Abstracts
M. Aschenbrenner: Residual properties of 3-manifold groups
Given a property P of groups, a group G is called residually P if for every non-identity element g of G there exists a normal subgroup N of G not containing g such that G/N has property P. One says that G is virtually P if there exists a finite index subgroup of G which is P. Since the 1980s it has been known that fundamental groups of Haken 3-manifolds are residually finite. I want to explain the result that every 3-manifold group, for all but finitely many primes p, is virtually a residually finite p-group. (Joint work with Stefan Friedl, University of Warwick.)
S. Azgin: Extremal Valued Fields
A valued field (K, v) is extremal if the values of every polynomial in several variables attain a maximum when evaluated at tuples from the valuation ring. This is a first-order property for valued fields and it was expected that valued fields whose elementary theory is understood via the Ax-Kochen and Ershov theorem would be extremal (e.g. henselian valued fields of residue characteristic 0). We illustrate examples where this expectation fails and provide an almost-complete classification of extremal valued fields. Joint with F-V. Kuhlmann, F. Pop.
F. Benoist: Semiabelian varieties over separably closed fields and maximal divisible subgroup
I will study the exactness of the functor which associates to a semiabelian variety G over a separably closed non perfect field K the maximal divisible subgroup p^\inftyG(K), and relate this question to some model theoretic properties of this group. This is joint work with E. Bouscaren and A. Pillay.
S. Carlisle: Model theory of real trees
A real tree is a metric space such that between any two points there is exactly one arc, and that arc is a geodesic segment. For example, the plane with the Paris metric is a real tree, as is the asymptotic cone of a hyperbolic group. I will summarize results about the continuous theory of real trees. For example, the class of real trees is axiomatizable in an appropriate continuous signature, and its theory has a model companion which is complete and stable, but not superstable.
Z. Chatzidakis: Canonical bases of types of finite SU-rank
I will speak about the CBP (canonical base property) for types of finite SU-rank. This property first appears in a paper by Pillay and Ziegler, who show that it holds for types of finite rank in differentially closed fields of characteristic 0, as well as in existentially closed difference fields. It is unknown whether this property holds for all finite rank types in supersimple theories. I will first talk about a reduction of the problem (which allows one to extend the Pillay-Ziegler result to existentially closed fields of any characteristic), and then derive some consequences of the CBP, in particular the UCBP, thus answering a question of Moosa and Pillay. If time permits, I will show an application of these results to difference fields.
Vinicius C.L.: Questions on Euler characteristics and definable endofunctions
In my research at Urbana with Lou van den Dries, I focused on Grothendieck semirings of categories of definable sets and functions, and on a related property of definable endofunctions. We also studied division rings whose vector spaces are pseudofinite. In this talk, I present some further questions about those subjects.
I. Goldbring: Rosiness and the Urysohn Sphere
In this talk, I will discuss joint work with Clifton Ealy on developing the theory of thorn-forking and rosiness in continuous logic. After some preliminary definitions, I will spend much of the talk proving that the theory of the Urysohn sphere is rosy (with respect to finitary imaginaries). We do this by showing that: 1) The Urysohn sphere is real rosy. 2) The Urysohn sphere admits weak elimination of finitary imaginaries. 3) Any real rosy theory which admits weak elimination of finitary imaginaries is rosy with respect to finitary imaginaries. It is known that the theory of the Urysohn sphere is not simple; in fact there is no known example of a simple, unstable essentially continuous theory. Thus, the Urysohn sphere is the first example of an unstable essentially continuous theory which admits a nice notion of independence.
A. Gunaydin: The real field with the rational points of an elliptic curve (joint work with P. Hieronymi)
We consider the model theoretic structure (R,E), where R is the real field and E is the group of rational points of an elliptic curve. We axiomatize this structure and show that it eliminates quantifiers up to existential formulas. As a by-product, we also prove that it has o-minimal open core, which is to say that all the open sets definable in (R,E) are already definable in the real field.
C.W. Henson: On separable categoricity of metric structures
We consider bounded metric structures for a countable continuous signature. First, some aspects of the underlying theory of separably categorical metric structures, including analogues of the classical Ryll-Nardzewski theorem, will be reviewed. Then several examples will be discussed, including some new ones obtained by performing an analogue of the Fraisse construction. Finally a number of open problems will be described.
P. Hieronymi: Defining the integers in expansions of the real field by a closed discrete set
Let D be a closed and discrete subset of the real number and f be a function mapping D^n onto a somewhere dense subset of the real numbers. Then the real field expanded by the function f defines the integers. As an application, we get that the real field expanded by two distinct cyclic multiplicative subgroups defines the integers.
H. Kikyo: Instability and generic automorphisms (joint work with A. Tsuboi)
For many unstable theories, the class of the generic automorphisms is not elemetary. Usually, this fact is proved as follows: First give a certain automorphism $\sigma_0$ and then show that the class of the generic automorphisms extending $\sigma_0$ is not elementary.
For the theory of the random graphs, given any automorphism $\sigma_0$, the class of the generic automorphisms extending $\sigma_0$ is not elementary.
For any theory with the strict order property, given any automorphism $\sigma_0$ such that there is $x$ with $x < \sigma_0^m(x)$ for some natural number $m \geq 1$, the class of the generic automorphisms extending $\sigma_0$ is not elementary. Here, $<$ is a definable transitive order.
However, there is a theory with the strict order property such that for many automorphisms $\sigma_0$, the class of the generic automorphisms extending $\sigma_0$ is elementary.
