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Timetable (MAAW03)

Model Theory, Algebraic and Analytic Geometry

Monday 11th July 2005 to Friday 15th July 2005

Monday 11th July 2005
08:30 to 09:30 Registration
09:30 to 10:00 Welcome INI 1
10:00 to 11:00 L van den Dries ([Illnois])
Asymptotic differential algebra and H-fields

The differential field of LE-series has very strong closure properties, and seems a perfect arena for asymptotic differential algebra. The theory of H-fields is an approach to develop a model theory for this differential field. I will discuss the field of LE-series, the subject of H-fields in general, and review the present state of an ongoing project on these matters in collaboration with Matthias Aschenbrenner and Joris van der Hoeven. The talk by Aschenbrenner will focus on recent progress.

INI 1
11:00 to 11:30 Coffee
11:30 to 12:30 Definable groups over valued fields INI 1
12:30 to 13:30 Lunch
14:30 to 15:30 J-M Lion ([Rennes 1])
The Haefliger theorem for foliations and o-minimal structures
INI 1
15:30 to 16:00 Tea
16:00 to 17:00 J-P Rolin ([Bourgogne])
Quasi-analytic solutions of differential equations and o-minimal structures

We present the results of a joint work with F. Sanz and R. Schaefke. Consider a non-oscillating trajectory of real analytic vector field. We show, under certain assumptions, that such a trajectory generates an o-minimal and model complete structure together with the analytic functions. The proof uses the asymptotic theory of irregular singular ordinary differential equations in order to establish a quasi-analyticity result from which the main theorem follows. As applications, we present an infinite family of o-minimal structures such that any two of them do not admit a common extension, and we construct a non-oscillating trajectory of a real analytic vector field in that is not definable in any o-minimal extension of the reals.

INI 1
17:00 to 18:00 M Aschenbrenner ([Illinois])
Solving linear differential equations over H-fields

Many basic questions about algebraic differential equations over H-fields remain open. I will talk about recent work (joint with Lou van den Dries and Joris van der Hoeven) which answer some of these questions in the case of linear differential equations.

INI 1
18:00 to 18:30 Wine Reception
18:45 to 19:30 Dinner at Wolfson Court (Residents only)
Tuesday 12th July 2005
09:00 to 10:00 R Pink ([ETH Zentrum])
A common generalisation of the conjectures of Andre-Oort, Manin-Mumford and Mordell-Lang

The formal similiarity between the Manin-Mumford and Mordell-Lang conjectures on the one hand, and the Andre-Oort conjecture on the other hand, suggests that a common generalization should exist for subvarieties of mixed Shimura varieties. We propose such a conjecture, explain why it implies all the stated conjectures, and explain its relation with existing results.

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INI 1
10:00 to 11:00 D Roessler ([Paris 7])
Two applications of automatic uniformity
INI 1
11:00 to 11:30 Coffee
11:30 to 12:30 Strongly minimial groups in the theory of compact complex spaces

A compact complex space is viewed as a first-order structure in the language where all analytic subsets of the cartesian powers are named. Anand Pillay and Thomas Scanlon have characterised all strongly minimal groups definable in such a structure as being either a simple complex torus or the additive/multiplicative group of the complex field. I will discuss joint work with Matthias Aschenbrenner and Thomas Scanlon in which we give a uniform version of this result thereby characterising strongly minimal groups in elementary extensions of compact complex spaces.

INI 1
12:30 to 13:30 Lunch
14:30 to 15:30 Aspects of the algebraic structure of groups definable in o-minimal structures

Let M be an o-minimal expansion of a real closed field. A definable group is a group that both the set and the graph of the operation are definable in M. Let G be a closed and bounded definable group. I will show the following:

(1) G is divisible if and only if G is definably connected.

(2) (Joint work with M.Edmundo) If G is abelian then the group structure of the torsion subgroups of G is determined.

Both proofs require the understanding of the o-minimal cohomology algebra of G.

I will also discuss the role played by the o-minimal Euler characteristic in aspects of the algebraic structure of definable groups.

INI 1
15:30 to 16:00 Tea
16:00 to 17:00 On the number and shape of asymptotic cones of semialgebraic groups

Asymptotic cones are a useful concept in order to study large scale geometric invariants of metric spaces. I will explain the construction of asymptotic cones and will indicate why the number of homeomorphism types of asymptotic cones of a Lie group depends on the continuum hypothesis.

INI 1
17:00 to 18:00 E Jaligot ([Lyon II])
Groups of finite Morley rank and genericity

The ultimate Algebricity Conjecture concerning groups of finite Morley rank postulates that simple groups of this class are algebraic. The weaker Genericity Conjecture postulates that they contain a generous Carter subgroup. These definable connected nilpotent subgroups of finite index in their normalizers exist in any group of finite Morley rank and they are a good approximation of maximal tori in the algebraic context. Such a subgroup is said to be generous if its conjugates form a generic subset of the ambiant group. I will explain a conjugacy theorem of generous Carter subgroups and show some striking consequences of the presence a generous Carter subgroup in a group of finite Morley rank.

