These mini-workshops are intended to bring people together for
two to three days to work on one particular project, while allowing
other visitors to the program to participate. Each mini-workshop
involves between five and ten core participants and typically focuses
on the understanding of a specific problem or solution thereof.
The information below is tentative and subject to change.
|
January
30-31, 2009
Mini-workshop on Expansions of the real field by multiplicative
groups
|
Ayhan Gunaydin
Chris Miller |
Lou van den Dries, Philipp Hieronymi and Michael
Tychonievich |
|
March
5-6, 2009
10:30 am
Mini-workshop on o-minimality for Certain Dulac Transition
Maps
Room 210
|
Tobias Kaiser
Patrick Speissegger |
We present the main ideas for proving o-minimality
of the expansion of the real field generated by all Dulac transition
maps near a non-resonant hyperbolic singularity of a planar
analytic vector field. We also show how the existence of (non-explicit)
uniform bounds on the number of limit cycles of certain (very
special) families of analytic vector fields can be obtained
from our approach.
Jean-Philippe Rolin, Dmitry Novikov, Sergei Yakovenko |
|
March
16-20, 2009
Mini-workshop on the Infinitesimal Hilbert's 16th Problem
March 16 -3:30 pm
March 17 -3:30 pm
March 18 -3:30 pm
March 20 -3:30 pm
|
Dmitry Novikov
Sergei Yakovenko |
Edward Bierstone, Andrei Gabrielov,
Boris Khesin, Askold Khovanskii
|
|
March 23-25,
2009
Mini-workshop on New Perspectives in Valuation Theory
Room 210
|
Franz-Viktor Kuhlmann
Florian Pop
Bernard Teissier |
In recent years new perspectives in valuation
theory have begun to appear as well as unexpected applications.
The two historical flows of valuation theory, namely the Henselian
and the Zariskian, are merging like never before in the development
of Berkovich geometry and the new approaches to resolution of
singularities, which now extend to the singularities of vector
fields. One begins to really be able to do analysis on spaces
of of valuations, leading to important new results on complex
analytic dynamical sytems stemming from a radically new point
of view on the use of valuations of the ring of holomorphic
functions. There is a new understanding of the structure of
spaces of valuations with a given center, exemplified by the
valuative tree of Favre-Jonsson, and also of the more global
aspects for which tropical geometry gives useful hints. The
purpose of the workshop is to gather experts who are contributing
to this new perspective so that they can strengthen their common
views and share problems and results.
Charles Favre, Mattias Jonsson, Daniel Panazzolo, Florian Pop,
Mark Spivakovsky |
|
Monday
March 23
|
3:30 pm - Bernard Teissier
Some recent developments in valuation theory
5:15 pm Askold Khovanskii
TBA |
|
Tuesday
March 24
|
1:30 p.m. Charles Favre
Valuation Spaces
3:30 p.m. Mark Spivakovsky
Desingularization of 3-dimensional vector fields by blowing-up
along non-singular centres
5:15 p.m. Franz-Viktor Kuhlmann
On local uniformization in positive characteristic |
|
Wednesday
March 25
|
10:30 am - Mattias Jonsson
More on valuation spaces
1:30 pm - Salih Azgim
Extremal Fields
3:30 pm - Florian Pop
On the space of intertia elements
|
|
April
3-4, 2009
Miniworkshop on Differential Kaplansky Theory
Room 210
|
Salma Kuhlmann
Mickael Matusinski
|
Let (K,<,d) be an ordered diffenrential field, and v the
natural valuation. We assume that d is compatible with
v, i.e. that v is a differential valuation in
the sense of M. Rosenlicht. Denote by k the residue
field and by (G,Ψ) the induced asymptotic couple; i.e.
G = v(K) is the value group endowed with the map Ψ(v(a))
:= v(a′: =a).
