Venue: Room 204, Hai Na Yuan #2, Zijingang Campus

Time: 16:00 - 17:20, November 3

Speaker: Habib Alizadeh (USTC-IGP)

Pretalk: Barcodes in geometry

Abstract: Barcode is a notion introduced first in Topological Data Analysis (TDA) and it captures the topology (holes) of a data set. In a different form this notion appeared long before TDA in the works of Morse where he uses functions on spaces to study their topology. Inspired by TDA and Morse, barcodes were re-introduced in geometry and are vastly studied with many significant applications, e.g., they can be used to detect periodic points of certain maps on manifolds; this is in the same spirit as the Morse inequality that states the number of critical points of a smooth non-degenerate function on a manifold is at least the sum of the Betti numbers of the manifold. In this talk we will define barcodes, using simple linear algebra that should be accessible by undergraduate students, and if time permits we will mention some applications of barcodes in geometry. 

Research talk: Spectral diameter of a symplectic ellipsoid

Abstract: Consider a diffeomorphism of an even-dimensional Euclidean space, that is compactly supported in a given convex open subset X, and preserves the standard symplectic form; in dimension two in particular it preserves the area. Using barcodes we define an invariant of such a diffeomorphism that we call the spectral norm of the diffeomorphism. The space of such diffeomorphisms of X equipped with the spectral norm has a finite diameter which is called the spectral diameter of X. This number defines a symplectic capacity for X, an object defined axiomatically in symplectic geometry to capture the “symplectic size” of a domain; the first symplectic capacities were defined by Gromov using pseudo-holomorphic curves. In this talk, we compute the spectral diameter for all symplectic ellipsoids and polydisks, and in dimension four, for all convex toric domains. Exact computations of symplectic capacities are useful in obstructing symplectic embeddings.