Title: Stability of homomorphisms, coverings and cocycles

Abstract:

A property is stable if objects that almost satisfy the property are close to objects that actually satisfy the property. In this talk we describe three stability problems:
1. Homomorphism stability: Are almost homomorphisms close to actual homomorphisms? 
2. Covering stability: Are almost coverings of a cell complex close to genuine coverings of it?
3. Cocycle stability: Are cochains with small coboundaries close to cocycles?
We then prove these problems are equivalent.

In the rest of the talk, we will:
1. Calculate the cocycle stability rate of some cell complexes. 
2. Apply the aforementioned equivalence to the study of cosystolic expansion of 2-dimensional simplicial complexes. 
3. Discuss the stability rates of random 2-dimensional simplicial complexes in the Linial--Meshulam model. 
4. Provide several open problems with implications on the study of sofic groups and the Aldous--Lyons conjecture. 

This talk is based on ongoing projects with Alex Lubotzky. 

 

The talk can be watched live via Panopto, and will be available later using the same address: https://huji.cloud.panopto.eu/Panopto/Pages/Viewer.aspx?id=4d670252-c2b3-4269-af6c-b06d008d7e01