The purpose of this workshop is to study a differential Kaplansky
theory in this setting. We want to achieve progress on the
following problem: Find necessary and su±cient conditions
on (K; <, d) so that: (i) the data (G, Ψ) allows
to define a derivation d on the field of generalized
series k((G)); (ii) the induced asymptotic couple is precisely
(G,Ψ); (iii) there is an order preserving di®erential
embedding of (K,<,d) in (k((G)),<,d);
(iv) the embedding may be chosen to be truncation closed;
i.e. the image of the embedding is closed under the operation
of taking initial segments of series. Partial progress has
been achieved on this topic, for example regarding item (i),
we have described the construction of "well-defined"
derivations on k((G)). Regarding item (iii), J.M.Aroca
and J. Del Blanco have considered the case of archimedean
value group. Other approaches to this problem are described
in the works of M. Aschenbrenner - L. v. D. Dries on H - fields,
and the works of J. v. D. Hoeven on Transseries.
J. Del Blanco Marana, Lou van den Dries
|
|
Friday
April 3
Room 210
|
10:00 am - Salma Kuhlmann, Presentation of the workshop
10:30-12 pm - M. Matusinski
Hardy type derivations on generalized series fields
We consider an arbitrary Hahn group of monomials and the corresponding
field of generalized series with real coefficients. First
we show how to construct well-defined derivations on such
fields. We then give a criterion for such a derivation to
be of Hardy type, that is to verify the same properties as
those in Hardy fields.
1:30-3 pm - L. van den Dries
What is an asymptotic differential field and when is it
differentially henselian
I will define asymptotic differential fields, propose a notion
of "differentially henselian" and indicate some
of its good properties.
3:30-5 pm - discussion
|
|
Saturday
April 4
Room 210
|
10:30-12 pm - discussion
3:00- 4:30 pm - J. del Blanco Marana
A differential Kaplansky immersion theorem on rank one valued
fields with real residue field
I will present the following result:
Let (K, v, d) be a differential valued field with archimedean
value group G = v(K -{0}) and residue field kv = R included
in K.
The following two statements are equivalent:
(i) The derivation d is functional.
(ii) There exists a differential analytic morphism (K, v, d)
->
(R[[X^G]], ord, ?), where ? is some monomial derivation on the
generalised series field R[[X^G]].
A functional derivation carries an abstract version of properties
that hold in the case of functions (continuity, L'Hospital's
rule), for instance germs in a Hardy field. The notion of monomial
derivation is a generalisation of the usual derivation for formal
power series. |
|
May 6-8, 2009
Mini-Workshop on (Co)Homology and sheaves in O-minimal and
Related Settings
Room 210
|
M. Edmundo, A. Piekosz and L. Prelli |
The workshop aims to introduce everyone
to the subject. Talk 1 will include an introduction to sheaves
with focus on sheaves on sub-analytic site and applications.
Talk 2 illustrates the use of sheaves to construct locally definable
spaces and the more sophisticated weakly definable spaces, which
when considered over an o-minimal expansion of a field, have
a well developed homotopy theory giving in particular different
kinds of homology and cohomology. Talk 3 focus on sheaves on
o-minimal site with applications to the theory of definable
groups.
M. Edmundo, P. Eleftheriou, A. Piekosz, L. Prelli, S. Starchenko.
|
|
Wednesday
May 6
Room 210
|
10:30 am - Luca Prelli
Sheaves on subanalytic sites and D-modules
|
|
Thursday
May 7
Room 210
|
10:30 am - Artur Piekosz
Locally definable and weakly definable spaces
Abstract: As they are defined using structure sheaves, locally
definable spaces and more sophisticated weakly definable spaces
are a natural generalization of both the subanalytic and the
definable o- minimal settings. If considered over an o-minimal
expansion of a field, they have a well developed homotopy theory,
which in particular gives different kinds of homology and cohomology.
The recommended reading for my lecture is the preprint arXiv:0904.4896
(http://arxiv.org/abs/0904.4896).
|
|
Friday
May 8
Room 210
|
10:30 am - Mario Edmundo
O-minimal sheaves and applications |
|
June 5-6,
2009
Mini-workshop on decidability in analytic situations
Room 230
|
Gareth O. Jones |
The workshop aims to understand the work of Macintyre
and Wilkie on the real exponential field, and the more recent
work of Macintyre on Weierstrass functions. The relation with
the constructive results of Gabrielov and Vorobjov would also
be investigated. The hope is that after careful study of these
papers, we would be able to prove further constructive model
completeness results for theories related to those above. If
this goes to plan, we would then combine the constructive model
completeness with recent work around Schanuel?s conjecture,
with the aim of proving unconditional decidability results for
certain analytic expansions of the real field.
