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## Upcoming talks and events:

# Regular Jónsson cardinals

For an infinite cardinal $\lambda$, we say that $\lambda$ is Jónsson if it satisfies the square bracket partition property $\lambda\to[\lambda]^{<\omega}_\lambda$. This means that, for every coloring $F\colon [\lambda]^{<\omega}\to\lambda$, there is some set $H\in [\lambda]^\lambda$ with the property that $\mathrm{ran}(F\upharpoonright [H]^{<\omega})\subsetneq\lambda$. It is rather well known that the consistency strength of "there is a Jónsson cardinal" lies above "$0^\sharp$ exists", but below the existence of a measurable. However, the question of what sorts of cardinals can be Jónsson has turned out to be rather difficult. The goal of this talk is to sketch a simplified proof of the result (due to Shelah) that if $\lambda$ is the least regular Jónsson cardinal, then $\lambda$ must be $\lambda\times\omega$-Mahlo. The advantage of the proof presented here is that the machinery employed can be easily generalized, and time permitting I would like to discuss how one might attempt to prove that such a cardinals must be greatly Mahlo.

# Foundations of cologic

The existence of a robust categorical dual to first-order logic is hinted at in (at least) four independent bodies of work: (1) Projective Fraïssé theory [Solecki & coauthors, Panagiotopolous]. (2) The cologic of profinite groups (e.g. Galois groups), which plays an important role in the model theory of PAC fields [Cherlin – van den Dries – Macintyre, Chatzidakis]. (3) Ultracoproducts and coelementary classes of compact Hausdorff spaces [Bankston]. (4) Universal coalgebras and coalgebraic logic [Rutten, Kurz – Rosicky, Moss, others]. In this talk, I will propose a natural syntax and semantics for such a dual “cologic”, in which “coformulas” express properties of partitions of “costructures”, dually to the way in which formulas express properties of tuples from structures. I will show how the basic theorems and constructions of first-order logic (completeness, compactness, ultraproducts, Henkin constructions, Löwenheim-Skolem, etc.) can be dualized, and I will discuss some possible extensions of the framework.

# Weihrauch reducibility and Ramsey theorems

Weihrauch reducibility is a common tool in computable analysis for understanding and comparing the computational content of theorems. In recent years, variations of Weihrauch reducibility have been used to study Ramsey type theorems in the context of reverse mathematics where they give a finer analysis than implications in **RCA _{0}** and they allow comparisons of computably true principles. In this talk, we will give examples of recent results and techniques in this area.

# Recursive Reducts of PA II

I will continue discussing work of Schmerl addressing the question of under what conditions a given reduct of PA has the property that for any non-standard model M of PA the restriction of M to the reduct must be non-computable.

# Coherent Systems of Finite Support Iterations

The method of matrix iterations was introduced by Blass and Shelah in their study of the dominating and the ultrafilter numbers. Since its appearance, the method has undergone significant development and applied to the study of many other cardinal characteristics of the continuum, including those associated to measure and category.

Recently, we were able to extend the technique of matrix iterations to a “third dimension” and so, evaluate the almost disjointness number in models where previously its value was not known. In addition, we obtain new constellations of the Cichon diagram (with up to seven distinct values). This is a joint work with Friedman, Mejia and Montoya.

# Bounding, splitting and almost disjointness can be quite different

The bounding, splitting and almost disjoint families are some of the well studied infinitary combinatorial objects on the real line. Their study has prompted the development of many interesting forcing techniques. Among those are the method of creature forcing, as well as Shelah’s template iteration techniques.

In this talk, we will discuss some recent developments of Shelah’s template iteration methods, leading to models in which the bounding, the splitting and the almost disjointness numbers can be quite arbitrary. We will conclude with a brief discussion of open problems.