Lancashire Yorkshire Model Theory Seminar
A regular series of meetings of the model theorists in Leeds, Manchester and Preston, supported by the London Mathematical Society
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| 10:10-10:30 | Arrival and Coffee |
| 10:30-11:30 | Christian d'Elbée (Leeds) Title: On Wilson conjecture for Lie algebras Abstract: Wilson conjecture states that every locally nilpotent omega-categorical p-group is nilpotent. In this talk, we will present some connections between the analogous of Wilson conjecture for Lie algebras and the work of Kostrikin and Zelmanov on the theory of n-Engel Lie algebras and the positive solution to the restricted Burnside problem. |
| 11:30-12:30 | Ricardo Palomino (Manchester) Title: Relative quantifier elimination for lattice-ordered modules of continuous semi-algebraic functions on a curve Abstract: In the late 1980s, Shen and Weispfenning proved, via relative quantifier elimination in a suitable 2-sorted language, that under a mild condition on a divisible abelian lattice-ordered group G (that is, a divisible abelian group equipped with a lattice order compatible with the group operation) of functions, the theory of G is completely determined by the theory of a distributive lattice canonically associated to G. In this talk I will give the relevant context and details of their result, to then explain how the ideas in their proof can be adapted to lattice-ordered modules M of continuous semi-algebraic functions on a curve by enriching their 2-sorted language with a new sort for a real closed valuation ring; as a consequence of the method, decidability of M is obtained whenever the base field is a recursive real closed field. |
| 12:30-14:15 | Lunch |
| 14:15-15:15 | Pietro Freni (Leeds) Title: Weakly immediate types and spherical completions of o-minimal structures Abstract: Let T be an o-minimal theory expanding RCF and T_convex its expansion by a non-trivial T-convex valuation valuation ring. T_convex is complete and weakly o-minimal [4]. If T is power bounded, every model of T_convex has a spherically complete immediate extension [1]. This cannot happen if T is exponential [2]. In fact, when T is power bounded every extension minimally realizing a weakly immediate type is immediate (a restatement of the rv-property, [3], [5]). If T is exponential, this fails, but weakly immediate types are still completely determined by their restriction to the o-minimal reduct and do not force proper extensions of the residue field. For lambda an uncountable cardinal, I define lambda-bounded wim-constructible extension as transfinite unions of a continuous chains of extensions built by considering at each successor step an extension minimally realizing a type given by an intersection of less than lambda many valuation balls in the previous extension. Since wim-constructible extensions satisfy an amalgamation property, for every theory T of o-minimal fields and every uncountable cardinal lambda, every model E of T_convex has a lambda-spherically complete lambda-bounded wim-constructible extension which is unique up to a non unique isomorphism over E and embeds over E in every lambda-spherically complete extension of E. [1] Elliot Kaplan. T-convex t-differential fields and their immediate extensions. Pacific Journal of Mathematics, 320(2):261–298, 2023 [2] Franz-Viktor Kuhlmann, Salma Kuhlmann, and Saharon Shelah. Exponentiation in power series fields. Proceedings of the American Mathematical Society, 125(11):3177–3183, 1997 [3] James Michael Tyne. T-levels and T-convexity. University of Illinois at Urbana-Champaign, 2003 [4] Lou Van Den Dries and Adam H Lewenberg. T-convexity and tame extensions. The Journal of Symbolic Logic, 60(1):74–102, 1995. [5] Lou Van Den Dries and Patrick Speissegger. The field of reals with multisummable series and the exponential function. Proceedings of the London Mathematical Society, 81(3):513–565, 2000. |
| 15:15-15:45 | Coffee and Tea |
| 15:45-16:45 | Adele Padgett (McMaster) Title: O-minimal definitions of the complex Gamma and Riemann zeta functions Abstract: I will discuss work with P. Speissegger in which we prove that the Gamma function and Riemann zeta function are o-minimal on certain unbounded complex domains. |
| 16:45- | Pub and dinner |
| 10:30-11:00 | Arrival and Coffee |
| 11:00-12:00 | Aris Papadopoulos (Leeds) Title: Zarankiewicz’s Problem and Model Theory Abstract: A shower thought that anyone interested in graph theory must have had at some point in their lives is the following: `How “sparse" must a given graph be, if I know that it has no “dense” subgraphs?’. This curiosity definitely crossed the mind of Polish mathematician K. Zarankiewicz, who asked a version of this question formally in 1951. In the years that followed, many central figures in the development of extremal combinatorics contemplated this problem, giving various kinds of answers. Some of these will be surveyed in the first part of my talk. So far so good, but this is a logic seminar and the title says the words “Model Theory"… In the second part of my talk, I will discuss how the celebrated Szemerédi-Trotter theorem gave a starting point to the study of Zarankiewicz’s problem in “geometric” contexts, and how the language of model theory has been able to capture exactly what these contexts are. I will then ramble about improvements to the classical answers to Zarankiewicz’s problem, when we restrict our attention to semilinear/semibounded o-minimal structures, Presburger arithmetic, and various kinds of Hrushovski constructions. The new results that will appear in the talk were obtained jointly with Pantelis Eleftheriou. |
| 12:00-13:00 | Shezad Mohamed (Manchester) Title: The uniform companion for theories of difference large fields with free operators Abstract: In 2005, Tressl showed that there is a theory of differential fields, called UC, such that whenever T is a model complete theory of large fields, T + UC is the model companion of T + "differential fields". This gave a uniform way of constructing model companions of differential fields, and put many of the well-known examples of differential fields inside a common framework. In this talk I will show how we can extend Tressl's result to the case of fields with free operators—fields equipped with a homomorphism from the field to its tensor product with some fixed algebra D. Fields with free operators have definable endomorphisms, and thus constructing the uniform companion must be done relative to the underlying difference field. We will introduce the necessary generalisations of field-theoretic notions to the difference case in order to facilitate this. |
| 13:00-14:30 | Lunch |
| 14:30-15:30 | David Evans (Imperial) Title: Some questions on homogeneous structures Abstract: I will discuss some questions about countable structures which are homogeneous in a finite relational language and their automorphism groups. The questions - about closed normal subgroups, weak elimination of imaginaries and reducts - are not new, but I want to discuss the relationship between them and also their relationship to some other old questions on finite covers and permutation modules. |
| 15:30-16:00 | Coffee and Tea |
| 16:00-17:00 | Michael Wibmer (Leeds) Title: Differential Galois groups Abstract: The absolute differential Galois group of a differential field K is the automorphism group of the differential field extension of K generated by all solutions of linear differential equations with coefficients from K. We will discuss the structure of this group for K a one-variable function field. This is based on joint work with A. Bachmayr, D. Harbater, J. Hartman and R. Feng. |
| 17:00- | Pub and dinner |
| 10:30-11:00 | Arrival and Coffee at the Atrium Bridge (first floor of the Alan Turing building) |
| 11:00-12:00 | Nadja Hempel (Düsseldorf) Title: Pushing Properties for NIP Groups and Fields up the n-dependent hierarchy Abstract: 1-dependent theories, better known as NIP theories, are the first class of the strict hierarchy of n-dependent theories. The random n-hypergraph is the canonical object which is n-dependent but not (n−1)-dependent. We proved the existence of strictly n-dependent groups for all natural numbers n. On the other hand, there are no known examples of strictly n-dependent fields and we conjecture that there aren't any. We were interested which properties of groups and fields for NIP theories remain true in or can be generalized to the n-dependent context. A crucial fact about (type-)definable groups in NIP theories is the absoluteness of their connected components. Our first aim is to give examples of n-dependent groups and discuss a adapted version of absoluteness of the connected component. Secondly, we will review the known properties of NIP fields and see how they can be generalized. (Joint work with Chernikov.) |
| 12:00-13:00 | Jinhe Ye (Oxford) Title: Curve-excluding fields Abstract: Given $C$ a curve over $\mathbb{Q}$ with genus at least 2 and $C(\mathbb{Q})$ is empty, the class of fields $K$ of characteristic 0 such that $C(K)=\emptyset$ has a model companion, which we call CXF. Models of CXF have interesting combinations of properties. For example, they provide an example of a model-complete field with unbounded Galois group, answering a question of Macintyre negatively. One can also construct a model of it with a decidable first-order theory that is not "large'' in the sense of Pop. Algebraically, it provides a field that is algebraically bounded but not ``very slim'' in the sense of Junker and Koenigsmann. Model theoretically, we find a pure field that is strictly $NSOP_4$. |
