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Logic group

The union of all the seminar of the Logic group: Logic (Wed 3pm), Model Theory (Wed 2pm), Set Theory (Wed 1pm).

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Results 1 to 10 of 55

Rob Sullivan (Charles University, Prague) – Sharply $k$-homogeneous actions on Fraïssé structures

Date
@ Roger Stevens LT 14 (10M.14), online
Category

NOTES: unusual room.
Given an action of a group $G$ on a relational Fraïssé structure $M$, we call this action sharply $k$-homogeneous if, for each isomorphism $f : A \to B$ of substructures of $M$ of size $k$, there is exactly one element of $G$ whose action extends $f$. This generalises the well-known notion of a sharply $k$-transitive action on a set, and was previously investigated by Cameron, Macpherson and Cherlin. I will discuss recent results with J. de la Nuez González which show that a wide variety of Fraïssé structures admit sharply $k$-homogeneous actions for $k \leq 3$ by finitely generated virtually free groups. Our results also specialise to the case of sets, giving the first examples of finitely presented non-split infinite groups with sharply 2-transitive/sharply 3-transitive actions.

Joshua Losh (University of Leeds) – The Zarankiewicz Problem

Date
@ MALL 1
Category

Abstract: Arising from extremal combinatorics, the global Zarankiewicz problem seeks an upper bound on the number of edges of a finite $r$-hypergraph where the edge relation is induced by some fixed hypergraph that does not contain the compete $r$-hypergraph $K_{k,\dots,k}$  as a subgraph for some $k$. Special cases of this problem, where the edge relation comes from a definable set in a particular structure, have been of interest to model theorists. This talk will give an introduction to the problem as well as some recent results.

Maria-Romina Ivan (Cambridge/Stanford) – The game of cops and robbers can last any ordinal amount of time

Date
@ MALL 2, online
Category

The game of cops and robbers is played on a fixed graph, with the cop choosing a vertex to start at, then the robber chooses his, and then they take turns in moving to adjacent vertices. The game ends if the cop captures the robber (lands on its vertex). What graphs allow the cop to have a winning strategy, and how long does the game typically last, assuming optimal play? For finite graphs, the situation is very well understood — the cop-win graphs are precisely constructible graphs (constructed from a single vertex by repeatedly adding dominated vertices), and the capture time can be any finite ordinal (attained for example by finite paths).

In the infinite case, not much is known. In particular, there is no structural characterisation of cop-win graphs. What about the capture time? Is there an ordinal such that for any cop-win graph the sequence of moves of an optimal game is never that ordinal?

In this talk we will explore this question by showing that the answer is surprisingly 'no'.

Joint work with Tomas Flidr.

Angus Matthews (University of Leeds) – The Structure of Pregeometric Fields

Date
@ MALL 2
Category

NOTES: Unusual location.
Abstract: A pregeometric theory of fields is a theory T expanding ACF such that the model-theoretic algebraic closure acl satisfies the exchange property. Several important theories of fields satisfy this property, like ACF, RCF, ACVF, etc. Certain pregeometric fields have the additional property that we can expand the language with a generic derivation. We will discuss the structural implications of pregeometricness, and what we mean by a 'generic derivation'. Joint work with Antongiulio Fornasiero and Elliot Kaplan.