Anna Dmitrieva. Extensions of Z-group domains.
Given an algebraically closed field of characteristic zero with a surjective exponential map, it turns out that its kernel has to be elementary equivalent to the group of integers with the constant one. We call such structures Z-groups. As the first order theory of Z-groups is well studied, there exists a thorough classification of its models. However, in our case the kernel is not only a Z-group, but also an integral domain. In this talk we will discuss the general classification of Z-groups, the conditions under which a Z-group can be an integral domain and possible extensions of these Z-group domains.
Mirna Džamonja. Reasonable infinite structures.
A classical conclusion about the interaction between the combinatorial properties of infinite sets or models versus the finite ones, is that there is basically no connection, or if there is then it is an existential connection given by the compactness theorem for the first order logic. For example, one can prove the finite Ramsey theorem by using the infinite one and applying compactness. The connection in the other direction, building from our knowledge of the finite to say something about the infinite, is quite more mysterious.
Yet, there is a trick, which consists of looking at some subclass of infinite structures, where we know exactly how the structure was built from smaller, finite, pieces.For example, in model theory there is quite a lot of knowledge about pseudofinite structures, which are basically ultraproducts of finite structures. There is also a body of knowledge about structures constructed some version of Fraîssé limit, including the Hrushovski constructions.
In this talk we shall discuss another type of 'reasonable' infinite objects, which can be seen as being obtained from a family of finite structures organised along a simplified morass (an object that exists just in ZFC), to obtain an object of size $\aleph_1$.
Francesco Gallinaro. On Some Systems of Equations in Abelian Varieties.
Zilber's Quasiminimality Conjecture predicts that all subsets of the complex numbers that are definable in the language of rings expanded by a symbol for the exponential function are countable or cocountable. Building on Zilber's work, Bays and Kirby have proved that the Quasiminimality Conjecture would follow from the Exponential-Algebraic Closedness Conjecture, also due to Zilber, which predicts sufficient conditions for systems of exponential polynomial equations to be solvable in the complex numbers. Model-theoretic properties of other analytic functions, such as the exponential maps of semiabelian varieties, can be investigated along the same lines: establishing that some systems of equations are solvable would imply a strong tameness property for definable sets in the corresponding structures. In this talk I will discuss these conjectures, the interplay between them, and a specific case in the setting of abelian varieties, using homology and cohomology to show the existence of solutions of a certain class of systems.
Antonino Iannazzo. Differential Spectrum.
In joint work with Ivan Tomasic, we use methods of categorical logic to construct the classifying topos of differential rings and obtain the correct notion of the 'differential spectrum'.
Brian Tyrrell. Further afield and further, a field: some remarks on undecidability.
A theory $T$ is finitely undecidable iff every finite(ly axiomatised) subtheory of $T$ is undecidable. What fields have this property? Is it common? Is it useful? What can a single field sentence even express? I will partially answer these questions, and might convince you to try answering them yourself.