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Symmetry in Newcastle – 19th April 2021

Symmetry in Newcastle seminar is here again! The confirmed speakers for next Monday are Zoe Chatzidakis, CNRS – ENS, and Laura Ciobanu, Herriot-Watt University. Feel free to grab a beverage appropriate for your respective timezone and join us for a friendly chat during the break!

The talks will be recorded and made available on our YouTube channel https://tinyurl.com/zerodimensionalgroup and our website https://zerodimensional.group/. The running times of the talks, titles and abstracts are as follows

16:30 – 17:30 AEST (06:30 – 07:30  UTC) Zoe Chatzidakis

17:30 – 18:00 AEST (07:30 – 08:00  UTC) Break and chat

18:00 – 19:00 AEST (08:00 – 09:00  UTC) Laura Ciobanu

Speaker: Zoe Chatzidakis (CNRS – ENS)
Title: A new invariant for difference fields
Abstract: If (K,f) is a difference field, and a is a finite tuple in some difference field extending K, and such that f(a) in K(a)^{alg}, then we define dd(a/K)=lim[K(f^k(a),a):K(a)]^{1/k}, the distant degree of a over K. This is an invariant of the difference field extension K(a)^{alg}/K. We show that there is some b in the difference field generated by a over K, which is equi-algebraic with a over K, and such that dd(a/K)=[K(f(b),b):K(b)], i.e.: for every k>0, f(b) in K(b,f^k(b)).

Viewing Aut(K(a)^{alg}/K) as a locally compact group, this result is connected to results of Goerge Willis on scales of automorphisms of locally compact totally disconnected groups. I will explicit the correspondence between the two sets of results.

(Joint with E. Hrushovski)

Speaker: Laura Ciobanu
Title: Free group homomorphisms and the Post Correspondence Problem
Abstract: The Post Correspondence Problem (PCP) is a classical problem in computer science that can be stated as: is it decidable whether given two morphisms g and h between two free semigroups A and B, there is any nontrivial x in A such that g(x)=h(x)? This question can be phrased in terms of equalisers, asked in the context of free groups, and expanded: if the `equaliser’ of g and h is defined to be the subgroup consisting of all x where g(x)=h(x), it is natural to wonder not only whether the equaliser is trivial, but what its rank or basis might be.

While the PCP for semigroups is famously insoluble and acts as a source of undecidability in many areas of computer science, the PCP for free groups is open, as are the related questions about rank, basis, or further generalisations. However, in this talk we will show that there are links and surprising equivalences between these problems in free groups, and classes of maps for which we can give complete answers. This is joint work with Alan Logan.