| 13:00-13:30 | Problem session |
| 13:30-14:30 | Lunch |
| 14:30-15:30 | Problem session |
| 15:30-16:00 | Coffee and Tea at the Atrium Bridge (first floor of the Alan Turing building) |
| 16:00-17:00 | Problem session |
| 17:00- | Pub and dinner |
| 10:45-11:15 | Arrival and Coffee |
| 11:15-12:15 | Gabriel Ng (Manchester) Title: Differentially Henselian Fields Abstract: We say that a field equipped with a valuation and a derivation is differentially henselian if it is henselian as a pure valued field, and the derivation satisfies a certain ‘genericity’ condition. These are a special case of the topological fields with generic derivations as studied by Cubides Kovacsics, Guzy and Point. We will prove an ‘existential lifting’ lemma, from which we will deduce a number of other lifting properties, such as an Ax-Kochen/Ershov type result, and stable embeddedness results. |
| 12:15-13:15 | Vincent Bagayoko (Konstanz) Title: Growth order groups Abstract: I will introduce an elementary class of totally ordered groups, of which o-minimal geometry and differential ordered fields of generalized series (transseries or hyperseries) provide natural examples. |
| 13:15-14:30 | Lunch |
| 14:30-15:30 | Ibrahim Mohammed (Leeds) Title: Various contractions arising in Natural Ways Abstract: In the paper "Abelian Groups with Contractions", F.V Kuhlmann introduced the notion of contraction group. They consist of an ordered abelian group along with a unary map which collapses archimedean classes to a single point. The motivation behind them was to axiomatise the action of log on the value group of a non-standard model of R_exp, however there are a few other natural ways in which contraction groups arise. The first is the action of a hyperlogarithm (which can be thought of the composition of log \omega many times) on the value group of a transexponential ordered field. The other is the action of the hyperlogaithmic derivative on the same structure. In this talk I'll go through how contraction groups arise in these circumstances, and state various model theoretic results concerning them. |
| 15:30-16:00 | Coffee and Tea |
| 16:00-17:00 | Sebastian Eterovic (Leeds) Title: Likely Intersections Abstract: The past few years have seen many important results in Diophantine geometry concerning likely and unlikely intersections, and model theory has played a crucial role in this progress. In this talk I will introduce this topic and review some techniques from model theory that by now have become standard in the area. The main result I will present is a strong counterpart to the Zilber-Pink conjecture. Zilber-Pink predicts that if X is a proper subvariety of a special kind of variety S and X is not contained in a proper special subvariety of S, then the union of the unlikely intersections of X with the proper special subvarieties of S is not Zariski dense in X. We will see that the likely intersections of X (which are defined in a slightly more delicate way than one might naively expect) are Euclidean dense in X. This is joint work with Tom Scanlon. |
| 17:00- | Pub and dinner |
| 10:30-11:00 | Arrival and Coffee |
| 11:00-12:00 | Raymond McCulloch (Manchester) Title: Interdefinability of the exponential maps of abelian varieties Abstract: The model theory of $\mathbb{R}_{\exp}$ has been considered by model theorists for several decades. The Weierstrass $\wp$-function, a meromorphic function associated to a complex lattice, shares various properties in common with $\exp$ and its model theory has been considered by several authors. In this talk I shall discuss a result on the definability of Weierstrass $\wp$-functions due to Jones, Kirby and Servi before explaining how this may be extended to the exponential maps of abelian varieties. This is joint work with Gareth Jones and Jonathan Kirby. |
| 12:00-13:00 | Giuseppina Terzo (Naples) Title: Exponential ideals in an exponential polynomial ring Abstract: When we work with exponential polynomial rings over an exponential field some classical results fail, as Hilbert’s Basis Theorem and Nullstellensatz. We will present some results on exponential ideals in exponential rings, in order to see if some weak version of Nullstellensatz holds. Among E-ideals three categories stand out: E-prime ideals; E-ideals which are maximal as ideals, which we will call “strongly maximal”; E-ideals which are maximals among E-ideals, which we will call “E-maximal”. We want to study these three categories of E-ideals in an exponential polynomial ring over an exponential domain. We show that, except for the obvious implications, i.e. strongly maximal implies E-prime and E-maximal, the three notions are independent. (Joint work with Paola D’Aquino and Antongiulio Fornasiero) |
| 13:00-14:30 | Lunch |
| 14:30-15:30 | Vahagn Aslanyan (Leeds) Title: Existential Closedness and Zilber-Pink for the j-function with derivatives Abstract: I will discuss the Existential Closedness and Zilber-Pink conjectures for the modular j-function together with its derivatives. The former is about solvability of systems of equations involving j and its derivatives, and the latter is about unlikely intersections of algebraic varieties with certain special varieties associated to j and its derivatives. I will then explain how these two conjectures are related, and how that relation can be used to establish some partial results towards Zilber-Pink with Derivatives. |
| 15:30-16:00 | Coffee and Tea |
| 16:00-17:00 | Pablo Andújar Guerrero (Leeds) Title: O-minimal definable topology Abstract: We explore the extend of o-minimality as a framework for general tame topology, by addressing questions from set-theoretic topology and functional analysis in the context of o-minimal definable topologies. In the first part of the talk, we address the 3-element basis conjecture for uncountable first countable regular Hausdorff spaces, which states that it is consistent with ZFC that all such spaces have a copy of size ℵ₁ of the reals with the euclidean, Sorgenfrey, or discrete topology. We prove an o-minimal version of the conjecture. In the second part of the talk we present the result and proof from functional analysis that a compact Hausdorff topological space is metrizable if and only if the space of real valued continuous functions on it (with the uniform norm topology) is separable. We investigate the strengths and shortcomings of o-minimality in adopting similar proof schemes to reach affineness results. |
| 17:00- | Pub and dinner |
| 10:30-11:00 | Arrival and Coffee | |||||||||
| 11:00-12:00 | Anand Pillay (Notre Dame) Title: Groups definable in closed ordered differential fields Abstract: (joint with K. Peterzil and F. Point) We prove that a finite dimensional group definable in a model of CODF definably embeds in a semialgebraic group. The proof works in many contexts of a "nice" theory of fields equipped with a generic derivation. |
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| 12:00-13:00 | Deacon Linkhorn (Manchester) Title: Axiomatisations and model-completeness results for some linear orders in weak monadic second order logic Abstract: I will give an overview of some results concerning the weak monadic second order (WMSO) theories of certain classes of linear orders. In WMSO logic we have quantifiers ranging over the finite subsets of the universe, though I will work in a first order setup which captures this (expansions of atomic distributive lattices). In particular I will present results concerning the shared WMSO theory of finite linear orders, and the WMSO theory of a dense linear order. In both cases an axiomatisation will be supplied, and signatures suitable for a model-completeness result will be outlined. Time permitting, I will also explain how to transfer the model-completeness result for the WMSO theory of a dense linear order to the lattice of finite unions of closed intervals of a dense linear order. |
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| 13:00-14:30 | Lunch | |||||||||
| 14:30-15:30 | Student Presentations
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| 15:30-16:00 | Coffee and Tea | |||||||||
| 16:00-17:00 | Student Presentations
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| 17:00- | Pub | |||||||||
| 19:00- | Dinner at ZOUK |
| 10:30-11:00 | Arrival and Coffee | ||||||
| 11:00-12:00 | Tobias Kaiser (Passau) Title: Classes of Functions definable in ℝan,exp Abstract: There is a strong dividing line for functions definable in the o-minimal structure ℝan,exp, the expansion of the real field by restricted analytic functions and the exponential function. This dividing line originates from the latter. We consider respectively define natural intermediate classes of functions as log-analytic or so-called restricted log-exp-analytic functions to focus on this dividing line. We discuss various analytic properties for these classes. The key tool is given by preparation results. (Joint work with Andre Opris) |
