University of Leeds
Schedule
Notes
Some notes of the talks are available here. Please read the disclaimer in the first pages.
Wednesday, 25 January
10:00-11:00
Amador Martin-Pizarro - Equationality and chain conditions (Part 1/3)
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A complete first-order theory is equational if every definable set is a Boolean combination of instances of equations, that is, of formulae such that the family of finite intersections of instances has the descending chain condition. Equationality, as introduced by Srour and later studied together with Pillay, is a strengthening of stability. Typical examples of equational theories are the theory of an equivalence relation with infinite many infinite classes, completions of the theory of modules over a fixed ring, algebraically closed fields of some fixed characteristic, as well as differentially closed fields of characteristic 0 and separably closed fields of finite imperfection degree. So far, the only known "natural" example of a stable non-equational theory is the free non-abelian finitely generated group, as recently shown by Sela. Proving that a particular stable theory is equational is nontheless far from obvious, in general.
We will exhibit some of the properties of equational theories, as well as illustrate why some of the aforementioned examples are equational. If time permits, we will present a recent result, together with Martin Ziegler, on the equationality of the theory of proper pairs of algebraically closed fields. In characteristic 0, we show that definable sets are Boolean combination of certain definable sets, which are Kolchin-closed in the corresponding expansion DCF0.
11:30-12:10
Isabel Müller - A 2-ample geometry of finite rank
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In an attempt to classify the geometry of strongly minimal sets, Zilber had conjectured them to split into three different types: Trivial geometries, geometries which are vector space like and those which are field like. Hrushovski later refuted this conjecture by introducing a clever construction that had been modified and used a lot ever since. His counterexample to Zilbers conjecture provided a structure, which was not one-based, so could not be of trivial or vector space type, but nevertheless it forbade a certain point-line-plane configuration, which is present in fields. Hrushovski called that property CM-triviality and later Pillay, with some corrections by Evans, defined a whole hierarchy of new geometries, on which’s base we find non-one basedness (1-ample) and non-CM-triviality (2-ample) and on which’s top we find fields, being n-ample for all n.
Recently, Baudisch, Pizarro and Ziegler and independently Tent have provided examples proving that this ample hierarchy is strict. While their examples are omega-stable of infinite rank, it remained open if one can find geometries of finite rank which are at least 2-ample but do not interpret a field. In this talk we will now introduce an almost strongly minimal structure which is 2-ample, but not 3-ample, using a Hrushovski-like construction. This is a joint work with K. Tent.
12:20-13:00
Daoud Siniora - Hall's universal group and ample generics
Philip Hall's universal locally finite group is the Fraisse limit of the class of all finite groups. We use some of the techniques which emerged from the work of Hodges, Hodkinson, Lascar and Shelah on the small index property, and later the work of Kechris and Rosendal on generic automorphisms of homogeneous structures, to show that Hall's group admits ample generics.
14:30-15:30
Boris Zilber - Model theory and unabelian geometry
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I will start with a motivation of what algebraic and model-theoretic properties an algebraically closed field of characteristic 1 is expected to have. Then I will explain how these properties can be obtained by Hrushovski’s construction and then formulate very precise axioms that such a field must satisfy. The axioms have a form of statements about existence of solutions to systems of equations in terms of a ’multi-dimansional’ valuation theory and the validity of these statements is an open problem to be discussed.
16:00-16:40
Millette Tseelon-Riis - The Generalised Shift Graph
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Erdős defined the Shift Graph in 1968, and Avert, Luczac and Rodl expanded on this definition in 2014. I will prove some new results about a selection of these graphs.
16:50 - 17:30
Sebastian Eterovic - A Schanuel property for the j function
I will present a generic transcendence property for the modular j function in the spirit of Schanuel’s conjecture. Such a result was already proved for the exponential function (Bays-KirbyWilkie 2010) using differential algebra, pregeometries and o-minimality. I will show that, thanks to the Ax-Schanuel theorem for j (Pila-Tsimerman 2015), the methods used for the exponential can be adapted to work for j.
