Harmonic Analysis and Partial Differential Equations
           Centre Emile Borel, Institut Henri Poincaré, Paris, France
April 22-26, 2002 














Last update: April 15, 2002
 

This conference is organized jointly by the European Networks Harmonic Analysis and Harmonic Analysis and Related Problems (the TMR network HA is finishing while the IHP network HARP is beginning), the Centre Emile Borel during the special program on  Heat Kernels, Random Walks, Analysis on Manifolds and Graphs . Registration is mandatory.

A short page is on the site of the ihp 
 
 

List of Speakers
List of Titles
Programme
Program of the young reseachers' day
Abstract
Organizing Committee
Registration
Practicalities
More information (hotels...)

E-mail to Organizing Committee:  auscher@mathinfo.u-picardie.fr
E-mail to secretary: heat@ihp.jussieu.fr

The list of speakers will include
       
G. ALEXOPOULOS  Université de Paris-Sud P.-G. LEMARIE-RIEUSSET Université d'Evry
N. BOURNAVEAS University of Edimburgh Y. MARTEL Université de Cergy-Pontoise
G. DAVID Université de Paris-Sud V. MAZ'YA Linköping University
T. de PAUW Université de Paris-Sud S. MEDA University La Bicoca
L. ESCAURIAZA University of Bilbao Y. MEYER ENS Cachan
B. FRANCHI Université de Bologne D. MÜLLER Universität Kiel
P. GERARD Université de Paris-Sud D. ROBINSON Australian National University, Canberra
W. HANSEN University of Bielefeld P. SJÖGREN Göteborg University
W. HEBISCH University of Wroclaw P. TCHAMITCHIAN Université d'Aix-Marseille III
A. HULANICKI University of Wroclaw A. VOLBERG Université de Paris VI

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List of Titles
 
G. ALEXOPOULOS  Heat kernels on semi-simple Lie groups
N. BOURNAVEAS Low regularity solutions of the Dirac-Klein-Gordon equations
G. DAVID Recents results on analytic capacity
T. De PAUW Size minimizations and approximating problems
L. ESCAURIAZA Unique continuation and parabolic equations
B. FRANCHI Rectifiability and perimeter in step 2 Carnot groups
P. GERARD Non-linear Schrödinger equations on compact manifolds
W. HANSEN Normalized solutions of Schrödinger operators with potentially bounded measures
W. HEBISCH Singular integrals on Iwasawa AN groups
A. HULANICKI Estimates for the Poisson kernel and the Harris measure for NA groups
P.-G. LEMARIE-RIEUSSET Espaces de Lorentz et Navier-Stokes : le problème des solutions auto-similaires de Leray
Y. MARTEL Blow-up phenomenon in the energy space for the critical generalized KdV equation
S. MEDA On Holomorphic spectral multipliers for the Ornstein-Uhlenbeck operator
V. MAZ'YA The Schrödinger and the relativistic Schrödinger operators on the energy space: boundedness and compactness criteria
Y. MEYER Improved Gagliardo-Nirenberg inequalities
D. MÜLLER Mehler-type kernels and local solvability
D.ROBINSON Analysis on Lie groups of polynomial growth
P. SJÖGREN Some maximal operators for the Ornstein-Uhlenbeck semigroup with complex time parameter
P. TCHAMITCHIAN Hardy Sobolev spaces on strongly Lipschitz domains of Rn and square roots of elliptic operators
A. VOLBERG Bellman functions, heat extension and sharp (weighted) estimates of Riesz transforms and their combinations

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Programme

The Scientific Program will start on  Monday, April 22 at 9 a.m. and will end on Friday, April  at 5.30 p.m. All talks will take place in the Amphithéâtre Hermite at the Institut Henri Poincaré.

Monday will be devoted to talks by young researchers from the European Networks HA and  HARP (See Below)

Preliminary Schedule of Invited Talks (Subject to modifications):
 
 
  Tuesday  Wednesday Thursday  Friday 
9:30 -10:00 Coffee Coffee Coffee Coffee
10:00 -10:50 A. HULANICKI G. ALEXOPOULOS L. ESCAURIAZA B. FRANCHI
11:00 - 11:50 D. MÛLLER  D. ROBINSON P. GERARD T. De PAUW
  Lunch Lunch Lunch Lunch
14:10 -15:00 P. SJÖGREN Y. MEYER W. HANSEN P. TCHAMITCHIAN
15:10 -16:00 S. MEDA V. MAZ'YA Y. MARTEL N. BOURNAVEAS
16:00 - 16:30 Coffee Break Coffee Break Coffee Break Coffee Break
16:30 -17:20  A. VOLBERG P.-G. LEMARIE  W. HEBISCH G. DAVID

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Program of the young researchers' day (Monday, April 22).

