Titles and abstracts of the talks
Lena Wallner, The Consistency of the Axiom of Determinacy from Large Cardinals.
For a set A of reals we let Gω(A) be the game where two players alternately play natural numbers for ω-many times. By doing so they form a real since we identify the reals with ω^ω. We say that player 1 wins if this reals is an element of A. Otherwise payer 2 wins. We say that Gω(A) is determined whenever one of the players has a winning strategy. The axiom of determinacy (AD) states that Gω(A) is determined for every set of reals A. AD came up in the early 60s when it was realized that the axiom leads to regularity properties for sets of reals. About 25 years later Woodin proved that AD is consistent. He showed that AD holds in L(R) from ω-many Woodin cardinals and a measurable cardinal above. This talk will be about a proof of this statement. It includes a simpler version of the Derived Model Theorem which relies on the determinacy of universally Baire sets. The second method is genericity iterations by Neeman which we are going to use iteratively. These tools are fundamental in modern research on inner model theory and determinacy.
William Chan, Decomposing the Power Set of ω2 into ω1 Many Pieces. slides
This talk will introduce the notion of regularity and cofinality of (possibly nonwellorderable) sets under determinacy. Since ω2 is a weak partition cardinal which is not a strong partition cardinal, not much is known concerning the structure of the cardinality of the power set of the second uncountable cardinal and its cofinality. This talk will show a very weak form of regularity for the power set of ω2: If the power set of ω2 is partition into ω1 many disjoint pieces, then there is at least one piece which does not inject into the collection of bounded subsets of ω2. This is joint work with Stephen Jackson and Nam Trang.
Ido Feldman, Sums of Triples in Abelian Groups. slides
In 1974, Hindman proved that considering the semigroup (ℕ, +), for any partition ℕ = S0 ⊎ S1, there exists an infinite X ⊆ ℕ such that the set of its finite sums, is monochromatic, that is, contained in one of the cells.
In contrast, in 2016 Komjáth showed that, for the group (ℝ, +) there exists a partition ℝ = S0 ⊎ S1 such that, whenever X ⊆ ℝ is uncountable, not only is the set of finite sums not monochromatic, but already the set FS2(X) := {x + y | {x, y} ∈ [X]²} is not monochromatic.
These results motivate a general investigation of additive Ramsey theory in the spirit of the classical partition calculus, and which in fact for some cases are a strengthening of the classical partition calculus. This talk will concentrate on two reductions one can apply to the additive problem in order to get the those partition relations from classical Ramsey theory. In particular, we will try to answer the following:
under what conditions can one prove that for every Abelian group G of size ℵ2, constructing a coloring c : G → ℤ such that, for every uncountable X ⊆ G and every integer k, there exist three distinct elements x, y, z of X such that c(x, y, z) = k.
This is a joint work with Assaf Rinot. For further reading the article is available here: https://londmathsoc.onlinelibrary.wiley.com/doi/10.1112/mtk.12200
Adam Kwela, Katětov Order on Ideals and its Applications. slides
If I and J are ideals on ω then we say that I is below J in the Katětov order (I ≤K J) if there is f : ω → ω such that f⁻¹[A] ∈ J for all A ∈ I.
During the talk I will present several applications of ≤K in studies of set-theoretic and topological objects such as bounding numbers, ultrafilters and sequentially compact spaces. For instance, I will characterize (under CH) ideals I such that the class of I-ultrafilters coincides with P-points (following Baumgartner, an ultrafilter U on ω is an I-ultrafilter if for every function f : ω → ω there is A ∈ U with f[A] ∈ I).
Moreover, I will discuss connections of Katětov order with descriptive properties of ideals, answering questions of M. Hrušák and of G. Debs and J. Saint Raymond.
This work is partially joint with Rafał Filipów and Krzysztof Kowitz.
Dominik Adolf, Diamond Scales in Inner Models.
Lambie-Hanson and Rinot ("Knaster and friends III: subadditive colorings") utilized the property $\square\diamond(\vec{\lambda})$ to establish combinatorial principles on successors of singular cardinals. In this talk we will show that the weaker property $\diamond(\vec{\lambda})$ holds at most places in canonical inner models. To do so we will have to introduce the concept of the "diamond scale". Time permitting we will discuss steps to show that the stronger property holds as well, introducing the associated "diamond-square scale".
Bartosz Wcisło, A new construction of models with failures of DC. slides
Recently, Friedman, Gitman, and Kanovei, described a construction of a model whose real numbers satisfied countable choice, but in which there is a Π¹₂ -definable counterexample to the scheme of Dependent Choice, thus showing that the Axiom of Choice as a scheme in second-order arithmetic does not imply the scheme of Dependent Choice. In our recent work (in progress), joint with Sandra Müller, we extend their technique to construct models with definable failures of Dependent Choice containing large cardinals.