17:50 - 18:30
Alexander Jones - A semantic approach to pathological satisfaction classes
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Satisfaction classes, defined over first order Peano Arithmetic, are sets of (Gödel codes of) sentences that follow Tarskian compositional rules for truth, given by the theory CT −. For example, if S is a satisfaction class, φ ∈ S, and ψ ∈ S, then φ ∧ ψ ∈ S. Whilst these are good definitions of truth for the standard sentences of arithmetic, these satisfaction classes can be thought of as inadequate definitions of truth for nonstandard models of arithmetic. There is a satisfaction 2 class S which contains the sentence (0 = 1∨(0 = 1∨· · ·∨(0 = 1). . .)) where there are a nonstandard number of disjuncts 0 = 1. These sentences are known as pathological. I will introduce a new notion of what it means to be pathological using Robinson’s external notion of semantic entailment for nonstandard sentences. I will then prove that closing satisfaction classes under this notion results in a non-conservative theory of truth CT− + I∆0, which arises by similarly natural proof-theoretic considerations due to results by CieÅ›liÅ„ski. This leads to a new semantic conception of the theory CT− + I∆0 as well as a minimal adequacy definition on truth for nonstandard models of arithmetic.
Thursday, 26 January
10:00-11:00
Amador Martin-Pizarro - Equationality and chain conditions (Part 2/3)
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Part 2 of the ongoing minicourse. See abstract above, in part 1.
11:30-12:10
Christian d’Elbée - Minimal (unstable) expansions of (Z, +, 0)
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The two structures (Z, +, 0, <) and (Z, +, 0, |p) (where x|py ⇐⇒ vp(x) ≤ vp(y)) are strict expansions of (Z, +, 0). Take (Z, +, 0, . . .) a reduct of (Z, +, 0, <) which is a strict expansion of (Z, +, 0) then (Z, +, . . .) defines <, possibly with parameters. The same holds for (Z, +, 0, |p). In that sense, they are minimal expansions of (Z, +, 0). G. Conant proved in [1]: (Z, +, 0, <) is a minimal expansion of (Z, +, 0). We propose another proof of the result of Conant, as well as a proof that (Z, +, 0, |p) is a minimal expansion of (Z, +, 0). In [2] is proven that (Z, +, 0) has no stable dpminimal strict expansions. As both (Z, +, 0, <) and (Z, +, 0, |p) are dp-minimal, it suffice to consider an unstable reduct (Z, +, 0, . . .). Besides, with the cost of getting in a saturated model, we will only need to study 1-dimensional definable sets. While the result of Conant will follows pretty quickly with this approach, the case of (Z, +, 0, |p) will need a good understanding of 1-dimensional definable sets and their arithmetic. This is joint work with E. Alouf.
[1] G. Conant. There are no intermediate structures between the group of integers and Presburger arithmetic. May 2016. Available at https://arxiv.org/pdf/1603.00454.pdf.
[2] G. Conant, A. Pillay. Stable groups and expansions of (Z, +, 0). January 2016. Available at https://arxiv.org/ pdf/1601.05692.pdf.
12:20-13:00
Nicholas Wentzlaff-Eggebert - A new class of analytic (Zariski-) structures
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We will examine Gelfand-Naimark duality from the point of view of generalized Zariski Geometries. In particular we will show quantifier elimination for compact Hausdorff spaces in their natural Zariski language. Moreover we will examine questions of definability in perfectly normal topological spaces, and show how a tweak in the language may improve their stability properties.
14:30-15:30
Zoé Chatzidakis - Difference fields and applications to algebraic dynamics
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In this talk I will speak about algebraic dynamics and how one can apply the model
theory of difference fields to solve some problems. All definitions will be given.
16:00-16:40
Petr Glivický - Linear fragments of Peano arithmetic and discretely ordered modules
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We give a survey of our results (partialy a joint work with P. Pudlák) on linear arithmetics – linear fragments of Peano arithmetic (PA). For a cardinal κ, the κ-linear arithmetic LAκ is a theory extending Presburger arithmetic (in the language (0, 1, +, <)) by κ unary functions of multiplication by distinguished (nonstandard) elements (called scalars) and containing the full scheme of induction for its language.
We give a classification of all definable sets in models of LA1 and, as a corollary, show that LA1 is a tame theory – model complete, decidable, NIP, having recursive nonstandard models...
On the other hand we prove that LA2 (as well as any LAκ with κ > 2) is model theoretically wild. As a manifestation of this fact we show that there is a model M of LA2 in which an infinitely large initial segment of Peano multiplication (i.e. a multiplication · such that (M, ·) is a model of PA) is 0-definable. Consequently, the theories LAκ with κ > 1 are not model complete nor NIP.
