1. The Grothendieck property in C_p-theory

An important theorem of Grothendieck used in functional analysis essentially says that countably compact subspaces of certain function spaces are compact. J. Iovino has applied this to stability and definability in Model Theory. Arhangel’skii vastly generalized

Grothendieck’s theorem but left open some fundamental problems. We show they are undecidable. Clovis Hamel and I applied the work of Iovino and Arhangel’skii to Gowers’ problem on the definability of pathological Banach spaces. He will speak on this later this semester.

2. K-sigma-projective sets - a new topological generalization of descriptive set theory

I became interested in sigma-projective sets of reals when they appeared in my study with C. Eagle, C. Hamel, and S. Muller of the number of non-isomorphic countable models of countable theories in second order logic. I had been studying with I. Ongay-Valverde various possibilities for generalizing descriptive set theory beyond Polish spaces, following the path of Frolik, Jayne & Rogers, etc., but assuming determinacy axioms consistent with ZFC. Our conclusion is that the K-sigma-projective spaces - a natural generalization of the K-analytic spaces - are a useful venue to work in.

3. Counting the number of equivalence classes of sigma-projective equivalence relations and applications to second order logic

This is ongoing work with Eagle, Hamel, Muller, and now J. Zhang. We consider a second order version of Morley’s theorem on the number of countable models. This will be the first of several talks on this subject, Connections to work of Foreman, Magidor, Shelah, and Woodin on determinacy and large cardinals will appear.