Thematic
Program on Ominimal Structures and Real Analytic Geometry
Workshop Abstracts

Supported
by 

Andreas Fischer (Fields
Institute)
Definable versions of theorems by Kirszbraum and Helly
Kirszbraun's Theorem states that every Lipschitz map form $S$
to $\mathbb{R}^n$, where $S\subseteq \mathbb{R}^m$, has an extension
to a Lipschitz map defined on $ämathbb{R^m}$ with the same
Lipschitz constant. Its proof relies on Helly's Theorem: every
family of compact subsets of $\mathbb{R}^n$, having the property
that each of its subfamilies consisting of at most $n+1$ sets
share a common point, has a nonempty intersection.
We sketch the proof of versions of these theorems valid for definable
maps and sets in arbitrary definably complete expansions of ordered
fields.
Andrei Gabrielov (Purdue
University)
Triangulation of monotone ominimal families
Let $S_t,\,t\in(0,1]$, be a family of compact sets in a compact
$K\subset R^n$, such that $S_u\subset S_t$ for $0<t<u$.
If the family $S_t$ and the compact $K$ are definable in an ominimal
structure, we construct a triangulation of $K$ such that, for
each open simplex $\sigma$, the family $\sigma\cap S_t$, for small
values of $t$, is homeomorphic to one of the ``standard'' combinatorial
families.
Tobias Kaiser (University
of Passau)
On the resolution of singularities for power series with perturbation
by logarithmic terms
In [1] it was shown that transition maps at nonresonant hyperbolic
singularities of analytic vector fields in the plane are definable
in an ominimal structure. The proof uses the fact that the transition
maps allow an asymptotic expansion and live in a quasianalytic
class (see [2]). In the nonresonant case the asymptotic expansion
is given by certain generalized power series. Based on this fact
it was possible to define an appropriate generalization of this
quasianalytic class to several variables and to develop a blowup
algorithm for resolution of singularities to obtain ominimality
in the spirit of previous work in ominimal structures (see for
example [3]). In the resonant case the asymptotic expansion is
given by generalized power series where the monomials are additionally
perturbed by logarithmic polynomials. The desired goal would be
to develop a blowup algorithm for resolution of singularities
also in this case to obtain ominimality of the transition maps
in the resonant case and of related topics (see [4], [5]).But,
the presence of logarithmic terms complicates the situation enormously.
In this talk I discuss a possible approach towards a blowup algorithm.
Literature:
[1]T.Kaiser, J.P.Rolin and P.Speissegger: Transition maps at
nonresonant hyperbolic singularities are ominimal, Preprint,
2006,
arXiv:math/0612745, J. Reine Angew. Math. to appear
[2]Y. S. Ilyashenko, Finiteness theorems for limit cycles, Translations
of Mathematical Monographs 94 (1991)
[3]L.Van den Dries, and P. Speissegger: The real field with convergent
generalized power series, Trans. Amer. Math. Soc. 350 (1998),
43774421 [4]T. Kaiser: The Riemann mapping theorem for semianalytic
domains and
ominimality. Proc. Lond. Math. Soc. (3) 98, No. 2, 427444 (2009)
[5]T.Kaiser: The Dirichlet problem in the plane with semianalytic
raw
data, quasi analyticity, and ominimal structure. Duke Math. J.
147, No.
2, (2009), 285314.
Krzysztof Kurdyka
(Université de Savoie)
On the structure of gradient extremal set of generic functions
For a real valued smooth function on a Riemannian manifold by
"gradient extremal set" we mean all critical points
of the restriction to the fiber of the square of norm of the gradient.
For a generic (Morse) function this is set is a union of smooth
curves which intersect "transversally" at critical points.
A part of the gradient extremal set gradient which correspond
to the local minima is call a "Talweg" or "ridge
and valley lines". It can be used to estimate the length
of gradient trajectories and possibly should organise dynamics
of the gradient flow.
JeanMarie Lion (Université
de Rennes 1)
Pfaffian sets
We show that the Pfaffian closure of an ominimal structure with
analytic cell decomposition is model complete.
David Marin (Universitat
Autonoma de Barcelona)
Unfolding
of resonant saddles and the Dulac time
In this work we study unfoldings of planar vector fields in a
neighborhood of a resonant saddle. We give a C k temporal normal
form for the unfolding. That is, a normal form with respect to
the conjugacy relation. Using our temporal normal form we determine
an asymptotic development, uniform with respect to the parameters,
of the Dulac time of a resonant saddle. Conjugacy relation instead
of weaker equivalence relation is necessary when studying the
time function. The Dulac time of a resonant saddle can be seen
as the basic building block of the total period function of an
unfolding of a hyperbolic polycycle. (joint work with Pavao Mardesic
and Jordi Villadelprat)
Chris Miller (Ohio
State University)
Expansions of ominimal structures by trajectories of definable
planar vector fields
An expansion of the real field is said to be ominimal if every
definable set has finitely many connected components. Such structures
are a natural setting for studying "tame" objects of
real analytic geometry such as nonoscillatory trajectories of
real analytic planar vector fields. More generally, ominimality
is preserved under expanding an ominimal structure by nonoscillatory
trajectories of definable planar vector fields. But what happens
when ominimality is not preserved? In some cases, we see the
best behavior that one could reasonably expect, while in others
the worst possible, and we do not know at present of any other
outcomes. I will make all this precise in a survey of the current
state of the art.
