Workshop on model theory of finite and pseudofinite structures

School of Mathematics, University of Leeds

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 events

There will be a social dinner on Wednesday evening (April 11).


Monday, April 9

09.30-09.55 Registration
10.00-11.00 David Evans  (Imperial College)
11.00-11.30Coffee break
11.30-12.10 Zaniar Ghadernezhad  (University of Freiburg)
12.20-13.00 Anja Komatar  (University of Leeds)
14.30-15.30 Zoé Chatzidakis  (École Normale Supérieure)
15.40-16.10 Tingxiang Zou  (Université Claude-Bernard - Lyon 1)
16.10-16.40Coffee break
16.40-17.20 Ivan Tomašić  (Queen Mary College)

Tuesday, April 10

10.00-11.00 Artem Chernikov  (UCLA)
11.00-11.30Coffee break
11.30-12.10 Gabriel Conant  (University of Notre Dame)
12.20-13.00 Sylvy Anscombe  (University of Central Lancashire)
14.30-15.30 Vera Koponen  (Uppsala University)
15.40-16.10 Ove Ahlman  (Uppsala University)
16.10-16.40Coffee break
16.40-17.20 Lotte Kestner  (Imperial College)

Wednesday, April 11

10.00-11.00 Isaac Goldbring  (UC-Irvine)
11.00-11.30Coffee break
11.30-12.10 Aleksander Ivanov  (Silesian University of Technology)
12.20-13.00 Daniel Palacín  (Hebrew University)
14.30-15.10 Martin Bays  (University of Munster)
15.40-16.00Coffee break
16.00-17.00 School Logic Colloquium: Alex Kruckman  (Indiana University)
20.00Conference dinner


  • 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.

  • 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.

  • 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.

  • 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.

  • 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.

  • 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.

  • 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.