The conferences will start at 10:00AM after registration and coffee.
 
A. ALFONSECA (Universidad Autonoma de Madrid) An almost-orthogonality principle for families of maximal operators along directions
S. DUBOIS (Université de Picardie-Jules Verne) Solutions to the Navier-Stokes equations in L3,\infty and energy inequalities
V. FISCHER (Université de Paris-Sud) Maximal inequalities for homogeneous spheres on type h groups
G. FURIOLI (Università di Genova) On the Cauchy problem for the Schrödinger equation on the Heisenberg group
G. GIGANTE (Università degli Studi di Bergamo) Bessel functions and oscilatory integrals associated with the Schrödinger equation
D. LEVIN (Université de Cergy-Pontoise) On isoperimetric dimensions of product spaces
R. LILJENDAHL (Göteborg University)  The maximal operator for some non-doubling measures
V. OLEVSKII (University of Edimburgh) On estimates for Schur multipliers in Sp
E. SASSO (Università di Genova) Functional calculus for the Laguerre operator

PLANNING OF MONDAY APRIL 22:
 

  Monday
9:30-10:00 Coffee
10:00-10:30 G. FURIOLI
10:40-11:10 V. FISCHER
11:20-12:00 A. ALFONSECA
12:10-12:40 R. LILJENDAHL
  Lunch
14:10-14:40 G. GIGANTE
14:50-15:20 S. DUBOIS
15:30-16:00 E. SASSO
16:00-16:30 Coffee Break
16:30-17:00 D. LEVIN
17:10-17:40 V. OLEVSKII

ABSTRACTS

A. ALFONSECA: This is a joint work with F. Soria and A. Vargas. Given a set of angles W, we study the relation between the maximal function defined on a basis of rectangles forming an angle belonging to W with the x-axis, and the maximal functions associated to subsets of W. We obtain an almost-orthogonality result in L2, and an extension involving a square function for Lp with p other than 2. We give applications to related problems, in particular we give a simple proof of a recent result by N. Katz.

S. DUBOIS: We give simple conditions guaranteeing that solutions to the Navier-Stokes equations fulfil complete energy equalities. Combining this with a uniqueness and persistency result, we prove that solutions in C([0,T*[,X), where X = L3,\infty(R3), with initial data in the closure in X of the Schwartz class and in L2(R3) fulfil complete energy inequalities with a loss of energy at the possible singularities.

V. FISCHER:   fischer.tex On each group N = AÅ^Z of Heisenberg type (where Z is the center), let n=(X,Z) ®| n |=(| X |4+| Z |2)[1/4] denote the chosen homogeneous quasi-norm, and let M .f (n) be the supremum of the averages of f taken over all homogeneous spheres (for the previous quasi-norm) centered at n . We prove results on the Lp(N)-boundedness of f® M .f.

G. FURIOLI: We present some recent results obtained in a joint work with A. Veneruso. We consider the Schrödinger operator related to the full Laplacian DF on the Heisenberg group Hn and prove new Strichartz inequalities for the solution of the Cauchy problem ut = iDF u +f,     f  in L1((0,T), L2( Hn))) with initial data u(0,x)=u0  in L2( Hn). A key point consists in estimating the decay in time of the L\infty norm of the free solution; this requires a careful analysis due to the non homogeneous nature of the full Laplacian.

G.GIGANTE: (This is joint work with Fernando Soria, from Universidad Autónoma de Madrid (Spain).)  In this talk I would like to discuss the magnitude of some oscillatory integrals involving Bessel functions. These integrals arise in the problem of determining the minimal  Sobolev smoothness  a  required to bound  a  maximal square function for solutions to the Schrödinger equation in Rn+1+ with initial datum f in Ha(Rn).