Each model of a linear arithmetic naturally corresponds to a discretely ordered module over the ordered ring generated by the scalars. Our results on LA2 thus yield a non NIP ordered module answering negatively the question of Chernikov and Hils whether all ordered modules are NIP.
16:50 - 17:30
Alfonso Ruiz Guido - Towards a model theory of algebraic stacks
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Groupoids of algebraic varieties correspond to generalized imaginary sorts (in the sense of Hrushovski) in the theory of Zariski Geometries. Such a groupoid is an atlas for an algebraic stack, is there a Zariski geometry that captures the geometry of the algebraic stack? I will report some progress on this question.
17:50 - 18:30
Panel session
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An opportunity to ask questions to established researchers in logic: Dugald Macpherson, Amador Martin-Pizarro, Jaroslav NešetÅ™il, and Boris Zilber.
Friday, 27 January
10:00-11:00
Amador Martin-Pizarro - Equationality and chain conditions (Part 3/3)
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Part 3 of the three part minicourse. See abstract above, in part 1.
11:30-12:10
Nadav Meir - Infinite products of ultrahomogeneous structures
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We will define the “lexicographic product” of two structures and show that if both structures admit quantifier elimination, then so does their product. As a corollary we get that nice (model theoretic) properties such as (ultra)homogeneity, stability, NIP and more are preserved under taking such products.
It is clear how to iterate the product finitely many times, but we will introduce a new infinite product construction which, while not preserving quantifier elimination, does preserve (ultra)homogeneity. As time allows, we will use this to give a negative answer to the last open question from a paper by A. Hasson, M. Kojman and A. Onshuus who asked ‘Is there a rigid elementarily indivisible* structure?”
* A structure M is said to be elementarily indivisible structure if for every colouring of its universe in two colours, there is a monochromatic elementary substructure N of M such that N is isomorphic to M.
12:20-13:00
Jean Cyrille Massicot - Around Hrushovski’s stabilizer theorem.
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A K-approximate subgroup is a subset X of G such that X2 is contained in at most K translates of X. The Stabilizer Theorem constructs a subgroup H of the group generated by X, with H type-definable and of bounded index. This allows not only for classification resultst in the study of approximate subgroups, but also for explicit construction of an X00.
However, this construction uses an expansion L∗ of the language L of approximates subgroups. We will show how to obtain a subgroup H which is L-type definable, using Udi’s theorem together with a result of Schlichting about families of subgroup, in a model-theoretic version of Ben Yaacov and Wagner, and an old theorem of Beth on definability.
14:30-15:30
Jaroslav Nesetril - Structural limits and dichotomy of sparse classes
Structural limits are extension of various graph limits. Yet on the full generality one can obtain limiting distributions. The existence of nice limit objects (called modelling) is then equivalent to sparse dichotomy (nowhere- vs somewhere-dense). This is a joint work with Patrice Ossona de Mendez (Paris and Prague).
16:00 - 16:40
Thomas Kirk - Ellis groups of type spaces concentrating on a group G, and their relation to G/G00
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This talk will focus on some recent results regarding the interaction between topological dynamics and model theory. I will recap the definition of G00 and state various results regarding the conditions for which it exists. I will then define the notion of a G-flow, followed by Ellis Semigroups and their closed left ideals, showing further that minimal such ideals exist. To link these notions with model theory, I will show how to construct the semigroup from G acting on it’s type space, finding an interesting interaction between the minimal ideals, which we call Ellis Groups, and the model theoretic invariant quotient G/G00.
16:50 - 17:30
Jan Hubicka - Ramsey properties of Hrushovski classes
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A class K of finite structures is Ramsey if for every choice of A and B in K there exists C in K such that for every coloring of its substructures isomorphic to A with 2 colors there exists an isomorphic copy of B in C where all copies of A are monochromatic. It is well known that under mild assumptions Ramsey classes are amalgamation classes but converse does not hold. Majority of amalgamation classes can be however be expanded into a Ramsey class by additional relations such that every structure has only finitely many possible expansion. Hrushovski amalgamation construction provides an interesting example where such expansion does not exist. We discuss the example and show that Ramsey expansion still exists and it can be shown to be, in a certain sense, optimal. This is joint work with Jaroslav Nesetril and David Evans.