Keisler randomization and higher order VC-dimension
Abstract: A randomization of a first-order structure M, introduced by Keisler, is a structure M^R in continuous logic whose elements are the "random" elements of M. One can think of it as a continuous structure whose types correspond to probability measures on the space of types of the original structure. Randomization preserves certain model-theoretic tameness properties, e.g. stability and NIP. The latter was demonstrated by Ben Yaacov via developing aspects of the VC-theory (Vapnik-Chervonenkis) in the continuous setting, connected to earlier work of Talagrand and others. A more general hierarchy of n-dependent theories was introduced by Shelah, with the case n=1 corresponding to NIP: a theory is n-dependent if the edge relation of an infinite generic (n+1)-hypergraph is not definable. We will discuss n-dependence in continuous logic and demonstrate that n-dependence is also preserved by Keisler randomization: the main point is that the average of a family of uniformly n-dependent functions is n-dependent. Our proof relies on structural Ramsey theory and multidimensional de Finetti-type results (and provides in particular a new proof in the NIP case). Joint work with Henry Towsner.
Tuesday March 29, 2022 at 4:00 PM in 636 SEO