D. LEVIN: It is well-known that dimensions of Euclidean spaces add up, if one considers their product, \BbbRd=\BbbRm ?\BbbRn, d=m+n. For Riemannian manifolds, the notion of dimension is more delicate, e.g. the topological dimension does not reflect their geometry at infinity. However, one may introduce an isoperimetric dimension through isoperimetric inequalities. The dimension introduced in this way is not a number but a family of functions indexed by a parameter p, 1 < p < ¥. Our main result generalizes the addition of dimensions in the euclidean case using the notion of the isoperimetric dimension.

R. LILJENDAHL: We consider the Hardy-Littlewood maximal operator with respect to non-centered balls and a non-doubling measure. We give some results on the boundedness of the maximal operator when the measure only depends on the first coordinate.

V. OLEVSKII: We give sharp estimates for the Schur multiplier norm of some classes of kernels, through their variation. We shall also discuss the relation with the commutative case (Rubio de Francia's inequality).

E. SASSO: In this talk we shall present results on the boundedness of spectral multipliers associated to the multidimensional Laguerre operator La. It is well known that, for special values of a, the analysis of the Laguerre operator can be interpreted as the analysis of the Ornstein-Uhlenbeck operator acting on "polyradial" functions. Exploiting this relation, we prove that if M is a bounded holomorphic function on the sector Sp={z in C: | argz| < arcsin|2/p-1|}, for 1 < p < \infty, and satisfies suitable Hörmander type conditions on the boundary, then the spectral operator M(La) is bounded on Lp with respect to the Laguerre measure. We also obtain a weak type (1,1) estimate for multipliers of Laplace trasform type, as well as a necessary condition for multipliers whose norm is dilation invariant. Our results are the countepart, for the Laguerre operator, of results of Garcia-Cuerva, Mauceri, Meda, Sjógren, Torrea for the Ornstein-Uhlenbeck operator.

back to program of Monday April 22
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Abstracts

N. BOURNAVEAS:  We shall discuss the Klainerman-Machedon programme of low regularity local solutions of nonlinear wave equations and their relation to global solutions, with particular attention to the Dirac-Klein-Gordon equations. List of Titles

G. DAVID: Nous essaierons de décrire certains aspects des résultats de Xavier Tolsa sur la capacité analytique g. Si K est un ensemble compact du plan complexe, g(K) mesure combien de fonctions analytiques bornées on peut construire sur le complémentaire de K. Tolsa montre que g(K) est équivalente à la quantité g+(K) où l'on ne considère que les fonctions analytiques bornées obtenues par convolution du noyau de Cauchy avec une mesure positive. Il en déduit une description géométrique des ensembles effaçables pour les fonctions analytiques bornées, et la semi-additivité de g. List of Titles

T. de PAUW: We will motivate and describe a common work with Robert Hardt. We consider Plateau type variational problems related to the size minimization of rectifiable currents. We realize the limit of a size minimizing sequence as a stationary varifold and a minimal set. Other examples of functionals to be minimized include the integral over the underlying carrying set of a power q of the multiplicity function, with 0 < q <1. Because minimizing sequences may have unbounded mass we make use of a more general object called a rectifiable scan for describing the limit. This concept is motivated by the possibility of recovering a flat chain from a sufficiently large collection of its slices. In case the given boundary is smooth and compact, the limiting scan has finite mass and corresponds to a rectifiable current. List of Titles

L. ESCAURIAZA:  I will describe some results in collaboration with Luis Vega and Javier Fernández related to unique continuation properties for solutions to parabolic equations. The references are

1. L. Escauriaza. Carleman inequalities and the Heat operator. Duke Math. J.,104, n.1 (2000), 113-127.

2. L. Escauriaza, L. Vega. Carleman inequalities and the heat operator II.Indiana Math. J. 50, n. 3 (2001), 1149-1169.

3. L. Escauriaza, J. Fernández. Unique continuation for parabolic operators. (to appear). List of Titles

B. FRANCHI: Carnot groups endowed with their Carnot-Carathéodory distance provide the simplest example of metrics spaces that are not Euclidean even locally (i.e. they are not Riemannian manifolds), but nevertheless have a rich structure, since they have intrinsic translations and dilations. The Heisenberg group Hn is the simplest nontrivial example of Carnot group. It is possible to introduce a notion of perimeter in Carnot groups, and in addition to prove an intrinsic isoperimetric inequality in this setting. It is then natural to ask whether it is possible to develop a geometric measure theory for Carnot group; in particular we are interested to analyse the structure of sets of locally finite intrinsic perimeter in Carnot groups, extending the classical result due to De Giorgi in the Euclidean setting. A key point in this program is the correct definition of regular hypersurfaces, since, because of distortion of the geometry at any scale, Carnot groups do not contain surfaces meant as deformations of Euclidean spaces, but we need an utterly new notion. Other problems arise also from the lack of sophisticated covering lemmata that hold in the Euclidean setting (Besicovitch covering lemma). Nevertheless, it is possible to carry out the proof of a structure theorem that is the exact counterpart of De Giorgi's theorem.List of Titles

