Sylvy Anscombe: In previous work with Fehm, and then Dittmann and Fehm, we found that the existential theory of an equicharacteristic henselian valued field is axiomatised using the existential theory of its residue field, conditionally, similar to an earlier theorem of Denef and Schoutens -- giving a transfer of decidability for existential theories. In this talk I'll describe parts of ongoing work with Fehm in which we describe a framework for interpreting one family of incomplete theories in another in order to find transfers of decidability in various settings. I will discuss some consequences for parts of the universal-existential theory of equicharacteristic henselian valued fields.
Neer Bhardwaj: We develop a variation of the Pila-Wilkie counting theorem, where we count rational points that approximate bounded complex analytic sets. A unique aspect of our result is that it does not depend on the analytic set in question. We apply this approximate counting to obtain an effective Pila-Wilkie statement for analytic sets cut out by computable functions. This is joint work with Gal Binyamini.
Alexi Block Gorman: There are compelling and long-established connections between automata theory and model theory, particularly regarding expansions of Presburger arithmetic by sets that are "recognized" by a DFA in a certain well-defined sense. Büchi automata are the natural extension of DFAs and NFAs to a model of computation that accepts infinite-length inputs. We say a subset X of the reals is Büchi-automatic if there some natural number r and some Büchi automaton that accepts (one of) the base-r representations of every element of X, and rejects the base-r representations of each element in its complement. In this talk, we will contrast the interaction of standard automata and Presburger structure with the interaction of Büchi automata and affine structure over the reals. We will further discuss how each of these fit into the framework of tameness in their respective settings, and what work has been done to characterize structures in which every definable set is recognized by an automaton, either of the standard or Büchi variety.
Zoé Chatzidakis: This is work in progress, joint with Nick Ramsey (Notre Dame).
A conjecture, now disproved by Chernikov, Hrushovski, Kruckman,
Krupinski, Pillay and Ramsey, asked whether any group with a simple
theory is definably amenable.
It is well known that the counting measure on finite fields gives rise
to a non-standard counting measure on pseudo-finite fields (the infinite
models of the theory of finite fields). It was unknown whether other PAC
fields possessed a reasonable measure, and in this talk, we will show
that some of them do. In a previous talk I announced some results on
perfect PAC fields with e-free absolute Galois group. Our methods didn't
seem to go further. However, inspired by a technique used by
W. Johnson in defining a measure on fields with several orderings, we
were able to extend the results to all bounded perfect PAC fields, and to
Frobenius fields. Here are the main steps of the proof:
1 - Define the measure when G(k) is the universal Frattini cover of a finite
group G.
2 - Do a limit argument to define the measure on all bounded PAC fields, by
showing that a small projective Galois group is the "ultraproduct" of
projective covers of finite groups.
3 - Apply ultrapowers to deduce that Frobenius PAC fields have a
measure. (Here G(k) is extremely homogeneous).
These results can then be used to show that groups definable in those
PAC fields are definably amneable. I will also discuss some open
questions.
The ultraproduct argument does not extend to arbitrary PAC fields:
Projective Galois groups can interpret any graphs, and many graphs are
not "pseudo-finite". We hope however to be able to push the results a bit
further.
Jan Dobrowolski: I will start by briefly explaining the relationship between amalgamation property, NIP of existential formulas, and the Intermediate Value Property of terms in certain (existentially closed) linearly ordered structures expanded by an automorphisms (e.g. ordered groups or ordered fields). Then I will sketch the proof of the Intermediate Value Property in the case when the underlying theory is that of ordered abelian groups (which hence gives the Amalgamation Property for the class of ordered abelian groups with an automorphism). This is a joint work (in progress) with Roasrio Mennuni.
James Freitag: We will give several recent applications of group theoretic aspects of binding group actions, mostly centered around transcendence results for differential equations. Joint work with Moosa and Jimenez.
Ayhan Gunaydin: The Beatty Sequence generated by an irrational r>1 is ([nr] : n>0), where [a] denotes the integer part of a real number a. We will first investigate the expansion of (Z, +) by the unary subset consisting of the terms of a Beatty sequence. We will mention a quantifier elimination result and an axiomatization for the theory of such an expansion. A decidability result will follow from this axiomatization. In the rest of the talk we will speculate on the decidability of the expansion of (Z,+,<) by a Beatty sequence. (The first part of the talk is joint work with Melissa Özsahakyan.)
Philipp Hieronymi: D-minimality was proposed by Chris Miller as a natural generalization of o-minimality in the setting of expansions of the real field. I will survey the subject and mention several open questions. I will also report on on-going work towards a Pila-Wilkie result for d-minimal structures (joint with Felix Jäger) and a general cell decomposition results (joint with Madie Farris).