Dmitry Novikov (Weizmann
Institute of Science)
Infinitesimal Hilbert 16th problem: recent progress
I'll describe a recent progress toward providing general constructive
solution of Infinitesimal Hilbert 16th problem.
Olivier Le Gal (University
of Toronto)
Ominimalty for solutions of analytic 2dimensional linear ODEs
Daniel Panazzolo (Universidade
de São Paulo)
The group of power and translations and its relation to generalized
Witt algebras
The group of maps generated by the powers $x \righrarrow x^r$
and translations $x \rightarrow x + a$ appears in numerous problems
in the analytic theory of differential equations and dynamical
systems. In this talk, we will discuss some recent results related
to two old questions: how fast the number of fixed points of a
"word" can grow in terms of its length? Is this group
a free product?
Adam Parusinski (Université
d'Angers)
Limits of gradient directions at a singular point
Let $f$ be a definable $C^1$ function defined in a neighborhood
of the origin in $R^n$. Using Lagrange specialization and the
deformation to the normal cone we describe geometrically the space
of limits at the origin of secant lines and gradient directions
of $f$, understood as a subset of $P^{n1} \times P^{n1}$.
We apply this description to study the gradient flow of $f$.
Christiane Rousseau
(Université de Montréal)
Orbital analytic classification of germs of families unfolding
a codimension 1 resonant saddle or saddlenode
We consider the equivalence problem for germs of 1parameter
analytic families of planar vectors fields unfolding a resonant
saddle or a saddlenode. We will describe a complete modulus under
orbital equivalence. Moreover, we will give sufficient conditions
for a “candidate modulus” to be indeed realizable as
the modulus of a germ of family. We will characterize the moduli
corresponding to families of real vector fields.
Tere Seara (Universitat
Politècnica de Catalunya)
Resurgence of inner solutions for analytical perturbations
of the McMillan map
In the study of the exponentially small splitting that occurs
in certain perturbations of the McMillan map a sequence of "inner
equations" has to be considered. An essential step in the
measure of the splitting is to know some special solutions of
these equations and to be able to give an asymptotic value of
their difference.
The present work relies on ideas from resurgence theory: we obtain
the desired solutions as BorelLaplace sums of the formal solutions,
studying the analyticity of their Borel transforms. Moreover,
using 'Ecalle's alien derivations we are able to measure the discrepancy
between different BorelLaplace sums.
(Joint work with P. Martin and D. Sauzin)
Masahiro Shiota (Nagoya
University)
Ominimal Hauptvermutung
Arguments on PL (=piecewise linear) topology work over any ordered
field in the same way as over the real number field, and those
on differential topology do over a real closed field in an o
minimal structure that expands (R,<,0,1,+,\cdot). It is known
that a compact definable set is definably homeomorphic to a polyhedron.We
show uniqueness of the polyhedron up to PL homeomorphism (ominimal
Hauptvermutung).We see also that many problems on PL and differential
topology can be translated to those over the real number field.
Lou van den Dries
(University of Illinois at UrbanaChampaign)
Asymptotic real differential algebra
Joris van der Hoeven
(Université ParisSud)
Machine computations with transseries
In our talk, we will discuss several ways to implement computations
with transseries in a computer algebra system. In particular,
we will explain the new technique of metaexpansion, which is
highly effective in practice, although merely heuristic from a
theoretical point of view.
Alex Wilkie (University
of Manchester)
Some Model Theory for Complex Analytic Functions
We consider complex analytic functions that are locally definable
in a polynomially bounded ominimal structure (for example any
polynomial in z, exp(z)). This restriction seems to place quite
severe constraints on the analytic sets defined by such functions
and provides a way in to Zilber's conjecture, ie if a subset of
the set of complex numbers is first order definable in the complex
exponential field then it is either countable or cocountable.
Yosef Yomdin (The
Weizmann Institute of Science)
Poincare Inversion, Moment inversion, Compositions, and Signal
Processing
We present some new results and question related to the CenterFocus
problem for Abel differential equation. There was recently a serious
progress in understanding some aspects of this problem. In particular,
F. Pakovich and M. Muzychuk completely solved the vanishing problem
for polynomial moments: this is an infinitesimal version of the
CenterFocus problem. On the other hand, some initial results
on the "Poincare inversion problem" have been obtained.
This problem asks for a characterization of all possible sequences
of Taylor coefficients of the Poincare first return mapping, and
for reconstruction of the original Abel equation from the given
sequence of its Poincare coefficients. This generalizes the classical
CF problem (where all the coefficients besides the first one
are zero). In the Poincare inversion problem, besides the usual
"Composition condition" which is a conjectural Center
condition, some other types of compositions arise. Closed connections
have been found also with the classical Moment inversion problem
(which naturally appears as the infinitesimal version of the Poincare
inversion). Very encouraging connections have also been found
with the "algebraic sampling" problem (i.e. the problem
of reconstructing a nonlinear model from a set of measurements),
as it appears in Signal Processing.
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