P. GERARD:  (his is a jointwork with Nicolas Burq and Nikolay Tzvetkov (Université de Paris-Sud)). I shall report on some recent results on the Cauchy Problem for the nonlinear Schrödinger equation (NLS) on compact manifolds. First we establish Strichartz-type estimates for the linear Schrödinger equation on arbitrary compact manifolds, with some loss of derivatives which turns out to be optimal on spheres. Then, using these estimates, we prove global existence of smooth solutions for defocusing NLS on arbitrary compact surfaces and for defocusing cubic NLS on arbitrary compact three-manifolds, generalizing a result due to Bourgain on standard tori. Finally, we show some new instability properties for NLS on spheres.List of Titles
 

W. HANSEN: Given a potentially bounded signed measure m on a Brelot space (X,H) with Green function G, it is well known that m-harmonic functions (i. e., in the classical case, finely continuous versions of solutions to Du - um = 0) may be very discontinuous. In this paper it is shown that under very general assumptions on G (satisfied for large classes of elliptic second order linear differential operators) normalized perturbation, however, leads to a Brelot space (X,[(H)\tilde]m) admitting a Green function Tm(G) which is locally comparable with G and has all properties required of G before. In particular, iterated perturbation is possible. Moreover, intrinsic Hölder continuity of quotients of harmonic functions with respect to the local quasimetric r:=(G-1 +* G-1)/2 yields r-Hölder continuity for quotients of m-harmonic functions as well.List of Titles

A. HULANICKI: hulanicki.tex  We treat noncoercive invariant second order degenerate elliptic operators on simply connected homogeneous manifolds of negative curvature. J.Wolf [Wolf] and E.Heintze [Hei] proved that such a manifold is isometric with a solvable Lie group S=NA, being a semi-direct product of a nilpotent Lie group N and A= R + and, moreover, for a H ÎA the Lie algebra of A the eigenvalues of AdH|N are all greater than 0. Conversely, every such a group equipped with a suitable left-invariant metric becomes a Riemannian manifold with negative curvature.

We consider on S a second order left-invariant operator
L= m
å
j=1
Yj2+Y, 
such that Y1,...,Ym generate S. Let p: S® A=S\slash N be the canonical homomorphism. dp(L) is an elliptic invariant operator on R+, hence
dp(L)=(aa)2-gaa
for a g Î R. L=Lg is coercive if and only if g ? 0.

Let mt be the semigroup of measures generated by Lg. If g? 0, then there is a unique (up to a constant) positive Radon measure ng on N such that
mtg*ng=ng,    t > 0. 
ng is bounded, while n0 is unbounded and it is called the Harris measure for mt. In a recent paper E.Damek A.Hulanicki and R.Urban [Revista Matemática Iberoamricana 17 (2001), 257-295] proved
C-1(1+|x|)-Q-g£ mg (x) £ C(1+|x|)-Q-g,  x Î N. 
In the talk we discuss the existence of the limit

lim
x®¥
|x|Qm0(x).

  List of Titles

P.-G. LEMARIE-RIEUSSET: The role of Lorentz spaces in the Navier-Stokes problem has been underlined by Yves Meyer, who gave a simple proof for the uniqueness in L3 in the 3D case. We shall give in this lecture an elementary proof of the results of Necas and Tsai on the non-existence of Leray's self-similar solutions with a profile in L3 (Necas - proof by L3 uniqueness) or fullfilling a local energy inequality in the neighbourhood of the blow-up (Tsai - proof by Lorentz spaces) ; in both cases, we avoid using the regularity criterion of Caffarelli, Kohn and Nirenberg.List of Titles

Y. MARTEL: For the critical generalized KdV equation we present a series of papers proving the existence of a class of solutions which blow up in finite time in the energy space, and giving a precise qualitative study of the blow up phenomenon (blow up profile, blow up rate).  List of Titles