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| 12:00-13:00 | Pantelis Eleftheriou (Leeds) Title: Groups definable in the disjoint union of two structures Abstract: We consider a group G=(G, ∙) definable in the disjoint union M of two structures X₁ and X₂. We give an example of such G, with each Xᵢ being o-minimal, which is not a direct product of Xᵢ-internal groups, i=1,2. We prove that if M has NIP (not the independence property), and G is abelian and has fsg (finitely satisfiable generics), then G = A₁ ∙ A₂, where Aᵢ is an Xᵢ-internal subset of G. If, moreover, each Xᵢ is o-minimal, we can get rid of the two assumptions in G. (Joint work with A. Berarducci and M. Mamino.) |
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| 13:00-14:30 | Lunch | ||||||
| 14:30-15:30 | Student Presentations
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| 15:30-16:00 | Coffee and Tea | ||||||
| 16:00-17:00 | Student Presentations
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| 15:30-16:00 | Coffee and Tea | ||||||
| 16:00-17:00 | Student Presentations |
| 10:30-11:00 | Arrival and Coffee |
| 11:00-12:00 | Talk by Lorna Gregory. Title: Model Theory of Modules over Prüfer Domains and Dimensions on Lattice Ordered Abelian Groups Abstract: Much of the model theory of modules over a Prüfer domain R is captured by its value group, that is, the group of fractional ideals of R ordered by reverse inclusion. This group is a lattice ordered abelian group and all lattice ordered abelian groups occur as the value group of a Prüfer, or even Bézout, domain. In this talk I will explain how the principal invariants of model theory of modules, m-dimension and breadth of the lattice of pp-formulae, can be calculated (or bounded) for Prüfer domains using their value groups. I will explain the intended meaning of these invariants and, if there is time, explain how to show that for Prüfer domains, they measure what they are supposed to measure. |
| 12:00-13:00 | Question proposal session by Jan Dobrowolski, Lorna Gregory and Vincenzo Mantova. |
| 13:00-14:30 | Lunch |
| 14:30-17:00 | Question answer session and discussion |
| 17:00- | Pub and dinner |
| 13.50 | Start of the Zoom meeting | |
| 14.00-14.50 | Martin Bays (Univerität Münster) | Elekes-Szabó, weak general position, and generic nilprogressions Abstract: By the Elekes-Szabó theorem, any ternary algebraic relation in characteristic 0 which has asymptotically large intersections with products of finite sets in "general position" must be, up to finite correspondences, the graph of addition in an abelian algebraic group. I will discuss some first steps towards understanding what happens when we try to relax this general position hypothesis, which in model theoretic terms is a kind of minimality. I will concentrate on the case that we already know the relation to be the graph of multiplication in an algebraic group, where (via Balog-Szemerédi-Gowers-Tao) one is really talking about the existence of approximate subgroups in some sort of general position. Our main result is that for a certain natural weak notion of general position, this precisely characterises nilpotence of the group. The proof involves a generic Mordell-Lang result for arbitrary commutative algebraic groups. This is joint work with Jan Dobrowolski and Tingxiang Zou. |
| 15.00-15.50 | Nicholas Ramsey (UCLA) | Binarity, Treelessness, and Generic Stability Abstract: A theory T is called binary if any two tuples have the same type if and only if all corresponding subtuples of length 2 have the same type. With this strong restriction on theories, it turns out that certain classification-theoretic dividing lines collapse: for example, we show that a binary NSOP_1 theory is simple, and a binary NSOP_3 theory is NTP_1. Motivated by these results, we develop the basics of neostability theory for the broader category of treeless theories. We show such theories come equipped with a natural notion of independence, defined in terms of generically stable partial types, which is meaningful in both simple and NIP theories. This is joint work with Itay Kaplan and Pierre Simon. |
| 14.50 | Start of the Zoom meeting | |
| 15.00-15.50 | Gareth Boxall (Stellenbosch University) | Some finiteness results concerning points on a curve with a power on a curve Abstract: Let C1, C2 ⊆ 𝔾mN(ℂ) be geometrically irreducible closed algebraic curves, with N ≥ 3. Suppose C1 is not contained in an algebraic subgroup of 𝔾mN(ℂ) of dimension 1 and C1 ∪ C2 is not contained in an algebraic subgroup of 𝔾mN(ℂ) of dimension 2. It is a conjecture that at most finitely many points x ∈ C1 have the property that there is a positive integer n such that xn ∈ C2 and [n]C1 ⊈ C2, where [n]C1 = {xn : x ∈ C1}. We prove some special cases of this conjecture. We build on work done by Bays and Habegger [3] in the case where C1 = C2 and make use of a height bound of Amoroso, Masser and Zannier [1]. We also apply work of Bays, Kirby and Wilkie [2] which gave an analogue of Schanuel’s conjecture for the operation of raising to an exponentially transcendental power. [1] Amoroso, F., Masser, D. and Zannier, U., Bounded height in pencils of finitely generated subgroups, Duke Math. J. 166 (no. 13): 2599–2642, 2017. [2] Bays, M., Kirby, J. and Wilkie, A., A Schanuel property for exponentially transcendental powers, Bull. London Math. Soc. 42 (no. 5): 917–922, 2010. [3] Bays, M. and Habegger, P., A note on divisible points of curves, Trans. Amer. Math. Soc. 367: 1313–1328, 2015. |
| 16.00-16.50 | Alexi Block Gorman (University of Illinois at Urbana-Champaign) | Definability on the Reals from Büchi Automata Abstract: Büchi automata are the natural analogue of finite automata in the context of infinite strings (indexed by the natural numbers) on a finite alphabet. We say a subset X of the reals is r-regular if there is a Büchi automaton that accepts (one of) the base-r representations of every element in X, and rejects the base-r representations of each element in its complement. These sets often exhibit fractal-like behavior—e.g., the Cantor set is 3-regular. There are remarkable connections in logic to Büchi automata, particularly in model theory. In this talk, I will give a characterization of when the expansion of the real ordered additive group by a predicate for a closed r-regular subset of [0,1] is model-theoretically tame (d-minimal, NIP, NTP2). Moreover, I will discuss how this coincides with geometric tameness, namely trivial fractal dimension. This will include a discussion of how the properties of definable sets vary depending on the properties of the Büchi automaton that recognizes the predicate subset. |
| 17.00-17.50 | Erik Walsberg (University of California, Irvine) | Largeness, the étale open topology, and tame topology of definable sets Abstract: Definable sets in algebraically, real, and p-adically closed fields are all well-behaved with respect to the Zariski, order, and valuation topologies, respectively. In recent work with Will Johnson, Minh Chieu Tran, and Jinhe (Vincent) Ye, we have introduced the étale open topology over an arbitrary field K. This agrees with the Zariski, order, valuation topology over an algebraically, real, p-adically closed field, respectively. More recent work with Jinhe suggests that definable sets in basically all known model-theoretically tame perfect fields are well-behaved with respect to the étale open topology. I will discuss this, assuming minimal background from algebraic geometry. |
| 18.00- | Logic Pub |
| 8.50 | Start of the Zoom meeting | |
| 9.00-9.50 | Annalisa Conversano (Massey University Auckland) | Nilpotent groups definable in o-minimal structures Abstract: Many authors in the past thirty years have shown strong analogies between groups definable in o-minimal structures and real Lie groups, especially in the compact case. For nilpotent groups, not necessarily definably compact, it is possible to find strong similarities even with the smaller class of real algebraic groups. Some recent results about this analogy will be presented in this talk, including the fact that linear algebraic groups are the only nilpotent Lie groups that can be defined in an o-minimal expansion of the real field. |
| 10.00-10.50 | Christian d'Elbée (Hebrew University of Jerusalem) | Title: Dp-minimal integral domains. Abstract: (joint with Yatir Halevi) As expected from the classification of dp-minimal fields, dp-minimal integral domains are close to be valuation domains, but not always. The are local, divided in the sense of Akiba, and every localisation at a non-maximal prime ideal is a valuation domain. Furthermore, a dp-minimal integral domain is a valuation ring if and only if its residue field is infinite or its residue field is finite and its maximal ideal is principal. I will present these results as well as some examples of dp-minimal domains which are not valuation domains. If time allows it, I will also talk about a generalisation of a result of Echi and Khalfallah on the prime spectrum of the ring of bounded elements of the hyperreals. |
| 11.00-11.50 | Tingxiang Zou (University of Münster) | Title: Geometric random graph Abstract: Geometric random graphs are graphs on a countable dense set of some underlying metric space such that locally in any ball of radius one, it is a random graph. The geometric random graphs on ℝⁿ and on circles have been studied by probabilists and graph theorists. In this talk we will present some model theoretic views. In particular, we will show that under some mild assumptions, the geometric random graphs based on a fixed metric space will have the same theory. We will also talk about some geometric properties of the underlying metric space that can be recovered from the graphs. This is a work in progress joint with Omer Ben-Neria and Itay Kaplan. |
