MODNET research workshop.
O-minimality: model theory and geometry

Haifa, 1- 4 September 2008.

A meeting of the research training network in Model Theory MODNET


H. Adler: Theories controlled by formulas of VC codimension 1.

This natural class of theories includes all strongly minimal, (weakly) o-minimal and C-minimal theories, and they are all dp-minimal.

G. Comte: Local invariants in real and p-adic subanalytic geometry.

We define two finite sequences of local invariants in real subanalytic geometry, one coming from the localization of curvature invariants, the other being "polar invariants". We show that each term of one sequence is a linear combination of terms of the other sequence. This relation has to be seen as the multi dimensional Crofton formula. We show that all these invariants are continuous along Verdier strata (constant in the well-known complex case). In the p-adic case, only one relation remains, the so-called local Crofton formula.

M. Edmundo: Around Pillay's conjecture for orientable definable groups.

We will talk about the work around the solution of Pillay's conjecture for orientable definable groups in arbitrary o-minimal structures with definable Skolem functions.

A. Hasson: 1-dimensional geometries in o-minimal theories.

We discuss the classification of combinatorial pre-geometries arising as 1-dimensional reducts of o-minimal theories. The talk will focus on the development of a fine intersection theory for plane curves arising in this context, which played a crucial role in the analysis.

G. Jones: Model completeness for certain Pfaffian structures.

I show that the expansion of the real field by a total Pfaffian chain is model complete in a language with symbols for the functions in the chain, the exponential and all real constants. In particular, the expansion of the reals by all total Pfaffian functions is model complete.

T. Kaiser: O-minimal structures, Hilbert 16, the Riemann Mapping Theorem and the Dirichlet problem

We investigate the connection of o-minimality and the following three concepts from analysis.
Hilbert 16: What can be said about the number of limit cycles of a polynomial vector field in the plane? Riemann Mapping Theorem: A proper simply connected domain in the plane can be mapped biholomorphically onto the unit ball. Dirichlet problem: The following PDE with boundary value problem is considered. Given a domain U and a continuous boundary function h the Dirichlet solution for h is harmonic in U and can be continuously extended by h to the boundary of U.
Important tools in the work on Hilbert 16 are Poincare return maps resp. transition maps at singular boundary points of the vector field. In the context of the Riemann Mapping Theorem we are interested in the case that the domain is globally semianalytic. In the context of the Dirichlet problem we are interested in the case that the domain is in the plane and that both domain and boundary function are globally semianalytic. The main result is the following.
There is an o-minimal structure such that transition maps at non- resonant hyperbolic singular points, the Riemann map and the Dirichlet solution (under an additional condition on the angles at singular boundary points in both cases) are definable in this structure.
In the first part of the talk we introduce the three concepts from analysis. A key step in Ilyashenko's work on Hilbert 16 is the introduction of a certain quasianalytic class in which transition maps at hyperbolic singularities live. We show that both the Riemann map and the Dirichlet solution (with semianalytic raw data) can also be realized in this quasianalytic class. In the second part we show how we obtain an o-minimal structure from this quasianalytic class. The main difficulty is the right definition of the quasianalytic classes in several variables. In the third part we discuss possible extensions of the above theorem.

K. Kurdyka: Trajectories of horizontal gradients of polynomials

We consider a class of non-holonomic codimension 1 distributions on affine space. This class contains contact structure, Heisenberg and Martinet distributions. We equip these distributions with a subriemannian metric by fixing polynomial vector fields which give orthonormal basis for the distribution. Thus for a given polynomial on the affine space we obtain its horizontal gradient with respect to the subriemannian metric. We show that the behaviour of trajectories of horizontal gradient differs from the know results of Lojasiewicz on the Riemannian gradient of polynomials, in particular Lojasiewicz's gradient inequality does not hold. But we show that for generic polynomials the trajectories of the horizontal gradient approaching critical set have limits.

Joint work with S-T. Dinh and P. Orro.

P. Kowalski: On Peterzil's Question

In 2005 Kobi Peterzil asked a question whether Zilber's Dichotomy holds in strongly minimal structures definable in o-minimal ones. I will report on the progress which has been made so far in answering this question. I will focus mostly on my joint work with Assaf Hasson.

D. Marker: O-minimality, the independence property and VC-dimension.

D. Novikov: Infinitesimal Hilbert 16th problem

Limit cycles appearing as perturbations of cycles of Hamiltonian planar vector fields correspond in first approximation to zeros of Abelian integrals. Infinitesimal Hilbert 16th problem (IHP16) asks for an upper bound for the number of these zeros. Generalization of the problem deal with integrable systems (pseudo-abelian integrals) or with higher order approximations (which correspond to iterated integrals) or both. Together with G. Binyamini and S. Yakovenko we provide a constructive answer to IHP16 depending on degrees only. I'll also discuss the generalizations.

A. Onshuus: The stable/unstable dichotomy and the geometry of types in o-minimal structures.

When studying the Zilber-geometry of types in theories interpretable in o-minimal structures, it becomes clear that the best way to do it is to separate the "stable geometry" from the "unstable" and treating each case separately. I will give a quick survey on how this can be done and some of the results and conclusions one can get towards a complete Zilber-type characterization of types by looking at their geometry.

A. Rambaud: O-minimality and quantifier elimination in some exponentially bounded classes.

Let E be a well-closed class of real restricted functions (closed by sums, products, compositions, implicit functions and appropriate factorizations). We define a notion of 'exponential degeneration'; we prove that if E is not exponentially degenerate, then the complete theory of R equipped with E admits quantifier elimination and is o-minimal. These results are a generalization to exponentially bounded classes of the previous studies of quasi-analytic classes (in particular we define ”‘exponential stable families”’).

