:15 Nick's background talk

11:15-11:45 Break

11:45-12:30 Morgan's background talk

12:30-2 Lunch (catered)

2-2:45 Nick's research talk

2:45-3:15 Break

3:15-4 Morgan's research talk

Abstracts

• Nick Miller

Background talk: We will explain some of the ingredients that go into the research talk, focusing mostly on the technical tools necessary to understand the ideas of the proof. Topics are likely to include a subset of the following: Sunada constructions, elevations of curves in finite covers, maps between curve complexes/Teichmuller spaces in the presence of covers, and geodesic laminations.

Research talk: Unmarked simple length spectral rigidity for covers

A long studied problem in hyperbolic (or more generally, negatively curved) geometry is the extent to which a manifold M is determined by its collection of lengths of closed geodesics on M. For instance, Otal showed that any negatively curved metric on a surface is determined up to isometry by its marked length spectrum, that is, by the function which associates to each closed curve the length of the unique geodesic in its free homotopy class. By classic work of Fricke, the similar result is true if one restricts this function only to simple closed curves.

By celebrated constructions of Vignéras and Sunada, we now know that the corresponding statement is false when one forgets the marking, that is, there exist non-isometric surfaces which have the same collections of lengths of closed geodesics. In this talk, we will explore the extent to which surfaces arising from Sunada's construction can have the same collection of lengths of simple closed curves. Along the way we will also discuss some new general results about how simple lifts of curves can determine equivalence of covers. This represents joint work with Tarik Aougab, Max Lahn, and Marissa Loving.

• Morgan Weiler

Background talk: We will review the physical and dynamical motivation behind the definition of the standard symplectic form on C^n. We will then go on to discuss moment maps, which provide a combinatorial perspective on symplectic torus actions. Finally, we will introduce almost toric fibrations, a generalization of Lagrangian torus fibrations that allow us to construct symplectic embeddings.

Research talk: Infinite staircases of 4D symplectic embeddings

The ellipsoid embedding function of a symplectic manifold measures the amount by which the symplectic form must be scaled in order to fit an ellipsoid of a given eccentricity. It generalizes the Gromov width and ball packing numbers. In 2012 McDuff and Schlenk computed the ellipsoid embedding function of the ball, showing that it exhibits a delicate piecewise linear pattern known as an infinite staircase. Since then, the embedding function of many other symplectic four-manifolds have been studied, and not all have infinite staircases. We will classify those symplectic Hirzebruch surfaces whose embedding functions have an infinite staircase. This project unites several players in symplectic geometry: obstructions coming from cobordism maps in Floer theory and symmetries of solutions to Diophantine equations, and constructions coming from moment maps and Lagrangian torus fibrations. Continued fractions will make a starring appearance. Based on work with Magill and McDuff and work in progress with Magill and Pires.

Organizers: Franco Vargas Pallete and Bena Tshishiku