V. MAZ'YA:  mazya.tex   (This is a joint work with I. Verbitsky.)  We give a complete characterization of the class of functions (or, more generally, complex-valued distributions) Q such that the following inequality holds:
| ó
õ

Rn

|u(x)|2Q(x)dx const ó
õ

Rn

u(x)|2dx,
where the constant is independent of u Î C¥0(Rn). Similar inequalities are proved for the inhomogeneous Sobolev space W12(Rn). In other words, we establish a criterion for form-boundedness of Q relative to the Laplacian D under no a priori assumptions on Q. For the Schrödinger operator L=-D+Q, our criterion describes the class of admissible perturbations Q such that L:L12(Rn)® L2-1(Rn). We also establish similar boundedness and compactness criteria for the relativistic Schrödinger operator. List of Titles

S. MEDA:  meda.tex Let g be the Gauss measure on Rd and L the Ornstein-Uhlenbeck operator
- 1
2
D+ x·Grad
(D is the standard Laplacian on Rd), which is self adjoint in L2(g). We will discuss a couple of results concerning the Lp(g) boundedness of spectral multipliers for L.

More specifically, for every p in (1,\infty) such that p is not equal to 2, set fp* = arcsin|2/p-1|, and consider the sector Sfp* = {z in C: |argz| < fp*}. The first result is that if M is a bounded holomorphic function on Sfp*, whose boundary values on Sfp* satisfy suitable Hörmander type conditions, then the spectral operator M(L) extends to a bounded operator on Lp(g) and hence on Lq(g) for all q such that |1/q-1/2|<|1/p-1/2|. The result is sharp, in the sense that L does not admit a bounded holomorphic functional calculus in a sector smaller than Sfp*.

This is in striking contrast with some classical situations, such as that of spectral multipliers for the standard Laplacian on Rd, where a well known ``non-holomorphic functional calculus'' can be developed.

Then, we will discuss a recent result concerning the problem of finding necessary conditions on M that imply that M(L) is bounded on Lp(g) for some p not equal to 2.List of Titles

Y. MEYER: We aim at discussing the new proof by Michel Ledoux of the improved Gagliardo-Nirenberg inequalities. These inequalities allow to interpolate between the space BV of functions with bounded variation and the Besov space B\infty-1,\infty. This new proof is no longer based on some wavelet analysis of the space BV of functions with bounded variations. It is instead very much in the spirit of the Littelwood-Paley theory `à la Stein'. This new proof yields better constants as the dimension tends to infinity. List of Titles

D. MÜLLER:  mûll er.tex Let Vj be left-invariant vector fields on the Heisenberg group Hn, such that [Vj,Vj+n]=U,  1 <  j < n) are the only nontrivial commutation relations, and consider the operator
LA,a= 2n
å
j,k=1
ajkVjVk+iaU,
where A=(ajk) is a complex symmetric matrix satisfying ReA > 0, and where a in C. Such operators may be considered as generalizations of sub-Laplacians. In a previous joint paper with F. Ricci, we had studied the questions of local solvability of such an operator under the condition that LA,a satisfies a cone condition in the sense of Sjöstrand and Hörmander, i.e. that |ImA| > C ReA. If we denote by S the Hamiltonian map associated to LA,a, then the cone condition implies that S has no real eigenvalue, with the possible exception of the value 0, and that the Jordan block Sr of S associated to the eigenvalue zero satisfies Sr2=0. It turned out recently that the cone condition can be considerably relaxed, by just assuming that the spectrum of S is as described and that Sr2=0. If then Sr  is not equal to 0, one can prove that LA,a is locally solvable for every a. The proof will rely on explicit formulas for Mehler-type kernels arizing when exponentiating LA,a. Finally, I shall give an example where Sr is 3-step nilpotent, which indicates that the situation is presumably very different when Sr is nilpotent of a step higher then 2. List of Titles

D. ROBINSON: In this talk we give a brief review of estimates on heat kernels of complex, subelliptic, second-order, right invariant operators acting on Lie groups of polynomial growth. The emphasis is on asymptotic estimates and global properties. The main features of the analysis are structure theory of the Lie groups, homogenization theory and De Giorgi estimates. List of Titles

