Geometric Analysis and Topology Seminar

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Systolic freedom and rigidity modulo 2

Speaker: Alexey Balitskiy, IAS

Location: Warren Weaver Hall 512

Videoconference link: https://nyu.zoom.us/j/95759976381

Date: Friday, May 5, 2023, 11 a.m.

Synopsis:

Given a closed $n$-dimensional Riemannian manifold $M$, let us define its systole as the length of a shortest loop that does not bound a surface in $M$; and its cosystole as the smallest area of an $(n-1)$-dimensional submanifold that does not bound an $n$-dimensional domain (with mod 2 coefficients). Answering a question of Gromov, in 1998 Freedman exhibited the first examples of manifolds in which the product of the systole and the cosystole cannot be bounded from above by the volume of $M$; this manifests the phenomenon of systolic freedom. In a joint work with Hannah Alpert and Larry Guth, we showed that Freedman's examples are almost as "free" as possible, by bounding the systolic product by the volume raised to the power of $1+\epsilon$. I will give an overview of the systolic freedom phenomenon (including its connection to quantum error correcting codes), and outline our proof, which is based on the Schoen--Yau--Guth--Papasoglu minimal surface method.