Logic Seminar

Location: MALL
Title: Finite Grothendieck ring

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 $\mathbb{Z}/N\mathbb{Z}$.

Location: MALL
Title: Tame fields and beyond

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.

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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.

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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 $\omega$.

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