Time and Venue
The workshop will start on April 9 at 10.00, and ends on April 11 with the conference dinner.
All talks will take place at MALL, Level 8 of the School of Mathematics.
Coffee breaks will be in the space outside the common room on Level 9.
Social eventsThere will be a social dinner on Wednesday evening (April 11) at 8pm. The dinner will take place at the Comptoir Libanais.
Trinity Shopping Centre,
Inside EveryMan Cinema,
Monday, April 9
- 09.30-09.55 Registration
- 10.00-11.00 David Evans. (Imperial College)
Omega-categorical structures and Macpherson-Steinhorn measurability.
Partially answering a question of Elwes and Macpherson, we show that some of Hrushovski’s non-locally modular omega-categorical, supersimple structures are not MS-measuarable. The proof makes use of a higher amalgamation property in MS-measurable structures which follows from the Goldbring-Towsner version of the Hypergraph Removal Lemma. It would be interesting to know whether there is a more elementary proof.
- 11.00-11.30 Coffee break
- 11.30-12.10 Zaniar Ghadernezhad. (Imperial College - University of Freiburg)
Building countable generic structures with the algebraic closure property.
In this talk we introduce a new method of building countable generic structures with the algebraic closure property. This method generalizes the well-known Hrushovski constructions of building generic structures using pre-dimension functions and captures certian features of them. Using this method it is particularly easy to build a generic structure that its theory is not simple. Time permitting, we investigate TP_2 and NSOP of the non-simple generics that are obtained from this method.
- 12.20-13.00 Anja Komatar. (University of Leeds)
Ordering property and partial orders.
In structural Ramsey theory, often rather than considering a class of structures, we consider an ordered class. We extend the language of certain structures to include an order relation, and consider a class of ordered structures, that is , structures together with a total order of the points of the structures. This restricts the embeddings and isomorphisms in the ordered class to embeddings and isomorphisms that respect the total orders on structures. We will give an example showing that the class of partial orders is not Ramsey. It illustrates why focusing on specific ordered classes is necessary to obtain Ramsey classes. We will then give an example of an ordered class with Ramsey and ordering properties and explore the related topological dynamics results.
- 13.00-14.30 Lunch break
- 14.30-15.30 Zoé Chatzidakis (CNRS - École Normale Supérieure)
Non-existence of prime models of pseudo-finite fields.
Let F be a non-principal ultraproduct of finite fields, T its theory, and A a subfield of F which is relatively algebraically closed. It is known that any two models of T which contain A as a relatively algebraically closed subfield are elementarily equivalent over A.
Theorem: If A is not pseudo-finite, then T has no prime model over A.
The proof of this result is fairly easy when A is countable: one just uses that prime models over A are atomic. The proof when A is uncountable is more involved, as it involves constructing 2|A| non-isomorphic models of T which are of transcendence degree 1 over A. We also discuss the existence or non-existence of κ-prime models of T (i.e., κ-saturated models of T containing A and which A-embed into any κ-saturated model of T containing A), for regular uncountable κ such that κ <κ=κ.
- 15.40-16.20 Tingxiang Zou (Université Claude-Bernard - Lyon 1)
Coarse dimension and pseudofinite difference fields.
Consider the ultra-products of finite difference fields. We will show that when the field grows quickly enough compared to the automorphism, the definable sets will establish some nice behaviors. Namely, the coarse pseudofinite dimension of definable sets with respect to the size of the field will be integer-valued and continuous. We will also make a partial connection between the coarse dimension and an algebraic notion in difference algebra, that is the transformal transcendence degree.
- 16.20-16.50 Coffee break
- 16.50-17.30 Ivan Tomašić (Queen Mary College)
Graphons, Tao's regularity and difference polynomials.
We show that Tao's spectral proof of the algebraic regularity lemma for asymptotic classes has a very natural formulation in the context of graphons. We apply these techniques to study expander difference polynomials over fields with powers of Frobenius.
Tuesday, April 10
- 10.00-11.00 Gabriel Conant (University of Notre Dame)
Pseudofinite groups and VC-dimension.
We develop local NIP group theory in the context of expansions of pseudofinite groups. This includes a local type-definable connected component, uniqueness of the pseudofinite counting measure as a left-invariant measure on NIP formulas, and local generic compact domination. These results generalize to a local notion of NIP formulas with "fsg", and can also be used to prove arithmetic regularity lemmas for subsets of finite groups whose family of left translates has uniformly bounded VC-dimension.
- 11.00-11.30 Coffee break
- 11.30-12.10 Sylvy Anscombe (University of Central Lancashire)
Multidimensional exact classes, examples and questions.
We study classes of finite structures obeying a very tight constraint on the sizes of definable sets, refining the concept of a multidimensional asymptotic class. We study some examples and discuss open questions. This is joint work with Dugald Macpherson, Charlie Steinhorn, Daniel Wolf.
