UEA, London and Oxford
First meeting: 5th February 2016 at UEA
12:30-1pm Meet in the UEA Maths Common room: S1.20
1pm Ð 1:45 Mirna Dzamonja (UEA) room S3.05
2:00 Ð 2:45 Levon Haykazyan (Oxford) S0.31
2:45 Ð 3:30 Tea in room S1.20
3:30 Ð 4:15 Tim Zander (UEA) S3.05
4:15 Ð 5pm Ivan Tomasic (QMUL) S3.05
We plan to go for a drink and a meal afterwards.
All are welcome. There is some money for travel expenses, especially for PhD students. If you would like to attend, particularly if you want to claim travel expenses or you want accommodation, please email jonathan.kirby@uea.ac.uk.
Mirna Dzamonja (UEA): Logical perspectives of the theory of graphons
Abstract: Since 2006 the research of large networks and several other aspects of graph theory has been much influenced by the concept of graphon introduced by Lovasz and Szegedy. Graphons are certain limits of sequences of finite graphs and they are uncountable. This is in contrast with the well known class of countable limit graphs obtained as Fraõse limits of a class of finite structures. The limit graphon is a limit of a convergent sequence of finite graphs in the graphon space, which is the completion of the metric space consisting of the set of finite graphs endowed with the cut metric. The graphon space is a compact space and this fact is equivalent to strong forms of the Szemeredi Regularity Lemma from graph theory. However, graphons can also be interpreted as certain ultraproduct spaces. Elek and Szegedy developed an integral and measure theory of the ultraproduct of finite sets and obtained the Hypergraph Regularity Lemma and the Hypergraph Removal Lemma which form the basis of the graphon theory as a consequence. This work was taken forward by Aroskar and Cummings who have extended the methods of convergent sequences in ultraproducts to work for any relational structure and, furthermore, any fixed universal theory. This in particular means that they can deal with structures that omit a certain fixed finite substructure, for example the triangle-free graphs.
Our objective in the work in progress is to use the general theory of ultraproducts, to extending the methods of the theory of graphons in two directions: to more structures, and to more cardinals. Both of these aims are in the realms of set theory, model theory and the fine interaction between the two.
Levon Haykazyan
(Oxford): Quasiminimal and excellent groups.
Abstract: Quasiminimality and excellence are generalisations of strong minimality. It is natural to ask how much of the geometric stability theory is valid in these contexts. In this talk we will look at the question whether every quasiminimal and excellent group is commutative.
Tim Zander (UEA): Higher Amalgamation in stable theories
Abstract: Amalgamating types is a essential tool in stable and simple theories. For example the reason to consider imaginaries in stable theories was exactly done to obtain uniqueness of 2-Amalgamation over algebraic closed sets. The Independence Theorem in simple theory is nothing more than 3-Amalgamation over models. But n-Amalgamation for n > 3 can fail in stable theories. But if one expands the theory by certain finite covers we can still obtain n-Amalgamation over the empty set for every n. Now if forking is easy enough (SU-rank 1) this directly translates to Amalgamation over parameters. In general this does not need to hold.
Ivan Tomasic (QMUL): The fundamental group of a difference scheme
Abstract: The fundamental group of a difference scheme classifies its etale difference coverings. This study combines geometric insights from algebraic topology and geometry, recent achievements in difference algebra, as well as the study of symbolic dynamics. We will emphasise its model-theoretic interpretation as a space of `types' and discuss various applications. A number of these topics stems from joint work with Michael Wibmer.