| 12.00- | Logic Lunch |
| 14.50 | Start of the Zoom meeting | |
| 15.00-15.50 | Alexander Berenstein (Universidad de los Andes) | Title: Expansions of geometric theories as measurable structures Abstract: We say that a theory T is geometric if for any model $M\models T$ the algebraic closure satisfies the exchange property and T eliminates the quantifier $\exists^{\infty}$. Examples of these theories include SU-rk one theories and dense o-minimal theories. In this talk I will present the basic properties of these theories and some well known expansions like H-structures and lovely pairs. We will consider the special case where the underlying theory is measurable (in the sense of Macpherson and Steinhorn) of SU-rk one. Under these assumptions, the expansion as an H-structure can be studied as a generalized measurable structure whose dimension has values in $\omega^2$. This is joint work with García and Zou. |
| 16.00-16.50 | Gabriel Conant (Cambridge) | Title: NIP approximate groups and arithmetic regularity Abstract: I will present recent work with Anand Pillay on the structure of finite approximate groups satisfying a local NIP assumption. Our results can be seen as a unification of work of Breuillard, Green, and Tao on the structure theory of approximate groups, and the model-theoretic study of "tame” arithmetic regularity. |
| 17.00-17.50 | Samaria Montenegro Guzman (Universidad de Costa Rica) | Title: Definable groups in PRC fields Abstract: This is a joint work with Alf Onshuus and Pierre Simon. We will study the class of pseudo real closed fields (PRC-fields) from a model theoretical point of view. PRC fields were introduced by Prestel and Basarav as a generalization of real closed fields and pseudo algebraically closed fields, where we admit having several orders. We know that the complete theory of a bounded PRC field (i.e., with finitely many algebraic extensions of degree m, for each m > 1) is NTP2 and we have a good description of forking. In this talk we will focus in the groups with f-generic types definable in bounded PRC fields. The main theorem is that such a group is isogeneous with a finite index subgroup of a quantifier-free definable groups. This generalizes similar results proved by Hrushovski and Pillay for (not necessarily f-generic) groups definable in both pseudo finite fields and real closed fields. |
| 18.00- | Logic Pub |
| 10.30-11.00 | Arrival and coffee | |
| 11.00-12.00 | Isabel Müller (Imperial) | Stationary Independence and Symmetric Indivisibility Abstract: In 2012 Tent and Ziegler introduced the notion of a Stationary Independence Relation (SIR) and used this tool to study the normal subgroup structure of homogeneous structures. In 2016, we used the existence of an SIR to establish the universality of automorphism groups of the corresponding structures. Recently, in his PhD thesis, Meir introduced and studied lexicographic products of relational first order structures to answer questions about symmetric indivisibility. In this talk, we will introduce the notions and results mentioned above, show how they can be combined and state some open questions around the area. This is work in progress with Nadav Meir. |
| 12.00-13.00 | Bea Adam-Day (Leeds) | Membership Graphs of Models of Anti-Foundation Abstract: It is known that if we take a countable model of ZFC and symmetrise the membership relation, then we obtain the Random Graph. It turns out that doing so in Anti-Foundational set theory yields the ``Random Loopy Graph'': the Fra\"iss\'e limit of finite graphs with self-edges. However, if one instead considers the \emph{double-membership relation}, $x\in y\in x$, then the resulting graph is much more complicated. I will discuss properties of these graphs and their theories, presenting some results from two papers, joint with Peter Cameron and with John Howe and Rosario Mennuni. |
| 13.00-14.30 | Lunch | |
| 14.30-15.30 | Gareth Jones (Manchester) | Powers are easy to avoid Abstract: Suppose that a set is definable in the expansion of the real field by restricted analytic functions, and is also definable in the expansion of the real field by the restricted exponential function together with all real power functions. Then the set is definable using just the restricted exponential function. So additional exponents can be avoided. I will discuss the general result behind this, and how it can be seen as a polynomially bounded version of an old conjecture of van den Dries and Miller. This is joint work with Olivier Le Gal. |
| 15.30-16.00 | Coffee | |
| 16.00-17.00 | Sylvy Anscombe (UCLan) | A newish view of Cohen rings, complete discrete valuation rings, and NIP |
| 17.00- | Pub and Dinner | |
| 10.30-11.00 | Arrival and coffee | |
| 11.00-12.00 | Jan Dobrowolski (Leeds) | Elementary Equivalence Theorem for pseudo algebraically closed structures Abstract We generalise a well-known theorem saying that two PAC fields are elementarily equivalent if a suitable isomorphism of their Galois groups exists, to the context of pseudo algebraically closed structures. This is a joint work with D. Hoffmann and J. Lee. |
| 12.00-13.00 | Nadav Meir (Imperial) | Pseudo-finite sets, pseudo-o-minimality Abstract: Given a language L, the class of o-minimal L-structures is not elementary, e.g., an ultraproduct of o-minimal structures need not be o-minimal. This fact gives rise to the following notion, introduced by Hans Schoutens: Given a language L, an L-structure is pseudo-o-minimal if it satisfies the common theory of o-minimal L-structures. Of particular importance in pseudo-o-minimal structures are pseudo-finite sets. A definable set in an ordered structure is pseudo-finite if it is closed, bounded and discrete. Many results from o-minimality translate to pseudo-o-minimality by replacing finite with pseudo-finite. We will review the key role that pseudo-finite sets play in pseudo-o-minimality, as well as other first-order properties of o-minimality such as definable completeness* and local o-minimality**. Finally, we will see how pseudo-finite sets can be used to answer two questions by Schoutens, one of them is whether there is an axiomatization of pseudo-o-minimality by first-order conditions on one-variable formulae only. This also partially answers a conjecture by Antongiulio Fornasiero. * An ordered structure is definably complete if every bounded definable subset has a supremum. ** An ordered structure is locally o-minimal if, for every definable subset D and every point x, there is an interval I containing x such that the intersection of D and I is a finite union of intervals and points. |
| 13.00-14.30 | Lunch | |
| 14.30-15.30 | Francesco Parente (Leeds) | Saturated Boolean ultrapowers, Keisler’s order, and universality of forcing extensions Abstract: In this talk, I will discuss some recent results at the interface between model theory and set theory. The first part will be concerned with model-theoretic properties of ultrafilters in the context of Keisler’s order. I will use the framework of ‘separation of variables’, recently developed by Malliaris and Shelah, to provide a new characterization of Keisler’s order in terms of saturation of Boolean ultrapowers. Furthermore, I will show that good ultrafilters on complete Boolean algebras are precisely the ones which capture the maximum class in Keisler’s order, answering a question posed by Benda in 1974. In the second part of the talk, I will report on joint work with Matteo Viale in which we apply the above results to the study of models of set theory. In particular, our work aims at understanding the universality properties of forcing extensions. To this end, we analyse Boolean ultrapowers of $H_{\omega_1}$ in the presence of large cardinals and give a new interpretation of Woodin’s absoluteness results in this context. |
| 15.30-16.00 | Coffee | |
| 16.00-17.00 | Vincenzo Mantova (Leeds) | Some unconditional results on exponential-algebraic closure Abstract TBA |
| 17.00- | Pub and Dinner | |
| 10:30-11:00 | Arrival and Coffee |
| 11:00-12:00 | Talk by Piotr Kowalski. Title: Model theory of free operators in positive characteristics Abstract: This is joint work with Özlem Beyarslan, Daniel Hoffmann and Moshe Kamensky. We give algebraic conditions about a finite commutative algebra B over a field of positive characteristic, which are equivalent to the companionability of the theory of fields with ``B-operators'' (i.e. the operators coming from homomorphisms into tensor products with B). The notion of a B-operator includes derivations, endomorphisms and (truncated, non-iterative) Hasse-Schmidt derivations. We show that, in the most interesting case of a local B, these model companions admit quantifier elimination in the ``smallest possible'' language and they are strictly stable. We also describe the forking relation there. |
| 12:00-13:00 | Question proposal session |
| 13:00-14:30 | Lunch |
| 14:30-17:00 | Question answer session and discussion |
| 17:00- | Pub and dinner |
Venue: Foster Building, lecture theatre, UCLAN. The campus map can be found here.
