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.
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: 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.
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?
Location: MALL
Title: Jordan permutation groups and limits of treelike structures
A transitive permutation group $G$ on a set $X$ is a Jordan group if there is a subset $A$ of $X$ (a 'Jordan set') with $|A|>1$ such that the subgroup of $G$ fixing the complement of $A$ is transitive on $A$ (+ a non-degeneracy condition that if $G$ is $k$-transitive on $X$ then $|X \setminus A|\geq k$.) So if $X$ carries some first-order structure, this is a bit like saying all elements of $A$ realise the same type over $X \setminus A$. Work of Adeleke, myself and Neumann in the 1990s gave a kind of classification of Jordan groups which are 'primitive', i.e. preserve no proper non-trivial equivalence relation on $X$. Many key examples can be seen as Fraïssé limits.
I will discuss examples, and also sketch recent work in Asma Almazaydeh's thesis (and subsequent work with her) on certain mysterious $\omega$-categorical structures which are limits of treelike structures. This relates to earlier work with Meenaxi Bhattacharjee, and to a recent preprint of David Bradley-Williams and John Truss.
Location: MALL 1
Title: M.e.c. limits of homogeneous structures
In this talk, I aim to give a glimpse into the study of m.e.c. limits, especially those of homogeneous structures. This is centred upon a conjecture of MacPherson, Steinhorn, Anscombe, and Wolf, that a homogeneous structure has a m.e.c. limit if and only if it is stable. I’ll give some background, talk through some examples and outline a novel method of tackling this question for certain unstable homogeneous structures.
A. Let $T$ be a small theory, $p\in S(T)$ be a non-isolated type, $T_1:= T\cup p(\bar c)$ be a non-essential expansion. There are examples of small theories 1-2 such that:
$I(T, \omega)< \omega$, $I(T_1, \omega) = \omega$. (R. Woodrow, M.Peretyat'kin).
$I(T,\omega)= \omega$, $I(T_1, \omega)< \omega$ (B. Omarov)
Question. Is there a small theory such that $I(T, \omega) =2^{\omega}$, $I(T,\omega)\leq \omega$?
We discuss the usage of constant expansion on research of Vaught Conjecture.
B. Let $\mathfrak M$ be a model of a theory $T$. A finite diagram (S. Shelah) of $\mathfrak M$ is the collection of all $\emptyset$-definable complete types that are realized in $\mathfrak M$:
A dowry of a set $B\subseteq M$ is the following collection:
$$\mathcal D(B)=\{p\in S(T)\ | \text{ there exist }\mathfrak M\models T \text{ and } \bar b\in B \text{ such that } \bar b\in p(\mathfrak M)\}.$$
Conjecture. If $|\{ \mathcal D(\mathfrak M) \mid \mathfrak M\models T\}| >\omega$, then $|\{ \mathcal D(\mathfrak M) \mid \mathfrak M\models T\}| =2^{\omega}$, and, consequently, $I(T,\omega)= 2^{\omega}$.
Question. Let $P:=\{p_n\in S(T) \mid n<\omega\}$, $Q:=\{q_n\in S(T) \mid n <\omega\}$ be two families of non-isolated types such that for every $n<\omega$ there exists $\mathfrak M_n$ is realized first $n$ types from $P$ and is omitted first $n$ types from $Q$. Do there exist a countable model $\mathfrak M$ of $T$, such that it realizes each type from $P$ and omits each type from $Q$?
The criterion for the existence of such a theory will be shown.
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.
