Logic Workshop

Video

​

Video

​

Video

​

Video

​

Over half a century ago Hyman Bass proved that all flat left modules are projective precisely when the underlying ring satisfies the descending chain condition on right principal ideals. He called such rings left perfect. Gena Puninski noticed that this can be given a model theoretic proof. Every infinite descending chain of principal right ideals gives rise to a descending chain of (pp) formulas which, in turn, gives rise to a direct limit of finitely generated projective modules that is not projective. Such a module is flat and not projective, and called a Bass module.

I demonstrate how this construction is elementary model theory and at the same time generalizes to other classes of (pp) formulas and modules, which, among other things, yields a new proof of the late Daniel Simson’s result that all left modules are Mittag-Leffler iff the ring is left pure-semisimple (which, to model theorists, means that all left modules are totally transcendental).

I will emphasize the model theoretic ideas and explain the connection with the algebraic concepts. This is part of ongoing work with Anand Pillay.

Video

​

Video

​

I will discuss a theory developed in my joint paper with Junguk Lee and Slavko Moconja. One can view it as a variant of Kechris, Pestov, and Todorčević theory in the context of (complete first order) theories. I will discuss several 'definable' Ramsey-theoretic properties of first order theories and their dynamical characterizations. The point is that all the Ramsey-theoretic properties that we introduce involve 'definable colorings' and the dynamical characterizations are 'dynamical properties of the theories', i.e. they are expressed in terms of the action of the group of automorphisms of a monster (i.e. sufficiently saturated and homogeneous) model of the theory in question on the appropriate space of types. One of the basic results says that a theory has the definable Ramsey property iff it is extremely amenable (as defined by Hrushovski, Pillay and myself). But there are various other results, some of which are essentially new and may be surprising in comparison with the Kechris, Pestov, Todorčević theory. One of the motivations to study those properties was to understand better the so-called Ellis group of a theory (which was used by Pillay, Rzepecki, and myself to explain the nature of the Lascar Galois groups of first order theories and spaces of strong types, and led E. Hrushovski to some original development with striking applications to approximate subgroups). Using our dynamical characterizations, we obtain several criteria for profiniteness and for triviality of this Ellis group, with many examples where they apply. I will try to discuss it during my talk. If time permits, I may very briefly mention an abstract generalization of the above considerations and results, which also applies both to the context of definable groups as well as to the classical context of Kechris Pestov, Todorčević theory, leading to some new notions, results, and questions.

Video

​

One could make the claim that category theory is as foundational as set-theory or model-theory. So, we should be able to transfer from one perspective to the other. In this talk, I will consider one aspect of this meta-equivalence, by introducing a theory in a very simple, one-sorted(!) language, whose models are all categories admitting a terminal object (many categories do). Many categorical constructions then turn out to be first-order. But something even more strange happens: standard categories (like the category of Abelian groups) become actually universal models! I'll explain this apparent contradiction.

In the second part of the talk, I will concentrate on one particularly interesting category: that of compact Hausdorff spaces. I will show that we can recover the natural numbers $N$ and the reals $R$, or rather, (the isomorphism classes of) their compactifications $\bar N$ and $\bar R$, by parameter-free definitions, including their order relation, addition and multiplication. Moreover, in any category that is elementary equivalent to the category of compact Hausdorff spaces, the resulting objects are then a model of PA and a real closed field respectively. Full disclosure: while I have a complete proof for the first assertion, the second is still conjectural.

Apart from some basic model-theory, category theory and topology, everything else will be explained in the talk and so it should be accessible to many.

Video

​

The Heisenberg group G(F) of a field F is the group of upper triangular matrices in GL_3(F), with 1's along the diagonal and 0's below it. This group is obviously interpretable (indeed definable) in the field F. Mal'cev showed that one can recover F from G(F), and indeed that there is an interpretation of F in G(F) using two parameters. Any two noncommuting elements of G(F) can serve as the parameters, but Mal'cev was unable to produce an interpretation without parameters.

After introducing the notions of a computable functor and an effective interpretation, we will present joint work showing that there is an effective interpretation of each countable field in its Heisenberg group, without parameters, uniformly in F. (That is, the same formulas give the interpretation, no matter which field F we consider.) Moreover, from the effective interpretation we will then extract a traditional interpretation without parameters, in the usual model-theoretic sense. Finally we will note that, whereas Mal'cev's result actually gives a definition of F in G(F), there is no parameter-free definition of F there.

This work is joint with Rachael Alvir, Wesley Calvert, Grant Goodman, Valentina Harizanov, Julia Knight, Andrey Morozov, Alexandra Soskova, and Rose Weisshaar.

Video

​

Classically, an object is ordinal definable if it is the unique one satisfying a formula with ordinal parameters. Generalizing this concept, given a cardinal $\kappa$, I call an object $\lt\kappa$-blurrily ordinal definable if it belongs to an ordinal definable set with fewer than $\kappa$ elements (called a $\lt\kappa$-blurry definition). By considering the hereditary versions of this notion, one arrives at a hierarchy of inner models, indexed by cardinals $\kappa$: the collection of all hereditarily $\lt\kappa$-blurrily ordinal definable sets, which I call $\lt\kappa$-HOD. In a ZFC-model, this hierarchy spans the entire spectrum from HOD to V.

The special cases $\kappa=\omega$ and $\kappa=\omega_1$ have been previously considered, but no systematic study of the general setting has been carried out, it seems. One main aspect of the analysis is the notion of a leap, that is, a cardinal at which a new object becomes hereditarily blurrily definable.

In this talk, I will focus on the ZFC-provable structural properties of the blurry HOD hierarchy, which turn out to be surprisingly plentiful. So for the most part, the talk will be forcing-free. Time permitting, I may hint at the result of the equiconsistency between the least leap being the successor of a singular strong limit cardinal and the existence of a measurable cardinal, for which, admittedly, forcing is used in one direction.

Video

​