| 10.30-11.00 | Arrival and coffee | |
| 11.00-12.00 | Ulla Karhumaki (Manchester) | Definably Simple Stable Groups with Finitary Groups of Automorphisms Abstract We prove that infinite definably simple locally finite groups of finite centraliser dimension are simple groups of Lie type over locally finite fields. Then, we axiomatise some of the key properties of Frobenius maps under the name of finitary automorphism groups. This allows us to classify definably simple stable groups in the specific case when they admit such automorphism group. |
| 12.00-13.00 | Jonathan Kirby (University of East Anglia) | Existentially closed exponential fields Abstract: We characterise the existentially closed models of the theory of exponential fields. They do not form an elementary class, but can be studied using positive logic. We find the amalgamation bases and characterise the types over them. We define a notion of independence and show that independent systems of higher dimension can also be amalgamated. We extend some notions from classification theory to positive logic and position the category of existentially closed exponential fields in the stability hierarchy as NSOP_1. This is joint work with Levon Haykazyan |
| 13.00-14.30 | Lunch | |
| 14.30-15.30 | Alexander Antao (Manchester) | Partial Skolemization of the real exponential ordered field Abstract: A programme of model theory is finding "natural'" mathematical structures with the properties of quantifier elimination, or the next best thing, model completeness. Some examples expanding the set of reals:
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| 15.30-16.00 | Coffee | |
| 16.00-17.00 | Omar León Sánchez (Manchester) | Isolated types in totally transcendental theories |
| 17.00- | Pub and Dinner | |
| 11.00 - 12.00 |
Harry Schmidt (Manchester) | Mahler functions and Manin-Mumford for $G_m^n$
Abstract: I will report on work in progress on connections between algebraic independence of certain Mahler functions and the Manin-Mumford conjecture for the multiplicative group. |
| 12.00 - 13.00 |
Tom Kirk (UCLan) | Definable Topological Dynamics in Metastable Theories. Abstract: We consider a dynamical system where a definable group $G$ acts on the space of complete types $S_{G}(M)$. Specifically, we will take $G$ to be an affine algebraic group definable in a metastable theory and consider the minimal ideals of this action. We give a full description for the Minimal Flows, and Ellis Group, of $SL_2(\mathbb{C}((t)))$, and note that this is not isomorphic to $G/G^{00}$; providing a negative answer as to whether metastability is a suitable weakening of a since disproven conjecture of Newelski. Further, we discuss recent work in $ACVF$ where $G$ admits a stably dominated / $fsg$ group decomposition (possibly with non-trivial intersection) and give a description of the Ellis Group in this setting. |
| 13.00 - 14.30 |
Lunch | |
| 14.30 - 15.30 |
Julia Wolf (Cambridge) | The structure of stable sets in finite abelian groups. Abstract: We shall begin by explaining the idea behind the so-called "arithmetic regularity lemma" pioneered by Green, which is a group-theoretic analogue of Szemerédi's celebrated regularity lemma for graphs with wide-ranging applications. We will then describe recent joint work with Caroline Terry (University of Chicago), which shows that under the natural model-theoretic assumption of stability the conclusions of the arithmetic regularity lemma can be significantly strengthened, leading to a characterisation of stable subsets of finite abelian groups. In the latter part of the talk, we survey related work by various authors including Alon, Conant, Fox, Pillay, Sanders, Sisask, Terry and Zhao, further exploring this topic from both a combinatorial and a model-theoretic perspective. |
| 15.30 - 16.00 |
Tea/Coffee | Common Room |
| 16.00 - 17.00 |
Marcus Tressl (Manchester) | On closed ordered differential fields.
