Tuesdays, 16:00-18:00
Organized by Rahman Mohammadpour and Grigor Sargsyan

Updates:

  • This is an in-person seminar in room 106 at IMPAN in Warsaw


    • Tuesday, 10-02-2026

    Grigor Sargsyan

    • Title: An Introduction to Nairian Models
    • Abstract: No abstract

    Tuesday, 24-02-2026

    Lukas Koschat (TU Wien)

    • Title: Full models of \(\textsf{LSA}\)
    • Abstract: The main topic of the talk is the following recent result: assuming large cardinals in the region of three supercompact cardinals and letting \(\kappa\) be the least supercompact cardinal, in each forcing extension where \(\kappa\) is countable, the minimal inner model containing all universally Baire sets, \(L(uB)\), contains an full model of \(\textsf{LSA}\) (Largest Suslin Axiom). The Largest Suslin Axiom is a determinacy axiom of high consistency strength, and hence the result implies that in the given context \(L(uB)\) satisfies very strong determinacy axioms. The result is part of joint work with Sandra Müller and Grigor Sargsyan. The talk is aimed at a general set theory audience and thus a major part of the talk will be dedicated to introducing the relevant concepts, and explainingtheir importance to the study of consistency strength, as well as their connections to recent developments in the endeavour of forcing failures of square principles over models of determinacy.

    Tuesday, 10-03-2026

    Corentin Lagadec

    • Title: The strong Chang's conjecture in \(\mathbb{P}_{\textsf{max}}\) models
    • Abstract: I will present a proof that an improved version of the strong Chang's conjecture can be forced with \(\mathbb{P}_{\textsf{max}}\) over a model of determinacy.

    Tuesday, 17-03-2026

    Nam Trang (University of North Texas)

    • Title: Coherent Sequences Constructed over Models of Determinacy
    • Abstract: We construct various types of coherent sequences over models of determinacy. We use these sequences and core model induction arguments to calibrate the consistency strength of theories extending \(\textrm{MM}(\mathfrak{c})\) and \(\textrm{CH}\) + there is an \(\omega_1\)-dense ideal on \(\omega_1\) such as (1) \(\textrm{CH}\) + there is an \(\omega_1\)-dense ideal on \(\omega_1 + \neg \square_{\omega_1}\). (2) \(\textrm{CH}\) + there is an \(\omega_1\)-dense ideal on \(\omega_1 + \neg \square(\omega_2)\). This is joint work with M. Zeman.

    Tuesday, 24-03-2026. (!!! 16:45)

    Jouko Vaananen (University of Helsinki)

    • Title: New inner models from second order logics
    • Abstract: I define a new inner model based on replacing first order logic in Godel’s definition of the constructible hierarchy by the fragment of second order logic in which second order variables range over countable subsets of the domain. I compare this inner model to the previously studied inner models from extended logics, mainly \(C^*\) and \(C(aa)\). I will discuss large cardinals in this model and forcing absoluteness of the its theory. There are many open problems. Time permitting, I will also discuss the inner model \(\textrm{HOD}_1\), obtained in a similar way from the existential fragment of second order logic. This is joint work with Menachem Magidor.

    Tuesday, 31-03-2026

    Otto Rajala

    • Title: \(\textrm{GCH}\) in \(C(aa^+)\)
    • Abstract: I will talk about the inner model \(C(aa^+)\) obtained from the \(aa^{+}\)-quantifier which is a variant of the \(aa\)-quantifier. Hence, \(C(aa^+)\) can be viewed as a variant of the inner model \(C(aa)\) introduced by Kennedy-Magidor-Väänänen. The \(aa^+\)-quantifier says that there is a club of countable subsets of the given structure such that the subformula holds in the next admissible set built from the structure and any one of the subsets. In particular, I will present the proof of \(\textrm{GCH}\) in the model \(C(aa^+)\).