- 12.10-14.30 Lunch break
- 14.30-15.30 Vera Koponen (Uppsala University)
Simple homogeneous structures: the ternary case.
I will give an overview of recent results about ternary simple homogeneous structures (where ternary means that no relation symbol has arity greater than 3). The results give information about topics like SU-rank, triviality of dependence, consequences of being finitely constrained and a connection between the nature of constraints and definable equivalence relations. If M is homogeneous then C is a constraint of M if C does not belong to Age(M) but every proper substructure of C belongs to Age(M). M is finitely constrained if it has only finitely many constraints.
- 15.40-16.20 Ove Ahlman (Uppsala University)
Non-amalgamation classes and homogenizability
Homogeneous structures are highly symmetric, where any finite partial isomorphism can be extended to an automorphism. An old theorem by Fraïssé shows that each homogeneous structure can be matched with a unique (under some closure assumptions) class of finite structures which satisfies a structure-amalgamation property. To better understand homogeneous structures one can study the homogeneous structures which are reducts of non-homogeneous structures. These non-homogeneous structures are called homogenizable structures. Just like the homogeneous structures could be studied using classes of finite structures, so can the homogenizable structures, however these classes do not satisfy the amalgamation property.
In this talk I will present the current state of research on homogenizable structures. This will be viewed through their corresponding classes of finite structures which satisfy certain weaker versions of structure-amalgamation.
- 16.20-16.50 Coffee break
- 16.50-17.30 Charlotte Kestner (Imperial College)
Some model theory of bilinear forms.
I will give an introduction to the model theory of the structure (V, F, β) where V is a vector space over a finite field F, and β a bilinear form. In particular I will talk about different notions of independence on this structure.
Wednesday, April 11
- 10.00-11.00 Isaac Goldbring (UC-Irvine)
The ubiquity of pseudofinite metric structures.
In this talk, we show that it is much easier for a metric structure to be pseudofinite than for a classical structure. In particular, we prove that a perfect relational metric structure is always pseudofinite. I will discuss versions of this result for not necessarily relational languages. This is joint work with Bradd Hart.
- 11.00-11.30 Coffee break
- 11.30-12.10 Aleksander Ivanov (Silesian University of Technology)
Decidability of continuous theories of operator expansions of finite dimensional Hilbert spaces.
We study continuous theories of classes of finite dimensional Hilbert spaces expanded by a finite family (of a fixed size) of unitary operators. Infinite dimensional models of these theories are called pseudo finite dimensional dynamical Hilbert spaces. Our main results connect decidability questions of these theories with the topic of approximations of groups by metric groups.
- 12.20-13.00 Daniel Palacín (Hebrew University)
Product-free sets, a nonstandard approach.
Babai and Sós asked whether there exists a constant c>0 such that every finite group of order n has a product-free subset of size at least c·n: that is, a subset X that does not contain three elements x, y and z with xy = z. Gowers proved that the answer is no. In this talk I will give a model-theoretic interpretation of the existence of a large product-free set in the nonstandard finite setting, and obtain an alternative proof of the aforementioned result of Gowers.
- 13.00-14.30 Lunch break
- 14.30-15.30 Martin Bays (University of Munster)
General position, minimality, and geometry.
A (pseudo)finite subset of an algebraic variety is said to be in general position if it has small intersection with any proper subvariety. This can be seen as a notion of minimality, and so yields a geometry. I will describe the role this plays in some problems related to the Elekes-Szabó theorem. This is joint work with Emmanuel Breuillard.
- 15.30-16.00 Coffee break
- 16.00-17.00 Alex Kruckman (Indiana University)
The convergence of three notions of limit for finite structures.
The (Rado) random graph R arises as a limit of finite graphs in at least three ways. First, R is the Fraïssé limit of the family of all finite graphs, the unique countable graph up to isomorphism which is universal and homogeneous for finite graphs. Second, it is pseudofinite: its first-order theory Th(R) is the logical limit (e.g. via an ultraproduct), of the first-order theories of a family of finite graphs, and in fact R the unique countable model of this theory up to isomorphism. And third, it admits a random construction (the Erdős-Rényi construction on countably many vertices), which is naturally the limit of a uniform family of random constructions of finite graphs (the Erdős-Rényi construction on n vertices, for all n ). Stronger, these random constructions of finite graphs witness pseudofiniteness, in the sense that every sentence in Th(R) is true with arbitrarily high probability in sufficiently large random finite structures.
I will explain how the strong agreement of these three limit notions (as in the case of the random graph) can be viewed as a consequence of higher-dimensional analogues of the amalgamation property in classes of finite structures. And in recent joint work with Cameron Hill, I will show when we have this sort of strong agreement, an analysis of the "random construction" (formally an invariant probability measure on the space of countable structures) can shed light on the model-theoretic properties of the limit. In particular, I will describe a generalization of de Finetti's theorem which is a useful tool in understanding invariant measures.