Abstract: An ordered differential field is an ordered field K together with a derivation d:K -> K; no interaction of d with the order is assumed. Michael Singer has shown that the existentially closed ordered differential fields (denoted CODF) are axiomatisable with quantifier elimination in the language of ordered differential rings. I will give an introduction to CODFs and report on some recent developments in the model theory of CODFs and its generalizations. |
| 17.00- | Pub and Dinner |
| 10.30 - 11.00 |
Arrival and coffee will be in the Physics Research Deck. | |
| 11.00 - 12.00 |
Pablo Cubides Kovacsics (Caen) | On iso-definability of curves in Hrushovski-Loeser spaces
Abstract: Hrushovski and Loeser introduced a model-theoretic version of the analytification of a quasi-projective variety over a non-archimedean valued field. Their construction gives rise to a strict pro-definable set in general and to an iso-definable set in the case of curves. In this talk I will report on a joint work with Jérôme Poineau in which, focusing on the later case, we provide an alternative approach to endow the Hrushovsi-Loeser analytification of an algebraic curve with a definable structure. In particular, this allows us to get a complete description of the definable subsets of such curves. |
| 12.00 - 13.00 |
Laura Capuano (Oxford) | Unlikely intersections and o-minimality Abstract: The theory of o-minimality has made a huge remark in arithmetic geometry in the study of the so called “problem of unlikely intersections”, starting with the alternative proof of Manin-Mumford conjecture due to Pila and Zannier. One of the main ingredients of the proof is a result of Pila about counting rational points of bounded height on subanalytic surfaces, which is a special case and predates the celebrated Pila-Wilkie theorem. Since then, there has been a lot of work centred around the Zilber-Pink conjecture, and Pila-Zannier “strategy” has been used to prove several results in this area in many different settings. In my talk, I will give a general overview about these problems, with a special regard to questions of unlikely intersections inside tori and families of abelian varieties. |
| 13.00 - 14.30 |
Lunch | |
| 14.30 - 15.30 |
Zaniar Ghadernezhad (Imperial) | Minimality of automorphism groups of free homogeneous structures. Abstract: A topological group $G$ is called minimal if every bijective continuous homomorphism from $G$ to another Hausdorff topological group is a homeomorphism or equivalently, if $G$ does not admit a strictly coarser Hausdorff group topology; $G$ is called totally minimal if every continuous surjective homomorphism to a Hausdorff topological group is open. These minimality notions have been extensively studied in topological group theory and known for some cases for example the infinite permutation group and the unitary group. Automorphism groups of countable first order structures are topological groups and one could ask whether or not they are minimal. In an interesting work Ben-Yaacov and Tsankov proved that automorphism groups of stable, $\omega$-categorical structures are totally minimal. In this talk we investigate the minimality of automorphism groups of free homogeneous structures. This is a joint work with Javier de la Nues González. |
| 15.30 - 16.00 |
Tea/Coffee | |
| 16.00 - 17.00 |
Philip Dittmann (Oxford) | Recovering Arithmetic from Galois Theory - a Model-Theoretic Perspective
Abstract: A common task in field arithmetic is recovering information about a field, e.g. about its orderings and valuations, from Galois-theoretic data. Model-theoretic interpretability is one way to formalise such statements. I will present such an interpretation of Stone spaces of orderings and p-valuations in suitable Galois structures, applicable to wide classes of fields, for instance the class of all pseudo real closed and pseudo p-adically closed fields. An important part of this is finding a good model-theoretic language for Galois theory. |
| 17.00- | Pub and Dinner |
| 10.30-11.00 | Arrival and coffee | |
| 11.00-12.00 | Victoria Gould (York) | ℵ0-categoricity for semigroups Abstract may be found here. |
| 12.00-13.00 | Mike Prest (Manchester) | Nori motives and model theory Abstract: Homology and cohomology theories attach algebraic and numerical invariants to varieties and schemes. There are many such (co)homology theories and the idea (of Grothendieck) is that there should be a universal such theory - one through which all the others factor. This would be an abelian category of "motives" built from a suitable category of varieties. In the 90s Nori gave a construction of a category with some of the desired features. A recent paper of Barbieri-Viale, Caramello and Lafforgue gives a much more direct construction, using (topos-theoretic) model theory. Barbieri-Viale and I subsequently described an even more direct approach using (classical-style) model theory. In these approaches Nori motives are imaginary sorts in an appropriate language. I will talk about this and some current work with Luca Barbieri-Viale and Annette Huber. |
| 13.00-14.30 | Lunch | |
| 14.30-15.30 | Luck Darnière (Angers) | Lattices of closed semi-algebraic sets. Abstract: Let K be a p-adically closed field, X a semi-algebraic set of dimension d defined over K and L(X) the lattice of semi-algebraic subsets of X which are closed in X. We prove that the complete theory of L(X) is decidable (contrary to what happens over a real closed field) and eliminates the quantifiers in a certain language Lasc, the Lasc-structure on L(X) being an extension by definition of the lattice structure. We classify these structures up to elementary equivalence, and get in particular that the complete theory of L(K^d) only depends on d, not on K nor even on p. |
| 15.30-16.00 | Coffee | |
| 16.00-17.00 | Andrew Brooke-Taylor (Leeds) | Generalised model theory from a category theory perspective Abstract: Shelah introduced the framework of Abstract Elementary Classes to generalise model theory beyond the first order. Parallel to this, in category theory the notion of accessible categories was developed with a similar aim. It has recently emerged that the two approaches are intimately related to each other. In this talk I will give an overview of this connection, with a particular eye to how set theory can contribute to the mix. |
| 17.00- | Pub and Dinner | |
| 10.30-11.00 | Arrival and coffee | |
| 11.00-12.00 | Sam Dean (Glasgow) | Positive primitive formulas for sheaves Abstract: Others have defined what it should mean for a sheaf to sit inside another sheaf as a pure substructure. This is done in an algebraic fashion. But in the model theory of modules, we know well that this condition can be said in terms of pp formulas. Sheaves, not usually being 1st-order structures, can't obviously be approached like this. Even when we do get a nice class of sheaves which are (secretly) 1st-order structures, the answer we get for what the usual notion of purity means is geometrically wrong. I will give a notion of a pp formula for sheaves which fits with the geometric notion of purity, and explain what the remaining questions are. |
| 12.00-13.00 | Sasha Borovik (Manchester) | Permutation groups of finite Morley rank Abstract: I will introduce some basic concepts and ideas of this theory, and will survey latest results by Altinel, Berkman, Borovik, Deloro, and Wiscons (in various combinations of authors). |
| 13.00-14.30 | Lunch | |
| 14.30-15.30 | Gabor Elek (Lancaster) | Limits of finite graphs via ultraproducts Abstract: I will show how to obtain the Lovasz-Szegedy resp. Benjamini-Schramm graph limits of dense resp. sparse graphs using ultraproducts and ultralimits. |
| 15.30-16.00 | Coffee | |
| 16.00-17.00 | Gareth Jones (Manchester) | Pfaffian functions and elliptic functions Abstract: I will discuss work with Harry Schmidt in which we give a pfaffian definition of Weierstrass elliptic functions, refining a result due to Macintyre. The complexity of our definition is bounded by an effective absolute constant. As an application we give an effective version of a result of Corvaja, Masser and Zannier on a sharpening of Manin-Mumford for non-split extensions of elliptic curves by the additive group. We also give a higher dimensional version of their result. |
| 17.00- | Pub and Dinner | |
| 10:30-11:00 | Arrival and Coffee |
| 11:00-12:00 | Talk by Martin Hils. Title: Model theory of compact complex manifolds with an automorphism Abstract: One may develop the model theory of compact complex manifolds (CCM) with a generic automorphism in rather close analogy to what has been done for existentially closed difference fields, in important work by Chatzidakis and Hrushovski, among others. The corresponding first order theory CCMA is supersimple, and the Zilber trichotomy holds for "finite-dimensional" types of SU-rank 1. In the talk, I will present some results in CCMA in the spirit of geometric simplicity. Moreover, I will discuss the issue whether a given sort of CCM is stably embedded in CCMA. This is joint work with Martin Bays and Rahiim Moosa. |
| 12:00-13:00 | Question proposal session |
| 13:00-14:30 | Lunch |
| 14:30-17:00 | Question answer session and discussion |
| 17:00- | Pub and dinner |
| 10.30 - 11.00 |
Arrival and coffee | |
