Bounding the Vapnik-Chervonenkis density of definable families

Abstract: The notion of Vapnik-Chervonenkis density of a set system is a concept that arose more-or-less simultaneously in probability theory, in model theory, and in combinatorics. After giving some basic definitions and history, I will survey some old and new results on obtaining tight quantitative bounds on the VC-density of definable families in various structures and explain a common topological underpinning in the proofs of these results. The ``new result'' is a tight bound on the VC-density of definable families over algebraically closed valued fields and is joint work with Deepam Patel. The proof of the new result relies heavily in part on model-theoretic results of Hrushovski and Loeser on stable dominated types.

References.

1. “VC density of definable families over valued fields” Saugata Basu and Deepam Patel, to appear in the J. Eur. Math. Soc.

2. ''Vapnik-Chervonenkis density:from uniform convergence to Berkovich analytification and stably dominated types’’, blog article at saugatabasumath.wordpress.com.

algebraic geometryalgebraic topologydifferential geometryK-theory and homologylogicoptimization and controlprobability

Audience: researchers in the topic

Comments: Zoom link: sissa-it.zoom.us/j/98926079269 Meeting ID: 989 2607 9269

Real Geometry