Tamara Servi, Dan Miller, Andrei Gabrielov |
|
Friday
June 5
Room 230
|
10:00-11:30 Tamara Servi
On the decidability of the real field with a generic power
function part I
(joint work with G. Jones) In recent work we proved that, if
A is a real number not zero-definable in the real exponential
field, then the theory of the real field with the power function
x^A is decidable, relatively to an oracle for A. I will prove
this statement, and give a proof of the existence of a computable
generic real number.
13:30-15:00 Tamara Servi
On the decidability of the real field with a generic power
function part II |
|
Saturday
June 6
Room 230
|
10:00-11:30 Dan Miller
13:130-15:00 Andrei Gabrielov
Multiplicity of a Noetherian intersection and degree of nonholonomy
A differential ring of analytic functions in several complex
variables is called a ring of Noetherian functions if it is
finitely generated as a ring and contains the ring of all polynomials.
The multiplicity of an isolated solution of a system of $n$
equations $f_i=0$, where $f_i$ belong to a ring of Noetherian
functions in $n$ complex variables, can be expressed in terms
of the Euler characteristics of the generalized Milnor fibers
associated with this system. This provides an effective upper
bound on this multiplicity. In combination with constructive
resolution of singularities over the fields of characteristic
zero, this allows one to obtain an effective upper bound on
the complexity of the resolution of singularities defined by
Noetherian functions. For $n=1$, Noetherian functions are soultions
of a system of algebraic ordinary differential equations. The
upper bound on their multiplicity implies an effective upper
bound for degree of nonholonomy of a system of algebraic vector
fields, an important problem in control theory. |
|
June
8-10, 2009
Mini-workshop on Finiteness theorems for certain quasi-regular
algebras and Hilbert's 16th problem
Room 210
|
Abderaouf Mourtada |
On introduit une algèbre de germes de fonctions, dite
algèbre quasi- régulière d'Hilbert: ces
éléments sont quasi-analytiques et possèdent
une structure asymptotique de "type Hilbert". Cette
algèbre contient les compositions des déploiements
analytiques d'applications de Dulac pour les singularités
heyperboliques de champs de vecteurs du plan. L'étude
de la cyclicité des polycycles hyperboliques du plan
se ramène a l'étude de l'action sur cette algèbre
d'une certaine classe de dérivations dites "Dérivations
d'Hilbert". La désingularisation de telles dérivations
(dans l'algèbre quasi-régulière d'Hilbert!!)
fournit des dérivations irréductibles qui sont
hyperboliques, linéaires et diagonales. Les théorèmes
de finitude exposés sont relatifs a ces dérivations
irréductibles. Une première application globale
de ces résultats, dans le cadre du 16eme problème
d'Hilbert est la suivante: la cyclicité d'un cycle
singulier d'un champ Hamiltonien du plan, ne dépend
que du degré de l'Hamiltonien et de "la multiplicité
algébrique" de l'intégrale abélienne
associée, ce qui constitue la généralisation
naturelle du théorème de Khovanski- Varchenko.
Jean-Philippe Rolin, Patrick Speissegger
|
|
Monday
June 8
Room 210
|
10:30 am- Introducion des algebres
et derivation d'Hilbert.
3:30 pm- A la demande pour plus de details.
|
|
Tuesday
June 9
Room 210
|
10:30 am- Les theoremes de finitude
et une application localedans le cadre du 16eme probleme d'Hilbert.
3:30 pm- A la demande pour plus de details.
|
|
Wednesday
June 10
Room 210
|
10:30 am- Une application globale
dans le cadre du 16eme probleme d'Hilbert.
3:30 pm- A la demande pour plus de details.
|
| |
|