| 11.00 - 12.00 |
Isolde Adler(Leeds) | Testing logically defined properties on structures of bounded degree
Abstract: Property testing (for a property P) asks for a given input, whether it has property P, or is "far" from having that property. A "testing algorithm" is a probabilistic algorithm that answers this question with high probability correctly, by only looking at small parts of the input. Testing algorithms are thought of as "extremely efficient", making them relevant in the context of big data. We extend the bounded degree model of property testing from graphs to relational structures, and we discuss testability of first-order logic and monadic second-order logic in this model. This is joint work with Frederik Harwath. |
| 12.00 - 13.00 |
Omar León Sánchez (Manchester) | Conditions for finite-rank types to be isolated in omega-stable theories (and applications) Abstract: In omega-stable theories, isolated types play a crucial role; for instance, they are known to be dense in the (Stone) type space over any set of parameters. It is thus important to understand characterizations of such types in terms of more "algebraic" conditions. One potential condition is that of weak-orthogonality. We will see that weak-orthogonality together with analisability (to a given definable set) imply isolation. While this does not characterize isolated types, it does yield interesting applications in the theory of differential Hopf algebras and, more generally, Hopf-Ore extensions (these notions will be explained). This is joint work with J. Bell and R. Moosa. |
| 13.00 - 14.30 |
Lunch | |
| 14.30 - 15.30 |
Thomas Quinn-Gregson (York) | Homogeneity of Inverse Semigroups Abstract: The concept of homogeneity of relational structures has connections to model theory, permutation groups and combinatorics. A number of complete classifications have been obtained, including those for graphs, semilattices and posets. We may extend this definition by naming an arbitrary structure homogeneous if every isomorphism between finitely generated sub-substructures extends to an automorphism. The key to this extension is that connections with model theoretic properties such as quantifier elimination and $\aleph_0$-categoricity remain. An inverse semigroup $S$ is a semigroup in which every element has a unique inverse, that is, if $a \in S$ then there exists a unique $b \in S$ such that $a = aba$ and $b = bab$, which we denote as $a^{-1}$. It is clear that groups are inverse semigroups, as indeed are semilattices with binary operation of meet. Since an inverse semigroup can be viewed as either a semigroup or as a unary semigroup (a semigroup equipped with a basic unary operation), we obtain two concepts of homogeneity; homogeneous semigroups and homogeneous inverse semigroups. We discuss how the two concepts of homogeneity differ, and how the homogeneity of an inverse semigroup effects its substructure, in particular its semilattice of idempotents and maximal subgroups. We also consider the following question: Given a homogeneous group $G$, which homogeneous inverse semigroups contains $G$ as a maximal subgroup? This will be completely answered for the case where $G$ is finite and where $G$ is Hall's universal locally finite group. |
| 15.30 - 16.00 |
Tea/Coffee | |
| 16.00 - 17.00 |
Jan Dobrowolski (Leeds) | Polish structures
Abstract: The notion of a Polish structure is a purely topological concept (it can be thought of as a Hausdorff topological space X equipped with a continuous action of a Polish group G), which is, however, inspired by model theory. After introducing the basic concepts and explaining in what way some model-theoretic intuitions can be transferred to this setting, I will discuss the main directions of research and open problems related to that subject. |
| 17.00- | Pub and Dinner |
| 10.30-11.00 | Arrival and coffee | |
| 11.00-12.00 | Sonia L'Innocente (Camerino) | Irreducible generalised power series Abstract: A classical tool in the study of real closed fields are the fields K[[G]] of generalised power series (i.e., formal sums with well-ordered support) with coefficients in a field K of characteristic 0 and exponents in an ordered abelian group G. A fundamental result by Berarducci ensures the existence of irreducible series in the subring of K[[G]] consisting of the generalised power series with non-positive exponents. In this work, we are able to prove that all series in this subring can be factorized as a product of irreducibles and a "small" series, first in the case when the group is the additive group of real numbers and then, in the case of arbitrary groups. |
| 12.00-13.00 | Daoud Siniora (Leeds) | Ample Generics and Fraisse limits Abstract: The automorphism group of a countably infinite first order structure becomes a topological group when endowed with the pointwise convergence topology. We then can ask whether the automorphism group has 'ample generics'? By the work of Hodges, Hodkinson, Lascar, and Shelah, and later by Kechris and Rosendal, we can answer this question in the setting of homogeneous structures, or Fraisse limits, by examining the age of the structure. In this talk I will introduce ample generics, and an approach to show their existence for homogeneous structures. Moreover, I will discuss some group theoretic consequences of the existence of ample generics including the small index property, uncountable cofinality, and the Bergman property. |
| 13.00-14.30 | Lunch | |
| 14.30-15.30 | Davide Penazzi (UCLan) | Topological dynamics in the p-adic world Abstract: I will provide a brief introduction on topological dynamics and model theoretic applications, i.e. when the flow is that of a group G acting on its type space S_G(M). I will then focus on the study of the case of SL_2(Q_p). In a joint work with Pillay and Yao we determine its minimal flows, the Ellis group and the universal minimal flow. |
| 15.30-16.00 | Coffee | |
| 16.00-17.00 | Arno Fehm (Manchester) | Existentially definable henselian valuation rings with p-adic residue fields Abstract: Earlier joint work with Sylvy Anscombe gave us an abstract valuation theoretic condition characterizing those fields F for which F[[t]] is existentially 0-definable in F((t)). In this talk I will report on joint work with Sylvy Anscombe and Philip Dittmann in which the study of this condition leads us to some beautiful results on the border of number theory and model theory. In particular, I will present and apply a p-adic analogue of Lagrange's Four Squares Theorem. |
| 17.00- | Pub and Dinner | |
| 10.30 - 11.00 |
Arrival and coffee | |
| 11.00 - 12.00 |
Dario Garcia (Leeds) | Unimodularity unified
Abstract: Unimodularity was defined by Hrushovski, in his proof that a unimodular strongly minimal set is locally modular, thus generalising Zilber’s result thata locally finite strongly minimal set is locally modular. It was claimed in the same paper that unimodularity was equivalent to an a priori weaker notion known later as functional unimodularity. In an attempt to clarify the situation, Pillay and Kestner distinguished two types of functional unimodularity -one for definable sets and one for type-definable sets- and studied their relationship in the context of strongly minimal structures. In this talk, I will present joint with Wagner where we introduce yet another variant called correspondence unimodularity (for types and for definable sets) and present several results describing the relationship between the different concepts. For instance, we show the variants of unimodularity for types coincide in omega-stable theories, and all variants coincide for non-multidimensional theories where the dimension is associated to strongly minimal types (e.g. strongly minimal theories or groups of finite Morley rank). |
| 12.00 - 13.00 |
Franziska Jahnke (Münster) | Henselianity in the language of rings Abstract: (Joint work with Sylvy Anscombe) We consider four properties of a field K related to the existence of (definable) henselian valuations on K and on elementarily equivalent fields and study the implications between them. Surprisingly, the full pictures look very different in equicharacteristic and mixed characteristic. |
| 13.00 - 14.30 |
Lunch | |
| 14.30 - 15.30 |
Erick Garcia Ramirez (Leeds) | Tangent cones and stratifications in RCVF Abstract: I will talk about tangent cones of definable sets in real closed valued fields. A notion of 't-stratification' will be introduce too and I will then explain how a t-stratification of a definable set induces t-stratifications on tangent cones. I will also discuss further interests on this subject. |
| 15.30 - 16.00 |
Tea/Coffee | |
| 16.00 - 17.00 |
Antongiulio Fornasiero (Parma) | Non-elementary lovely pairs
Abstract: We present Lovely Pairs: expansions of a structure M with a predicate P for a "small" set (satisfying certain additional properties). Lovely pairs (a generalization of Poizat's "Belle paires") have been studied (in several contexts and under various names) for a long time. The prototypical cases are the real field R with P denoting the real algebraic numbers, or the complex field C with P a proper algebraically closed subfield. In the classical cases, P has always been an elementary substructure of M (the "elementary" lovely pairs). However, more recent works have considered other kind of structures that resemble lovely pairs, but where P is not an elementary substructure (e.g.: P a dense transcendence basis of R, or P a transcendence basis of C, or P a dense multiplicative subgroup of R* of finite rank). We will show that such "non-elementary" lovely pairs have much in common with the elementary ones. |
