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.
Sometimes more abstract concepts in general topology are considered as having little relation with areas outside of topology. In this talk we will explore a beautiful construction that unexpectedly links the notion of n-Hausdorffness and a special topology in the dynamical systems setting.
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 1
Title: Elimination of imaginaries in ordered abelian groups of bounded regular rank
NOTE time change: 4pm
In this talk I will present some results about elimination of imaginaries in pure ordered abelian groups. For the class of ordered abelian groups with bounded regular rank (equivalently with finite spines) we obtain weak elimination of imaginaries once we add sorts for the quotient groups $\Gamma/ \Delta$ for each definable convex subgroup $\Delta$, and sorts for the quotient groups $\Gamma/(\Delta+ \ell\Gamma)$ where $\Delta$ is a definable convex subgroup and $\ell \in \mathbb{N}_{\geq 2}$. We refer to these sorts as the quotient sorts. For the dp-minimal case we obtain a complete elimination of imaginaries if we also add constants to distinguish the cosets of $\Delta+\ell\Gamma$ in $\Gamma$, where $\Delta$ is a definable convex subgroup and $\ell \in \mathbb{N}_{\geq 2}$.
Location: MALL
Title: Adelic Geometry via Geometric Logic
NOTE time change: 3pm
On a syntactic level, the peculiarity of geometric logic can be seen from the choice of logical connectives used (in particular, we allow for infinitary disjunctions but do not allow negation), but there are deep ramifications of this seemingly innocuous move. One, geometric logic is incomplete if we restrict ourselves to set-based models, but is complete if we also consider models in all toposes (i.e. not just $\mathrm{Set}$) — as such, geometric logic can be viewed as an attempt to pull our mathematics away from a fixed set theory. Two, there is an intrinsic continuity to geometric logic, which is furnished by the definition of the classifying topos. Indeed, since every Grothendieck topos is a classifying topos of some geometric theory, this provides yet another way of viewing Grothendieck toposes as generalised spaces.
Both insights will inform the content of this talk. We shall start by giving a leisurely introduction to the theory of geometric logic and classifying toposes, before introducing a new research programme (joint with Steven Vickers) of developing a version of adelic geometry via topos theory.
The first step of this programme is to define the geometric theory of absolute values of $\mathbb{Q}$ and provide a point-free account of exponentiation. The next step is to construct the classifying topos of places of $\mathbb{Q}$, which incidentally provides a topos-theoretic analogue of the Arakelov compactification of $\operatorname{Spec}(\mathbb{Z})$. Interestingly, whereas the classical picture views the Archimedean place as a single point "at infinity", our picture reveals that the Archimedean place resembles a blurred interval living below $\operatorname{Spec}(\mathbb{Z})$. This raises challenging questions to our current understanding of the number theory, particularly in regards to reconciling the Archimedean vs. the non-Archimedean aspects.
Location: MALL 1
Title: Is o-minimality of the open-core an elementary property?
Let $M$ be a structure endowed with a dense linear order without endpoints. Its open-core is its reduced generated by the collection of all of its open definable sets (in any Cartesian power of $M$). We prove that the property of having an o-minimal open core isn't elementary. This is a joint work with Alexi Block-Gorman.
Location: MALL
Title: Functional interpretations and the contraction rule therein
Functional interpretations are generalisations of Goedel's Dialectica interpretation of Heyting arithmetic into a primitive-recursive functional system, which arose from a disbelief in universal quantifiers and integrated the ideas of functionals and proofs; the Dialectica interpretation shows the consistency of Heyting arithmetic (hence that of Peano arithmetic) with respect to that primitive-recursive functional system. We give an introduction to the idea of functional interpretations, and in the course, we will see how a seemingly innocuous inference rule called contraction (from A\vee A, deduce A) interferes with the interpretations.
Grothendieck rings were introduced in model theory in the early 2000s. They appear especially in motivic integration, where they are used to express formulas for certain counting functions in a uniform manner. There also is a dictionary of the combinatorial properties of a structure and of the algebraic properties of its Grothendieck ring. It wasn't known until recently wether there exist finite Grothendieck ring. In this talk, we will show that for any integer $N$, we can construct a structure whose Grothendieck ring is $\mathbb{Z}/N\mathbb{Z}$.
The strong existential closedness problem was introduced in Zilber's work on pseudoexponentiation. Since then, it has been naturally adapted to many situations in arithmetic geometry. In this talk I will introduce the problem, review some important Diophantine questions that are connected to it, and discuss some of the known results.
