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
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
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.
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 $\mathbb{Z}/N\mathbb{Z}$.
I will introduce the notion of "tame field" and sketch what is known and is not known about the model theory of tame fields. Further, I will introduce the notion of "defect" of finite extensions of valued fields and indicate why it is important for the model theory of valued fields with residue fields of positive characteristic. Tame fields are defectless fields, i.e., all of their finite extensions have only trivial defect.
Tame fields are the closest we have come to the open problem of the decidability of formal Laurent series fields over finite fields, such as $\mathbb{F}_p((t))$, where $\mathbb{F}_p$ is the field with $p$ elements; yet tame fields are perfect, while $\mathbb{F}_p((t))$ is not. We are still lacking a complete recursive axiomatization for $\mathbb{F}_p((t))$ with the $t$-adic valuation. It is known that a simple adaptation of the axiom system that works for $p$-adically closed fields is not complete.
In the year 2003 Yuri Ershov introduced the notion of "extremal valued field" and proved that $\mathbb{F}_p((t))$ is extremal, but this was wrong as the definition of the notion was incorrect. In joint work with Salih Azgin and Florian Pop, we corrected the definition, and thereby the proof, and partially characterized extremal valued fields. This work was then continued together with Sylvy Anscombe. It has remained an open problem whether adding the (very handy) axiom system expressing extremality to the aforementioned adapted system will render it complete.
Another important question is whether we can push things beyond the class of defectless fields. In joint work with Anna Rzepka I have studied the valuation theory of "deeply ramified fields", which admit only certain less harmful defects. Perfectoid fields belong to this (elementary) class of valued fields, as well as all perfect valued fields of positive characteristic, such as the perfect hull of $\mathbb{F}_p((t))$. Thus the theory of deeply ramified fields may offer some clues for the open problem of the decidability of the latter.
Location: EC Stoner seminar room 8.90
Title: An unbounded version of Zarankiewicz's problem
NOTE room change: EC Stoner seminar room 8.90
Zarankiewicz's problem for hypergraphs asks for upper bounds on the number of edges of a hypergraph that has no complete sub-hypergraphs of a given size. Let $M$ be an o-minimal structure. Basit-Chernikov-Starchenko-Tao-Tran (2021) proved that the following are equivalent:
(1) "linear Zarankiewicz bounds" hold for hypergraphs whose edge relation is induced by a fixed relation definable in $M$
(2) $M$ does not define an infinite field.
We prove that the following are equivalent:
(1)' "linear Zarankiewicz bounds" hold for sufficiently "distant" hypergraphs whose edge relation is induced by a fixed relation definable in $M$.
(2)' $M$ does not define a full field (that is, one whose domain is the whole universe of $M$).
This is joint work (in progress) with Aris Papadopoulos.
Location: MALL
Title: Cohesive powers of linear orders
A cohesive power of a computable structure is an effective analog of an ultrapower, where a cohesive set acts as an ultrafilter. We compare the properties of cohesive powers to those of classical ultrapowers. In particular, we investigate what structures arise as the cohesive power of $B$ over $C$, where $B$ varies over the computable copies of some fixed computably presentable structure $A$, and $C$ varies over the cohesive sets.
Let $\omega$, $\zeta$, and $\eta$ denote the respective order-types of $(\mathbb{N}, <)$, $(\mathbb{Z}, <)$, and $(\mathbb{Q}, <)$. We take omega as our computably presentable structure, and we consider the cohesive powers of its computable copies. If L is a computable copy of omega that is computably isomorphic to the standard presentation $(\mathbb{N}, <)$, then all of $L$'s cohesive powers have order-type $\omega + (\zeta \times \eta)$, which is familiar as the order-type of countable non-standard models of PA.
We show that it is possible to realize a variety of order-types other than $\omega + (\zeta \times \eta)$ as cohesive powers of computable copies of omega. For example, we show that there is a computable copy $L$ of omega whose power by any $\Delta_2$ cohesive set has order-type $\omega + \eta$. More generally, we show that it is possible to achieve order-types of the form $\omega +$ certain shuffle sums as cohesive powers of computable linear orders of type $\omega$.