| 17.00- | Pub and Dinner |
| 10:30-11:00 | Arrival and Coffee | |
| 11:00-12:00 | Fabrizio Barroero (Manchester) |
Unlikely intersections in families of powers of elliptic curves Abstract: Let E_t be the Legendre elliptic curve of equation Y^2=X(X-1)(X-t). In 2010 Masser and Zannier proved that, given two points on E_t with coordinates algebraic over Q(t), there are at most finitely many specializations of t such that the two points become simultaneously torsion on the specialized elliptic curve, unless they were already generically linearly dependent. One of the main ingredients of the proof is a result of Pila about counting rational points of bounded height on subanalytic surfaces, which is a special case and predates the celebrated Pila-Wilkie theorem. As a natural higher-dimensional analogue, we considered the case of n generically independent points on E_t with coordinates algebraic over Q(t). Then there are at most finitely many specializations of t such that two independent relations hold between the specialized points. Here one needs a more sophisticated counting theorem: relying on results of Pila, we estimate the number of points on some subanalytic surfaces lying on certain linear affine varieties defined by equations with rational coefficients of bounded height. This is joint work with L. Capuano. |
| 12:00-13:00 | Davide Penazzi (Preston) | Existence Theorems for Differential Equations Abstract: We build on the article "Existence Theorems for Systems of Implicit Differential Equations" of Grill, Knebusch and Tressl; where it was shown that given a polynomial differential ideal of R{X_1,...X_n} which is semireal, then there exists an analytic map c from an interval I in R to R^N such that c solves the differential equations of the ideal (i.e. f(c(t))=0 for all f in the ideal and t in I). Our work aims at obtaining similar results for differential equations with initial value conditions (IVPs) and in a more general context: when R is the ring of convergent power series in one variable, i.e. for differential equations with power series coefficients. I will outline the results we have obtained so far and some of the ideas behind them. |
| 13:00-14:30 | Lunch | |
| 14:30-15:30 | Ivo Herzog (Ohio State) | Universal *-regular rings Abstract: Using the model theory of modules, we prove that if (R,i) is a ring with involution, then there exists a morphism $u: (R,i) \to (R',i'),$ with R' a *-regular ring, that is universal, in the sense that any such morphism factors in a unique way through u. Recall that an involution i of a ring R is an anti-automorphism of order 2 and that a von Neumann regular ring R with involution is called *-regular if for all r in R, $i(r)r \neq 0.$ For a commutative ring equipped with the identity involution, the existence of a universal *-regular ring was proved by Olivier. Suppose that L is a split semisimple Lie algebra over a field k of characteristic 0. We will use a result from J.C. Jantzen's thesis together with a theorem of Harish-Chandra to prove that the universal enveloping algebra U(L) may be equipped with an involution i in such a way that the morphism of (U(L),i) into its universal *-regular ring is an embedding. |
| 15:30-16:00 | Coffee | |
| 16:00-17:00 | Vincenzo Mantova (Leeds) | Towards a composition on surreal numbers Abstract: In a recent work with Alessandro Berarducci, we have shown that surreal numbers admit the structure of a field of transseries with a compatible "simplest" derivation. This raises the question whether surreal numbers can also be interpreted as differentiable functions, forming in fact a non-standard Hardy fields closed under composition. I will present the early partial results on this problem, with both positive and negative answers. This is joint work with Alessandro Berarducci. |
| 17:00- | Pub and dinner |
| 10.30-11.00 | Arrival and coffee | |
| 11.00-12.00 | Daniel Wolf (Leeds) | R-macs and Lie coordinatisation Abstract: I will present the notion of an R-mac, a generalisation of the definition of an N-dimensional asymptotic class given by Elwes, Macpherson and Steinhorn in 2007. I will then go over my current efforts to try to adapt the work of Cherlin and Hrushovski on Lie Coordinatisation to the R-mac setting. Joint work with Sylvy Anscombe (UCLan), Dugald Macpherson (Leeds) and Charles Steinhorn (Vassar) |
| 12.00-13.00 | Rosie Laking (Manchester) | Pointed morphisms and the lattice of pp formulas |
| 13.00-14.30 | Lunch | |
| 14.30-15.30 | Edith Vargas-Garcia (Leeds) | An introduction to the reconstruction of the topology on monoids of the rationals Abstract |
| 15.30-16.00 | Coffee | |
| 16.00-17.00 | Alessandro Berarducci (Pisa) | Compact domination, o-minimal homotopy and Pillay's conjectures Abstract: I will report on work Hrushovski, Peterzil, Pillay and Simon on NIP theories and compact domination and develop it further, yielding a new proof of Pillay's conjectures via an o-minimal "nerve theorem". This is joint work with Alessandro Achille |
| 17.00- | Pub and Dinner | |
| 10:30-11:00 | Arrival and Coffee |
| 11:00-12:00 | Talk by Amador Martin-Pizarro. Title: Definable and interpretable groups in pairs of algebraically closed fields. Abstract: We will provide a characterisation of definable groups in a beautiful pair (K, E) of algebraically closed fields: every definable group projects, up to isogeny, onto the subgroup of E-rational points of some algebraic group defined over E with kernel an algebraic group. If time permits, we will discuss the characterisation of interpretable groups. |
| 12:00-13:00 | Question proposal session |
| 13:00-14:30 | Lunch |
| 14:30-17:00 | Question answer session and discussion |
| 17:00- | Pub and dinner |
| 10.30 - 11.00 |
Arrival and coffee | |
| 11.00 - 12.00 |
Dugald Macpherson (Leeds) | Pseudofinite dimension and pseudofinite structures
Abstract: I will discuss recent joint work with Garcia and Steinhorn on a notion of pseudofinite dimension for definable sets in pseudofinite structures, introduced by Hrushovski and Wagner and developed further by Hrushovski. In particular, I will discuss conditions on pseudofinite dimension which ensure that a structure is simple, or supersimple, or stable, or that forking can be characterised by dimension-drop. I will discuss examples, and some possible applications. |
| 12.00 - 13.00 |
Lovkush Agarwal (Leeds) | The 11 Reducts of the Generic Digraph Abstract: Given two structures M and N, we say that N is a reduct of M if, intuitively speaking, N is a less detailed version of M or if N is obtained from M by discarding information. In this talk, I will describe what the reducts of the generic digraph are and time permitting will describe some aspects of the proof. |
| 13.00 - 14.30 |
Lunch | |
| 14.30 - 15.30 |
Lorna Gregory (Manchester) | Interpretation functors, wild algebras and undecidability Abstract: In this talk I will present results about uniform interpretations between module categories over finite dimensional algebras. In particular, I will focus on attempts to prove a conjecture of Prest which says that if a finite-dimensional $k$-algebra is of wild representation type, a notion coming from representation theory, then it uniformly interprets $ Mod-k\langle x,y\rangle$ and hence has undecidable theory of modules. |
| 15.30 - 16.00 |
Tea/Coffee | |
| 16.00 - 17.00 |
Charlotte Kestner (UCLan) | Some model theory of bilinear forms
Abstract: I will give a short introduction to geometric stability theory and independence relations, focussing on the tree properties. I will then introduce one of the main examples for general measureable structures, the two sorted structure of a vector space over a field with a bilinear form. I will state some results for this structure, and give some open questions. |
| 17.00- | Pub and Dinner |
| 10:30-11:00 | Arrival and Coffee |
| 11:00-12:00 | Talk by Pierre Simon |
| 12:00-13:00 | Question proposal session |
| 13:00-14:30 | Lunch |
| 14:30-17:00 | Question answer session and discussion |
| 17:00- | Pub and dinner |
| 10.30-11.00 | Arrival and coffee | |
| 11.00-12.00 | Mike Prest (Manchester) | TBA |
| 12.00-13.00 | Immanuel Halupczok (Leeds) | Families of definable sets in the ordered group $\mathbb{Z}$ Abstract |
| 13.00-14.30 | Lunch | |
| 14.30-15.30 | Ronnie Nagloo (Leeds) | On Transformations in the Painlevé family Abstract |
| 15.30-16.00 | Coffee | |
| 16.00-17.00 | Ivan Tomašić (Queen Mary) | Applications of the twisted theorem of Chebotarev Abstract |
| 17.00- | Pub and Dinner |
| 10.30-11.00 | Arrival and coffee | |
| 11.00-12.00 | Charlotte Kestner (Preston) | NIP categories |
| 12.00-13.00 | Marcus Tressl (Manchester) | Externally definable sets in real closed fields |
| 13.00-15.00 | Lunch | |
| 15.00-16.00 | Tamara Servi (Lisbon) | Quantifier elimination for generalised quasianalytic classes. Abstract |
| 16.00-17.00 | Andres Aranda-Lopez (Leeds) | Supersimple homogeneous 3-graphs |
| 17.00- | Pub and Dinner |