SEEMOD is the South and East of England Model Theory Network, which has meetings at UEA, in London, Oxford, and Cambridge. It is supported by a Scheme 3 grant from the London Mathematical Society. The co-ordinators are Jonathan Kirby and Vahagn Aslanyan.
The talks will take place in Lecture room S0.31, which is next to the doors to the left when you enter the building.
12-1pm arrival and lunch in S1.20.
1-2pm Charlotte Kestner
2-3pm Nicholas Ramsey
3-3:30pm tea and coffee break in S1.20
3:30-4pm Mark Kamsma
4-4:30pm Alexis Chevalier
4:30-5:30pm Vincenzo Mantova
followed by drinks/dinner in Norwich city centre
All talks will be in S0.31.
Alexis Chevalier
Title: Piecewise Interpretable Hilbert Spaces
Abstract: We say that a theory T interprets piecewise a Hilbert space H when H is a direct limit of imaginary sorts of T. For example, when T has a definable measure, the L^2 space associated to this measure is piecewise interpretable. We will state a theorem which gives the fine structure of piecewise interpretable Hilbert spaces which satisfy a certain finiteness condition. The proof of this theorem draws on continuous logic and local stability theory. As an application, we will show how this generalises one of the main theorems in 'Unitary Representations of Oligomorphic Groups', Tsankov, 2011.
Mark Kamsma
Title: The Kim-Pillay theorem for Abstract Elementary Categories
Abstract: In Shelah's classification of first-order theories using combinatorial properties,
the notion of a stable theory is the most well-known. Stable theories are very well-behaved.
The notion of a simple theory is a generalisation of this. The theory of the random graph is the prototypical example of a
simple theory. In particular, one can develop the concept of forking independence in these theories. This is a
generalisation of linear independence in vector spaces, for example. The Kim-Pillay theorem gives us a way to characterise
simple theories based on the existence of a certain independence relation. It states that if an independence relation of a
certain form exists, the theory is simple and that this independence relation coincides with forking independence. For
example, for the random graph this can be applied to the independence relation that says that sets A and B are independent
over C if A ∩ B ⊆ C. We will recall all this in more detail at the start of the talk. After that, we will set up the
category-theoretic framework of AECats, where we can make sense of certain model-theoretic tools and definitions.
Taking inspiration from work of Lieberman, Vasey and Rosický, we can define what an independence relation is in this
framework. This then allows us to formulate a category-theoretic version of the Kim-Pillay theorem. We will finish by
looking at a few examples of where this framework applies, including positive logic and continuous logic. In particular,
we will recover the original Kim-Pillay theorem as one of these applications.
Nicholas Ramsey
Title: Kim-independence over arbitrary sets
Abstract: Kim-independence is the independence notion corresponding to dividing at a generic scale. In our work with Itay Kaplan, we showed Kim-independence is an independence notion that succeeds in extending almost all of the basic properties of non-forking independence in a simple theory to the broader context of NSOP_1 theories, incorporating many new examples of model-theoretic interest. These results, however, had one important caveat: it was necessary to work over models, because over models, in an arbitrary theory, one has a good notion of generic sequence, coming from, e.g. finitely satisfiable or invariant types, and it makes sense to talk about dividing with respect to such sequences. Over sets the situation becomes more complicated. Assuming every type has a global non-forking extension, we were able, with Jan Dobrowolski and Byunghan Kim, to lift the theory to arbitrary sets. We will survey this work, and its subsequent refinements (with Artem Chernikov and Byunghan Kim), as well as discuss the general question of what it might mean for Kim-independence to exist over arbitrary sets, without making additional assumptions on the behavior of forking.