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INI 1
18:45 to 19:30 Dinner at Wolfson Court (Residents only)
Wednesday 13th July 2005
09:00 to 10:00 Uniform first order definitions in finitely generated fields INI 1
10:00 to 11:00 Orbital integrals for linear groups INI 1
11:00 to 11:30 Coffee
11:30 to 12:30 An application of motivic integration to representations of p-adic groups INI 1
12:30 to 13:30 Lunch
19:30 to 18:00 Conference Dinner - Warburton Dining Hall, Lucy Cavendish College
Thursday 14th July 2005
09:00 to 10:00 Non-oscillating trajectories of vector fields and Zariski's local uniformization

A transcendent non-oscillating trajectory of an analytic germ of real vector field induces a structure of Hardy field for the meromorphic functions. It has a natural valuation associated to it. The study of this valuation allows to get a reduction of singularities of the vector field following the strict transform of the trajectory. More generally, for a holomorphic complex vector field, the above results can be generalized for a given valuation of the field of the meromorphic functions. We obtain in this way a local uniformization in the sense of Zariski, that should be globalized in dimension three, following the classical results of Zariski. The key of these results is a construction (due to J. Cano and Grigoriev-Singer) based on the Newton Polygon of a differential operator, that assures finiteness results on the valuation allowing the local uniformization.

INI 1
10:00 to 11:00 Homotopy types of fibres of pfaffian maps

We prove a tight upper bound on the number of different homotopy types of fibres of semialgebraic, quadratic or semi-Pfaffian maps in terms of formats of these maps. A similar argument leads to a tight upper bound on the sizes of finite bisimulations of Pfaffian dynamical and hybrid systems.

INI 1
11:00 to 11:30 Coffee
11:30 to 12:30 On the topological degree of functions definable in o-minimal structures INI 1
12:30 to 13:30 Lunch
14:30 to 15:30 Analytic Denef-Pas cell decomposition

We prove a Cell Decomposition Theorem for Henselian valued fields with analytic structure in an analytic Denef-Pas language. To accomplish this, we introduce a general framework for Henselian valued fields K with analytic structure, and we investigate the structure of analytic functions in one variable defined on annuli over K.

We also prove that, after parameterization, definable analytic functions are given by terms. The results in this paper pave the way for a theory of analytic motivic integration and analytic motivic constructible functions in the line of R. Cluckers and F. Loeser [Fonctions constructible et int\'egration motivic I, Comptes rendus de l'Acad\'emie des Sciences, 339 (2004) 411 - 416].

Related Links

INI 1
15:30 to 16:00 Tea
16:00 to 17:00 L Lipshitz ([Purdue])
Overconvergent real closed quantifier elimination

Let K be the (real closed) field of Puiseux series in t over the reals, R, endowed with the natural linear order. Then the elements of the formal power series rings R[[x_1,...,x_n]] converge t-adically on [-t,t]^n, and hence define functions [-t,t]^n to K. Let L be the language of ordered fields, enriched with symbols for these functions. We show that K is o-minimal in L. This result is obtained from a quantifier elimination theorem. The proofs use methods from non-Archimedean analysis.

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INI 1
18:45 to 19:30 Dinner at Wolfson Court (Residents only)
Friday 15th July 2005
09:00 to 10:00 Model theory of difference fields and some remarks on Galois groups

In this talk I will present a short survey of the results known to-date on generic difference fields. I will also discuss Galois groups of difference equations, and why the model-theoretic Galois group coincides with the classical Galois group defined using Picard-Vessiot extensions

INI 1
10:00 to 11:00 Small profinite groups INI 1
11:00 to 11:30 Coffee
11:30 to 12:30 Elementary equivalence vs isomorphism over large fields

During the last time there was some progress in tackling the problem of "elementary equivalence vs isomorphism" for function fields in both the arithmetical and the geometric situation. We will show that actually similar results can be obtained in the case of function fields over the so called large fields (under some supplementary "finiteness" hypothesis).

Related Links

INI 1
12:30 to 13:30 Lunch
14:30 to 15:30 TW Scanlon ([Berkeley])
Additive groups
INI 1
15:30 to 16:00 Tea
16:00 to 17:00 Betti numbers of definable sets

A spectral sequence associated with a surjective closed mapping allows one to provide upper bounds for the Betti numbers of a wide class of sets defined by formulas with quantifiers in terms of the Betti numbers of auxiliary sets defined by quantifier-free formulas. A review of the recent development in this direction will be presented

INI 1
18:45 to 19:30 Dinner at Wolfson Court (Residents only)
University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons