Location: MALL
Title: Forcing over choiceless models
Forcing over models of ZF set theory without the axiom of choice has been studied in particular for L(ℝ) in work of Steel, Van Wesep, Woodin and more recently Larson and Zapletal. However, the axiom of choice can fail in much stronger ways than in L(ℝ). For example, in Gitik’s celebrated model all uncountable cardinals are singular. Since virtually all known forcing techniques fail in this situation, it is interesting to understand what forcing does to such models. We develop a toolbox for forcing over arbitrary choiceless models. We further introduce very strong absoluteness principles and show their relation with Gitik’s model. This is joint work with Daisuke Ikegami and in part with W. Hugh Woodin.
Location: MALL
Title: Wetzel's problem and the continuum
In the early 60's, John Wetzel came up with the following question in his PhD thesis on harmonic functions: If $\mathcal{F}$ is a family of entire functions (functions that are holomorphic on the complex plane) which at each point attains at most countably many values, is $\mathcal{F}$ itself necessarily countable? This question makes sense considering the quite restrictive nature of holomorphic functions. Not much thereafter, Erdős could show that a negative answer to Wetzel's Problem is in fact equivalent to the continuum hypothesis. His argument shows that any family of entire functions, that attains at each point less values than elements of that family, must have size continuum. Recently Kumar and Shelah have shown that consistently such a family exists while the continuum has size $\aleph_{\omega_1}$. We answer their main open problem by showing that continuum $\aleph_2$ is possible as well. This is joint work with Thilo Weinert.
Fixpoint operators are tools to reason on recursive programs and infinite data types obtained by induction (e.g. lists, trees) or coinduction (e.g. streams). They were given a categorical treatment with the notion of categories with fixpoints. An important result by Plotkin and Simpson in this area states that provided some conditions on bifree algebras are satisfied, we obtain the existence of a unique uniform fixpoint operator. This theorem allows to recover the well-known examples of the category Cppo (complete pointed partial orders and continuous functions) in domain theory and the relational model in linear logic. In this talk, I will present a categorification of this result where the 2-dimensional framework allows to study the coherences associated to the reductions of λ-calculi with fixpoints i.e. the equations satisfied by the program computations steps.
Location: MALL
Title: Residue field domination in some theories of valued fields
A paraphrase of the Ax-Kochen-Ersov theorem for some theories of valued fields is that the elementary theory is determined by the theory of the value group and the residue field. At the level of types, the intuition is that a type should be controlled by its trace in each of the residue field and value group. In this talk, I will explore some ways in which this intuition can be made precise, and also some limitations to that preliminary intuition. I will try to give lots of examples to keep the discussion concrete.
Location: MALL 1
Title: Two New Inequalities for Cardinal Characteristics of the Continuum
NOTE date and time change: Thu 1pm
Over the last decades the theory of cardinal characteristics of the continuum has emerged as one among several important subfields of set theory. Some of the classical results in it precede the invention of forcing and arguably the aforementioned emergence. Open problems in this field have inspired the invention over ever more versatile constructions of forcing notions and much of the progress has consisted of proving the values of cardinal characteristics not to be ZFC-provably related. A recent outlier has been the celebrated result by Malliaris and Shelah that $\mathfrak{p}$ is equal to $\mathfrak{t}$. I had guessed that there might be more ZFC-provable relations between the hitherto defined characteristics and I am going to talk about what I found up to now. This is to say that I am going to present some ZFC-provable inequalities. In particular I am going to show that the evasion number is at most the subseries number.
These cardinal characteristics have been introduced in work by Blass, Brendle, Brian, and Hamkins and originate from Algebra and Analysis, respectively. The proof interpolates via the pair-splitting number which is due to Minami.
Location: MALL
Title: Big Ramsey degrees for internal colorings
In this talk, I will define what it means for a coloring of substructures of an ultraproduct structure to be "internal", and a notion of finite big Ramsey degree for internal colorings. I will also present a certain Ramsey degree transfer theorem from countable sequences of finite structures to their ultraproducts, assuming AC and some additional mild assumptions. The big Ramsey degree of a finite structure in an ultraproduct can differ markedly from its internal big Ramsey degree, as demonstrated by the example of the class of all finite linear orders, which I will explain.
This is joint work with Dana Bartošová, Mirna Džamonja and Rehana Patel.
Location: MALL 2
Title: Around definable types in valued fields
NOTE time change: 11am
Haskell, Hrushovski, and Macpherson showed that the theory ACVF of algebraically closed valued fields has elimination of imaginaries after adding the so-called "geometric sorts" to the language. The same result holds in $p$-adically closed fields ($p$CF) by work of Hrushovski, Martin, and Rideau. In the case of ACVF, one way to prove this is to encode imaginaries using definable types, and then encode definable types in the geometric sorts. While $p$CF does not have "enough" definable types to encode imaginaries, the encoding of definable types carries over. Surprisingly, the geometric sorts are unnecessary: any definable type in $p$CF has a code in the home sort (the field sort). This fact and its proof have some unexpected applications to definable groups and definable topological spaces in $p$CF. For example, certain quotient groups are definable rather than interpretable, and there is a unified notion of "definable compactness" for definable topological spaces. Parts of this talk are joint work with Pablo Andújar Guerrero.
Distality was introduced by Pierre Simon in 2012 as a property of NIP theories which captures "pure instability". It encompasses weakly o-minimal structures, the field of $p$-adics, and Presburger Arithmetic. We present the definitions of distality and distal cell decomposition. We describe recent applications of distal cell decomposition, and discuss an open problem.
Location: MALL
Title: Cardinal characteristics of the continuum
When speaking about subsets of the real number line, we may define some idea of smallness. When we do so, it is often the case that a countable union of small sets is not enough to cover the whole real line. Any list of countably many real numbers can be extended, so the shortest inextensible list of real numbers must be uncountable. Any countable collection of nowhere dense sets also does not cover the real line, so we need uncountably many of those to do that. The same holds for Lebesgue measure zero sets, strongly null sets, nowhere everywhere sets, and more. If aleph_0 isn't good enough then how many do we need?
