Logic

In classical set theory, Gödel's constructible universe 𝐿 enjoys strong absoluteness properties and remains unchanged in forcing extensions. However, Heyting-valued forcings portray a very different picture by introducing new non-classical ordinals (i.e. ordinals not linearly ordered by the membership relation), and thus new elements in 𝐿, in intuitionistic extensions that violate the law of excluded middle.

The method of incomparable codings is a family of approaches I developed in my PhD to use such ordinals to control (especially, enlarge) 𝐿. In arXiv:2601.23070 [math.LO], I proved the following theorem: for any set z in a ZFC universe, there is a Heyting-valued extension where its powerset 𝒫(ž) ∈ 𝐿. In this talk, we will provide a sneak peek of this mechanism by building the forcing extension needed for 𝒫(ω) ∈ 𝐿; if time allows, we will briefly talk about technicalities and new developments on how this extends to sets larger than ω.

Henselian valued fields have a central place in model theory: they provide natural examples of tame theories in the sense of classification theory, while also offering a rich setting in which to study complexity phenomena in model theory. In this talk, I will survey some fundamental model-theoretic results on Henselian valued fields and present new results from joint work with Paul Wang, in which we generalize results from the algebraically closed valued field setting to the broader Henselian context.

NOTES: Unusual time 16:00. Part of workshop on "Tame Geometry and Combinatorics".
Large fields are an interesting class of fields first introduced by Pop in the 90's for Galois-theoretic reasons. They have subsequently been studied for a variety of reasons. As logicians we are interested in large fields because essentially all known logically tame fields are large. I will discuss recent work on the model theory of large fields, joint with Will Johnson, Chieu-Minh Tran, and Jinhe Ye. Only minimal background in algebra will be assumed.

Ultraproducts of rings and their modules have been appearing in various places in representation theory and the theory of tensor categories.  We consider how this is reflected in the model theory of modules over the rings and, in particular, we describe the effect on the lattices of pp formulas, definable subcategories, Ziegler spectra and the associated abelian categories of pp-imaginaries.  We also look at what happens when we enrich the picture with a monoidal = tensor product structure.

The theory of ZFC without powerset is interpretable in the system $Z_2$ of second order arithmetic. The interpretation is usually done in two steps. The first is to interpret an intermediate theory of sets by considering well founded trees modulo an appropriate equivalence relation. The second is by defining the constructible universe $L$ in the intermediate system.

I will go over some of the ways such interpretations are suited in more general systems that may have uncountable sets and restricted separation. I will also talk about how it turns out that the weakest principle needed to carry out the tree interpretation can be characterized in terms of clopen games and transfinite recursion. This is a joint work with Emanuele Frittaion.

The Wilson conjecture asks whether any locally nilpotent omega-categorical $p$-groups are nilpotent. In this talk, I will present solutions to a few cases (e.g. 4-Engel 5-groups, or groups of exponent 4), which use methods at the intersection of group theory, model theory and computer algebra.

In this talk, I discuss some recent advances in the interchange of ideas between logic and the classification of C*-algebras, which are a specific subclass of operator algebras. Operator algebras are certain collections of bounded operators on Hilbert spaces. They give a strong foundation for understanding quantum mechanics, as well as non-commutative flavors of geometry, topology, and probability theory. A huge theme within operator algebras (and indeed in all mathematics) is that of classification, which asks the broad question of how we can tell objects apart or conclude they are the same.

In this talk, I will discuss a game-theoretic variant of the unital C*-classification theorem: we show that there is a transfer of strategies between Ehrenfeucht-Fraïssé games on classifiable C*-algebras and their invariants. The proof techniques are interesting in their own right: they involve an abstract descriptive set theoretic classification result, as well as a first-order language for functors.

This is based on joint work with Michał Szachniewicz and Mira Tartarotti.

The global Zarankiewicz's problem for hypergraphs asks for an upper bound on the number of edges of a hypergraph, whose edge relation is induced by a fixed hypergraph $E$ that has no sub-hypergraphs of a given size. Basit-Chernikov-Starchenko-Tao-Tran (2021) obtained linear Zarankiewicz bounds in the case of a semilinear $E$, namely $E$ definable in a linear o-minimal structure. We extend this theorem to a broader range of "linear-like" structures, in o-minimal, Presburger arithmetic and stability theoretic settings. Some of the methods involved include (a) a reduction of the problem to the case of arbitrary subgroups $E$ of powers of groups, and (b) an abstract version of Zarankiewicz's problem in the saturated setting.
Joint work with Aris Papadopoulos.

Thatcher and Wright showed that a property of trees of bounded degree is MSO-definable if and only if it is recognizable by a tree-automaton. In this talk we explore the question of when MSO-definability of a property of graphs is equivalent to the existence of a tree automata which, given a suitable expression encoding the input graph, recognizes the property. In this talk, I will survey the state of the art of the "definability equals recognizability" problem. For proving "definability equals recognizability" results the key step is to transduce a suitable tree-like decomposition of the input graph. I will present a new MSO-transduction which forms the core for transducing a particular type of graph decompositions.