Combinatorics Seminar: "A tale of two widths: lattice and Gromov"
Abstract: To a polytope P whose facet normals are rational one can associate two geometric objects: a symplectic toric domain X_P and a polarised toric algebraic variety Y_P, which can also be viewed as a potentially singular symplectic space. A basic invariant of a symplectic manifold X is its Gromov width: essentially the size of the largest ball that can be embedded in X in a way that respects the symplectic structure. A conjecture of Averkov-Hofscheier-Nill proposed a combinatorial bound for the Gromov width of Y_P, which I recently verified in dimension two with Julian Chaidez. I’ll discuss the proof, which goes via various symplectic and algebraic invariants with winsome combinatorial interpretations in the toric case. If there’s time, I’ll discuss ongoing work and new challenges for a similar result in higher dimensions. I will assume no substantial prior understanding of symplectic or algebraic geometry.
Host: Laura Escobar Vega