S. Shelah: On 2-dependent first order complete theories

Recall the definition of 2-dependent $T$ (where $T$ is 2-independent when some $\langle \varphi(\bar x,\bar b_m,\bar c_n):m,n < \omega\rangle$ is an independent sequence of formulas); see \cite[\S5]{Sh:863}(H). Though a reasonable definition, can we say anything interesting on it? Well, we prove the following result Let $G_A$ be the minimal type-definable over $A$ subgroup of $G$, for suitable $\kappa$, . Now if $M$ is $\kappa$-saturated and $|B| < \kappa$ then $G_{M \cup B}$ can be represented as $G_M \cap G_{A \cup B}$ for some $A \subseteq M$ of cardinality $< \kappa$. So though this does not prove ``2-dependent is a dividing line", it seems enough for showing it is an interesting property.

P. Speissegger: Exponential bounds for certain pfaffian structures, without knowing model completeness

It is possible to prove the exponential boundedness of certain pfaffian structures without knowing whether they are model complete. The key technique to do so is what we call "blowing-up along a distribution", which allows us to describe (up to finite union and projection) the frontier of a pfaffian set as a solution of some system of differential equations. From there, Hardy field tricks give us the exponential bounds.

Joint work with Jean-Marie Lion and Chris Miller.

G. Terzo: Schanuel Nullstellensatz for Zilber fields.

We work with the pseudo-exponential fields introduced by Zilber in [Z]. Zilber’s main result in [Z] is the identification of a class of exponential structures for which he proves categoricity in every uncountable cardinality, and he conjectures that the complex exponential field is the unique model of cardinality conitnuum. Our main result is a Nullstellensatz result for exponential polynomials over a pseudo-exponential field. We were inspired by an analogous result obtained by Henson and Rubel (see [HR]) for the complex exponential field. [HR] C. W. Henson and L. A. Rubel: Some applications of Nevanlinna Theory to Mathematical Logic: identities of exponential functions, in Transactions of the American Mathematical Society, Vol 282, No 1, March 1984, 1-32. [Z] B. Zilber: Pseudo-exponentiation on algebraically closed fields of characteristic zero, Annals of Pure and Applied Logic, Vol 132, No 1, 2004.

M. Tressl: Heirs and externally definable sets in polynomially bounded structures

I will characterize heirs of certain types of a polynomially bounded structure M and compute some heirs and coheirs of n-types of pure real closed fields. One consequences of these characterizations is a good understanding of the structure which expands M by all convex subsets of M. I conjecture that in the case where M is a pure real closed field, every externally definable subsets of M^n is definable in this expansion of M. This holds true for subfields of the real field by the Marker-Steinhorn theorem and, by a result of Francoise Delon, also for the field of generalized power series with coefficients and exponents in the real field.

I. Tyomkin: Tropical geometry

This is an introductory talk on tropical geometry and its applications to algebraic geometry. Tropical geometry is the geometry of piece wise linear manifolds, and it has been a topic for an intensive study during the last five years due to its applications to algebraic geometry and mirror symmetry. It turns out that in many cases one can assign a piece wise linear manifold (i.e. a tropical variety) to a one-parameter family of algebraic varieties, moreover in some cases one can reconstruct the algebraic family from its tropicalization (correspondence theorems). In this talk we will focus on a particular case of curves. We will explain the tropicalization process, and will discuss the similarities between the geometry of algebraic and tropical curves (e.g. intersection theory). If time permits we will briefly discuss the ideas behind the proofs of the correspondence theorems.

N. Vorobjov: Approximation of definable sets by compact families, and upper bounds on homotopy and homology.

We study upper bounds on topological complexity of sets definable in o-minimal structures over the reals. We suggest a new construction for approximating a large class of definable sets, including the sets defined by arbitrary Boolean combinations of equations and inequalities, by compact sets. The construction is applicable to images of such sets under a large class of definable maps, e.g., projections. Based on this construction we refine the previously known upper bounds on Betti numbers of semialgebraic and semi-Pfaffian sets defined by quantifier-free formulae, and prove similar upper bounds, individual for different Betti numbers, for their images under arbitrary continuous definable maps.

Joint work with A. Gabrielov.

A. Wilkie: On expansions of algebraically closed fields associated with polynomially bounded, o-minimal structures: pregeometries and a valuation inequality.

Let T be the theory of a polynomially bounded, o-minimal expansion of the real field. I assume for convenience that all exponents are rational. Let M be a model of T. Then the finite elements of M form a valuation subring, the residue field expands naturally to a model of T, and the rank of M (i.e. the minimal number of elements needed to generate M under the 0-definable functions) bounds the sum of the dimension of the value group (as a vector space over the field of rational numbers) and the rank of the residue field. In these talks I discuss the corresponding situation for the M-definable holomorphic structure on M[i] (in the sense of Peterzil-Starchenko). The first task here is to find the right notion of "holomorphic Skolem closure" on M[i]. This was done in my paper in the Newton Institute volume for the case where M is an expansion of the real field itself, but infinitesimals create new problems. On the other hand, the proof of the valuation inequality proceeds more naturally than in the real case because one has the Preparation Theorem available.
Applications to the quasi-minimality of certain expansions of the complex field will be mentioned.

See further notes on the subject here.