Gaming Models by Buildings

Abstract: In continuous model theory, as in the classical setting, if one has an appropriately sized unstable structure A in a countable language, then depending on the truth of CH, there's either a unique or 2^c many nonisomorphic ultrapowers of A as we vary the choice of ultrafilter on ω. A similar statement may be made in regards to ultraproducts and sequences of structures that exhibit an order property.

In a partial answer to a question of Gromov, Kramer et al. showed that there is a finitely presented group such that, depending on the truth of CH, this group has either a unique or 2^c many asymptotic cones up to homeomorphism. Asymptotic cones of metric spaces are realized as particular metric ultraproducts. The Kramer et al. paper does not formalize the obvious model theoretic connection, but does comment on the combinatorial-geometric structure of the asymptotic cones, which was known to Thornton (and independently to Kramer and Tent) and is a certain kind of building.

In this talk, we'll give a brief overview of work done by Luther to formalize this model theoretic connection. Special attention will be given to Ehrenfeucht-Fraïssé games and how the building structure can give us additional tools to develop a possible winning strategy for Player II in games between (what are potentially) non-homeomorphic asymptotic cones of certain symmetric spaces.

logic

Audience: researchers in the topic

Online logic seminar

Series comments: Description: Seminar on all areas of logic