    Tuesday, 07-04-2026 (!!! 17:00)

    Iljas Farah (York University)

    • Title: Conjugacy of trivial autohomeomorphisms of \(\beta N\setminus N\).
    • Abstract: An autohomeomorphism of the Čech--Stone remainder \(\beta N\setminus N\) is called trivial if it has a continuous extension to a map from \(\beta N\) into itself. Such map is determined by an almost permutation, which is a bijection between cofinite subsets of \(N\). By results of W. Rudin and S. Shelah, the question whether nontrivial autohomeomorphisms of \(\beta N\setminus N\) exist is independent from \(\textrm{ZFC}\). We will be considering the so-called rotary autohomeomorphisms. An autohomeomorphism is called rotary if it corresponds to a permutation of \(N\) all of whose cycles are finite. If all autohomeomorphisms are trivial, then the problem of their conjugacy is also trivial (in the usual sense of the word). However the Continuum Hypothesis makes the conjugacy relation nontrivial. While our results are somewhat incomplete, they suffice to decide whether for example the rotary autohomeomorphisms whose cycles have lengths \(2^{2n}\), for \(n\in N\), and \(2^{2n+1}\), for \(n\in N\), are conjugate. This is a joint work with Will Brian.

    Tuesday, 14-04-2026

    Piotr Borodulin-Nadzieja (University of Wrocław)

    • Title: On generalizations of ultrafilters
    • Abstract: The notion of an ultrafilter can be naturally generalized in different directions, e.g. one can see ultrafilters as particular instances of measures or of Boolean homomorphisms. I will overview some results concerning these generalizations

    Tuesday, 21-04-2026

    Sebastiano Thei

    • Title: Another proof of Kunen Inconsistency
    • Abstract: In our previous seminar, we learned many proofs of Kunen Inconsistency. We also proved that \(\textrm{HOD}\) can be close to \(V\), in the presence of an extendible cardinal. In this talk, we address two related questions:
      1. Does the HOD hypothesis imply that \(\textrm{HOD}\) has the cover property above the first strongly compact cardinal?
      2. Suppose \(j\) is an elementary embedding from an inner model \(M\) into \(V\). We know that \(M\) cannot be \(V\), but can \(M\) be cardinal correct?
      The first question is open, while the second one has a negative answer (Goldberg-T.). Both questions are addressed through the lens of PCF theory.

    Tuesday, 28-04-2026

    No talks

    Tuesday, 05-05-2026

    Rahman Mohammadpour

    • Title: Rado's Conjecture on successive cardinals
    • Abstract: An intersection graph is determined by a linearly ordered set, where the vertices are nonempty intervals and edges are pairs of intervals with nonempty intersections. Rado's Conjecture states that if G is an intersection graph, then the chromatic number of G is countable if and only if the chromatic number of every subgraph of G of size the first uncountable cardinal is countable. Rado's Conjecture was shown to be consistent and has been extensively studied by S. Todorcevic. It is straightforward and plausible to generalize Rado's Conjecture to higher cardinals. In his Mostowski lecture in Wrocław, 2024, Todorčević asked whether Rado's Conjecture can hold at two successive cardinals. I shall talk about Rado's Conjecture and its variants. Specifically, I shall report on joint work with M. Eskew where we show that it is consistent that Rado's Conjecture holds at all regular cardinals simultaneously.

    Tuesday, 12-05-2026

    Ronnie Chen (University of Warsaw)

    • Title: Borel combinatorics and countable model theory
    • Abstract: We consider the global structure of locally countable Borel combinatorial problems, by which we mean the construction of Borel first-order structures on the reals (or any standard Borel space) that live on non-interacting countable subsets. Examples include finding Borel n-colorings, perfect matchings, or spanning trees in a given locally countable Borel graph; or showing that such a graph may be induced by a Borel action of a given countable group. Experience has shown that such problems are usually solved by arguing as if working with a single countable structure, but with only "canonical" or "definable" operations allowed. We formulate a result making this precise: the class of all locally countable Borel combinatorial problems is equivalent (as a category) to the class of all \(L_{\omega_1\omega}\) theories that interpret a certain distinguished theory called \(T_{LN} \sqcup T_{sep}\). This talk is based on joint works with Alexander Kechris and Rishi Banerjee.