Franziska Jahnke: Deeply ramified fields are a generalization of perfectoid fields and were introduced by Gabber and Ramero. As deeply ramified fields may admit immediate extensions, there is no hope for a classical Ax-Kochen/Ershov Theorem, and hence they have eluded model-theoretic machinery so far. In this talk, we present an AKE Theorem for certain perfect deeply ramified fields with a distiguished element t down to the pointed value group and "thickened" residue field. In particular, our Theorem applies to any perfectoid field, choosing a (pseudo)uniformizer for t. As a consequence, we obtain that the perfect hull of the henselization of F_p(t) is an elementary substructure of the perfect hull of F_p((t)). The results are joint work with Konstantinos Kartas.
Jonathan Kirby: The complex field, equipped with the multivalued functions of raising to each complex power, is quasiminimal: every definable subset is countable or co-countable. This proves a conjecture of Zilber and providing evidence towards his stronger conjecture that the complex exponential field is quasiminimal. This is joint work with Francesco Gallinaro.
Piotr Kowalski: This is joint work with Özlem Beyarslan. In our previous work (published as "Model theory of fields with virtually free group actions", Proc. London Math. Soc., (2) 118 (2019), 221-256), we used an erroneous argument in the proof of Theorem 3.6 saying that if G is a finitely generated virtually free group, then the theory of G-actions on fields has a model companion. In our recent paper (available on https://arxiv.org/abs/2210.00800 [arxiv.org]), we show a "strong negation" of the statement from Theorem 3.6 above, that is, we show that if G is an infinite finitely generated virtually free group, then the theory of G-actions on fields has a model companion if and only if G is free. In this talk, I will present some results and conjectures regarding the companionability of the theory of group actions on fields and (time permitting) discuss some related proofs.
Angus Macintyre: The emphasis is on definability , refined decidability. and uniformity in n. There is systematic use of the work of Ax-Kochen -Ershov, Ax,
P.J.Cohen, Denef -Pas and me. The case when n is a power of a prime (not necessarily standard) is basic, and one is able to give explicit axioms, and refine known periodicity
results using quite deep uniformities in rationality of p-adic Poincare series, and then use explicit recurrence relations.
The case of highly composite n is difficult, and uses a recent result with D’Aquino on a kind of converse to Feferman-Vaught to get hold of axioms.
With Derakhshan , using related adelic ideas, we recently solved positively a 60 year old problem of Ax on the decidability of the class of all Z/nZ.
Rahim Moosa: I will report on work with Jason Bell and Matthew Satriano contributing to our understanding of "quantifier-free nonorthogonality to the fixed field" in algebraic dynamics. One of the consequences of our results is an analogue in algebraic dynamics of the recent observation in differentially closed fields (due to Remi Jaoui and myself) that if a finite rank autonomous type is nonorthogonal to the constants then already its Morley square is not weakly orthogonal to the constants. While our results do not use any model theory of ACFA, the questions asked and the approach taken are motivated by model-theoretic considerations.
Anand Pillay: (Joint with David Meretzky.) If K is a Picard-Vessiot closed differential field and G a linear differential algebraic group over K, then G(K) is Kolchin dense in G, and if moreover G is finite-dimensional then G(K) = G(K^diff). Likewise for torsors for G. For an arbitrary differential field K we give closer relations between Picard-Vessiot extensions of K and torsors for suitable finite dim. linear differential algebraic groups of K. We also suggest differential field analogues of the notion of "boundedness" of fields.
Françoise Point: pdf
Patrick Speissegger: Gareth Jones and I have been suspecting for some time that the commonly used notion of pfaffian function is not equivalent to Khovanskii’s original definition. Jim Freitag’s recent proof that Klein’s j-function is not pfaffian (in the “common” sense) confirms this suspicion, and it implies also that the germs of pfaffian functions at 0, say, are not closed under taking implicit functions. Upon close inspection of Khovanskii’s pfaffian functions, we obtain what we call “nested pfaffian functions”. The germs at 0 of the latter are closed under taking implicit functions, which implies that the j-function is nested pfaffian. Together with Siegfried van Hille, we are currently exploring other possible closure properties of this class of germs.
Mariana Vicaria: One of the most striking results of the model theory of henselian valued fields is the Ax-Kochen/Ershov principle, which roughly states that the first order theory of a henselian valued field that is unramified is completely determined by the first order theory of its residue field and the first order theory of its value group.
Our leading question is: Can one obtain an Imaginary Ax-Kochen/Ershov principle?
In previous work, I showed that the complexity of the value group requires adding the stabilizer sorts. In previous work, Hils and Rideau-Kikuchi showed that the complexity of the residue field reflects by adding the interpretable sets of the linear sorts. In this talk we present recent results on weak elimination of imaginaries that combine both strategies for a large class of henselian valued fields of equicharacteristic zero. Examples include, among others, henselian valued fields with bounded galois group and henselian valued fields whose value group has bounded regular rank ( with an angular component map).