P. SJÖGREN:  sjögren. tex  Let g be the Gaussian measure in  Rd. The Ornstein-Uhlenbeck semigroup Ht,  t > 0, defines a bounded operator Hz on Lp(g) when t is replaced by a complex variable z staying in a closed set Ep. We examine the maximal operator H*pf=sup{ |Hzf|: z in Ep}. For p < 2 it is known that H*p is not of strong type nor of weak type p,p; however, it will be of weak type if a neighbourhood of a certain critical point is deleted from Ep in the definition of H*p. We shall consider instead the case p > 2, and prove that H*p is of weak type but not of strong type p,p. But if a neighbourhood of the origin is deleted from Ep in the definition of H*p, the resulting operator turns out to be of strong type. List of Titles

P. TCHAMITCHIAN: (joint work with P. Auscher and E. Russ) Let D be a strongly Lipschitz domain of Rn. Hardy spaces on D have been introduced by Miyachi, by Jonsson, Sjögren and Wallin and by Chang, Krantz and Stein. We investigate spaces of functions on D whose gradients belong to these Hardy spaces on D. In particular, we characterize these spaces via suitable maximal functions. Then, using this characterization, if L=-divAGrad is an elliptic second order divergence operator on D subject to the Dirichlet or the Neumann boundary condition, we compare the norms of L1/2f and Gradf in suitable Hardy spaces on D, depending on the boundary condition, under some technical assumptions on L. This solves the square root problem for L is this context. List of Titles

A. VOLBERG:  volberg.tex  Quasiregular map is a W12 solution of the Beltrami equation f\bar z-mfz = 0, where the function m (called Beltrami coefficient) satisfies ||m||\infty = k < 1. These maps self-improve, becoming of class W1p(k) with p=p(k) > 2. In particular, they are continuous, discrete, open,...

There were two questions (going sometimes by the name of problems of Gehring and Iwaniec):

1) What is the value of p(k)? In other words, what is the best smoothness of of the solution of the Beltrami equation, which is a priori in W12? This problem has been solved by Kari Astala.

2) What is the least smoothness, which allows the self-improvement? In other words, what is the smallest q=q(k) < 2 such that the solution of the Beltrami equation, which is a priori in W1q, will be necessarily in W12 (and, thus, continuous, discrete, open,...)?

Nicely, the exponents p, q turns out to be dual: 1/p+1/q = 1. And as it has been suggested by Gehring and Iwaniec p=1+1/k, q = 1+k.

On the level of sestimates those questions can be reduced to ``calculating" the norms of a certain combination of second order Riesz transforms on the plane. This combination is called the Ahlfors-Beurling transform.

Namely, we will be using the harmonic analysis Bellman function and the heat extension to explain how the Stochastic Optimal Control approach can be useful in estimating the Ahlfors-Beuling transform and many othe types of Riesz transforms.   List of Titles
 
 

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Organizing Committee
     
P. AUSCHER Université de Picardie
G. BESSON Université Joseph Fourier
A. BONAMI Université d'Orléans
T. COULHON Université de Cergy-Pontoise
A. GRIGOR'YAN Imperial College, London
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Registration form

Registration is mandatory and free. This will give you access to computer facilities and to the library of the Institute.
To register for the conference,  get back to the page on the ihp site and click on  registration form.

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Practicalities

The Conference will take place at the Institut Henri Poincaré: this is located
11, rue Pierre et Marie Curie
75231 Paris Cedex 05
tel: 33 1  44  27 67 89
fax: 33 1 43 25 40 67

How to get to the Conference Center and a list of hotels: here

Money exchange: French official currency is now the Euro, which is worth approximately 0.9 US Dollars.

Local transportation: Enjoy the many possibilities of Paris transportation : metro, RER, bus. The best if weather is appropriate
is your feet.

How to telephone in France: There are very few public phones that operate with cards or Visa. It is more convenient and also cheaper to use telephone cards. You can find them at most places selling newspapers or cigarettes.

Meals: Finding food will be your responsability. You'll find many restaurants (typical or exotic), bars, fast foods around the Institute.

Computer facilities: Registered participants to the conference can use machines (10 Mac and 5 PC), in common rooms.  Some are equipped with Linux.

Library: Registered participants to the conference have access to the library of the Institute.

Weather:  A French saying: "en Avril, ne te découvre pas d'un fil." It means, be careful, the temperature could be colder than you think. But one can expect nice, sunny and warm days.
 

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