P. Kowalski: Algebraic independence in positive characteristic
We discuss versions of Schanuel Conjecture over a non-Archimedean field of positive characteristic. The role of the exponential map is played by an additive power series, for example the exponential map of a Drinfeld module.
K. Krupinski: Small compact $G$-groups and $G$-rings
A compact $G$-group [$G$-ring] is a pair $(H,G)$ where $G$ is a Polish group acting continuously on a compact group $H$ [ring $H$] as a group of automorphisms. We say that it is small if there are countably many orbits on $H^n$ for every $n \in \omega$. These objects belong to a much more general class of small Polish structures that I defined and studied from the model theoretic point of view. In particular, I introduced certain topological independence relation that has similar properties to those of forking independence in simple theories, and defined a counterpart of Lascar U-rank, which I call NM-rank. The structures of ordinal NM-rank are called nm-stable.
After a short introduction on Polish structures, I will discuss conjectures, theorems, examples and counterexamples concerning the structure of small compact [nm-stable] $G$-groups and $G$-rings (some of them obtained together with F. Wagner). I will also recall what we do know about the structure of groups and rings in the particular case of small profinite structures introduced by Newelski. The whole topic is related to the studies of the structure of simple, $\omega$-categorical groups and rings.
J. Maříková: The o-minimal Hauptvermutung
Recently M. Shiota announced the proof of the o-minimal Hauptvermutung: If two simplicial complexes are definably homeomorphic in some o-minimal field, then they are piecewise linearly homeomorphic. We shall sketch Shiota's proof and discuss some consequences.
C. Miller: Expansions of o-minimal structures on the real field by trajectories of linear vector fields
Let $\mathfrak R$ be an o-minimal expansion of the field of real numbers that defines the restriction of complex exponentiation to a bounded nonempty open box. Let $\mathcal G$ be a collection of images of maximal solutions of vector fields given by linear transformations on various $\mathbb R^n$. Then the expansion of $\mathfrak R$ by the elements of $\mathcal G$ either is d-minimal or defines $\mathbb Z$, and the expansion of $(\mathfrak R,e^x)$ by the elements of $\mathcal G$ either is o-minimal or defines $\mathbb Z$. This is obtained via a more detailed statement about expansions of the real field by elements of $\mathcal G$.
A. Mofidi: Quantification via second order operators (joint work with S.-M. Bagheri)
(Contributed Talk)Metric model theory is a logical framework for studying metric structures. Similarly, integral logic is a logical framework for measure structures. In these logics supremum and integral are respectively used as quantifiers. In general, quantifiers are second order function operating on the first order ones. In this paper, we set up a general logical framework, containing the above ones, for studying mathematical structures equipped with some sort of quantifier. Indeed, we regard quantifiers as part of the structures by interpreting them as second order functions. This enables us to impose arbitrary axioms describing their properties. In particular, we axiomatize the quantifiers supremum and integral. We also gives some new examples of mathematical structures which are equipped with a quantifier. We also investigate some basic model theoretic properties such structures in this logic.
R. Moosa: A strengthening of internality
Working in stable theories I will introduce a natural strengthening of the notion of internality that is inspired by Moishezon morphisms in complex geometry. As an example of what this might be useful for I will discuss the "uniform" canonical base property and its consequences for finite rank types. I will also speculate on the existence of a "coreduction" to the nonmodular minimal types.
K. Peterzil: Central extensions of definable groups in o-minimal structures (joint work with Hrushovski and Pillay)
Let G be a definable group in an o-minimal expansion of a real closed field, and let H be a definable central extension of G. We prove, under various additional assumptions, that H can be interpreted in the pure group structure of G (e.g. this is the case if G is semisimple, namely has no infinite normal abelian subgroup).
One of the corollaries is that every definably compact, definably connected group H in an o-minimal strucuture is elementarily equivalent to a compact real Lie group. This in turn implies that such a group H is the almost direct product of its center and its (definable) commutator subgroup.
D. Pierce: Interactions of rings
The derivations of an associative ring compose a Lie ring, as well as a module over the associative ring. I shall discuss various results, some due to Özcan Kasal, inspired by consideration of the interaction between these two kinds of rings. I shall observe how associative rings and Lie rings are unique in possessing a certain kind of representation theorem.
P. Poitevin: Paving Nakano spaces by finite-dimensional sublattices
Nakano spaces are generalizations of the classical spaces $L_p(\mu)$ in which the exponent $p$ is allowed to vary randomly with the underlying measure space. The essential range of a Nakano space is the essential closure $K$ of the range of its random exponent $p$. Using the fact that any positively and contractively complemented sublattice of a Nakano space with bounded essential range $K$ is itself a Nakano space with essential range included in $K$, we provide an intrinsic Banach lattice characterization of Nakano spaces with essential range $K$. This characterization helps write down explicit axioms for the class of Nakano spaces with essential range $K$ and is analogous to that of the classical spaces $L_p(\mu)$ as Banach spaces that are paved by the finite-dimensional spaces $l_p^n$ almost isometrically.
S. Süer: Differential Schemes and Differential Galois Theory (joint work with D. Blázquez-Sanz)
We will give a differential scheme theoretic characterization of Pillay's generalized strongly normal extensions in the spirit of Kovacic's scheme theoretic characterization of strongly normal extensions. The advantage of using schemes is that the theory works for partial differential field extensions possibly of infinite transcendence degree.
E. Vassiliev: Weakly locally modular geometric structures (joint work with A. Berenstein)
We analyze linearity-like conditions in geometric theories using lovely pairs. In particular, we study weakly locally modular geometric structures, a common generalization of one-based supersimple SU-rank 1 structures and linear o-minimal structures.