Nicolas Roy's Curriculum Vita - Institut für Mathematik - Humboldt ...

NicolasRoy’s
CurriculumVita
Personal
Date of birth: 13Feb. 1974
Birthplace: MaisonsLaffittes, France
Nationality: French
Maritalstatus: Married, 2 children
Address: GeometricAnalysisGroup,InstitutfürMathematik,HumboldtUniversität,Rudower
Chaussee 25,Berlin D-12489,Germany
Email: roy@math.hu-berlin.de
Web: http://www.math.hu-berlin.de/∼roy/
Positions
Since Oct. 2006. Assistant position inthe GeometricAnalysisGroup,Institut fürMathematik,
Humboldt Universität, Berlin
Oct. 2003-Sep. 2006. Post-doctoral position inthe GeometricAnalysisGroup
Teaching experience
Assistant ”Mitarbeiter”, Humboldt Universität zuBerlin
09-10 Tutorial Linear Algebra1
09-10 Seminar Introduction toDynamicalSystems
08-09 Tutorial Linear Algebra1&2
07-08 Tutorial Linear Algebra2
06-07 Tut. DifferentialGeometry1
06-07 Tut. Analysison manifolds
06-07 Tut. Analysis1
ATER, Mathematics department, Grenoble
02-03 Tut. Introduction todynamicalsystems,DEUG SMb 1
02-03 Tut. Mathematicalmethodsof physics,IUTGénieMeca 2
Moniteur,Physics department, Grenoble
01-02 Practical Electrostaticsand Magnetostatics, DEUGSMa 1et MIAS1
01-02 Tut. Mathematicalmethodsof physics,IUPGénieélectrique 2
00-01 Pract. Electrostaticsand Magnetostatics, DEUGSMa 1etMIAS1
00-01 Tut. Mathematicalmethodsof physics,IUPGénieélectrique 2
99-00 Pract. Electrostaticsand Magnetostatics, DEUGSMa 1etMIAS1
99-00 Tut./Pract. Modelisationof physicalsystemsand numerics,Licence Physique
Recherche
Other
97-99 Instructor ofcomputer science and French language, asconscientious objector,
Grenoble
98-03 Music teacherin my own Jazzschool CEMA,Grenoble. Harmony, pianojazz,
improvisation, eartraining...
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Education
PhD in Mathematics, Sept. 2003, Institut Fourier, Grenoble (France), under supervision of
Y. Colin deVerdière, on deformationsof classicaland semi-classicalcompletelyintegrable systems
Master degree in Theoretical Physics, 1997, Ecole Normale Supérieure de Lyon (France).
Master thesis on Conformal transformations of (pseudo)Riemannian manifolds
UndergraduateandgraduatestudiesinPhysics,1992-1997,UniversitéJosephFourier,Grenoble
Practicalperiods
1997, Institut dePhysique Nucléaire de Lyon, Lyon, 3months
1996, Institut deFísica d’AltesEnergies, Barcelona, 6 months
1995, European Synchrotron Radiation Facility, Grenoble, 2 months
1993, Institut desSciences Nucléaires, Grenoble, 2 months
Invitedtalks
7th Sep. 2009,"Quantum dynamics”conference, Lyon, France
14th May. 2009, "Dynamicalsystems”seminar, Freie Universität, Berlin
21stApr. 2009,"Nonlineardynamics”seminar, Freie Universität, Berlin
16th Mar. 2009,"Symplecticand contactgeometry"seminar,Brussels
21stJan. 2009,Workshop "Resonancesin physics and mathematics", Marseille (France)
14th May. 2008, ”Geometric analysis and spectral theory” seminar, Humboldt Universität,
Berlin
13th Dec. 2006, ”Geometric analysis and spectral theory” seminar, Humboldt Universität,
Berlin
19th Jan. 2004,”Mathematical physics” seminar, Technische Universität, Berlin
3rd Dec. 2003, ”Geometric analysis and spectral theory” seminar, Humboldt Universität,
Berlin
12thNov. 2003,Workshop”SpectralproblemsforSchrödinger-typeoperatorsII”,Humboldt
Universität, Berlin
13th May2003,Topology seminar, AarhusUniversity, Aarhus(Denmark)
Research visits
Feb. 2009,MFO Oberwolfach (Allemagne), 2weeks, grant ”Research in Pairs”
Mai2006,Roma (Italy), 1 week, invited byC. Liverani.
Administrative duties
19-25 Jul. 2008, co-organiser and webmaster of the workshop ”Symplectic Field Theory 3”,
Berlin
24-26 Apr. 2008, co-organiser and webmaster of the conference”Mathematical Physics and
Spectral Theory”,Berlin
2-3Nov2007,co-organiserandwebmasteroftheconference”GeometricAnalysisandSpectral
Theory”,Berlin
2006-2009,Organiser of the ”Geometricanalysis”seminar
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Attended conferences and summerschools
1999-2009, many international mathematics workshops and conferences : Munich, Strasbourg,
Berlin, Bruxelles, Peyresq, Lille, Leipzig, Nantes, Giens, Stare Jablonki, Cargèse, Bordeaux,
Villetaneuse,Montpellier,Reims,...
Languages
Mother tongue : French
Fluently spoken andwritten languages : German,English, Polish
Music in general andjazz in particular
Trekking, climbing, skiing, biking
Computers
Personal Interests
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Published papers
Listof publications 1
[7] N. Roy and V. Humilière, The geometry of the space of fibrations. To appear in Ann.
Global Anal. Geom. (2009)
[6] N. Roy, F. Faure and J. Sjöstrand, Semi-classical approach for Anosov diffeomorphisms
and Ruelleresonances. Open Math. Journal (2008),vol. 1,35–81
[5] N. Roy, A semi-classical K.A.M. theorem. Comm. Partial Differential Equations 32
(2007), no. 5, 745–770
[4] N. Roy and F. Faure, Ruelle-Pollicottresonances for real analytic hyperbolicmaps. Nonlinearity
19(2006), no. 6, 1233–1252
[3] N. Roy, Intersections of Lagrangian submanifolds and the Mel’nikov 1-form. J. Geom.
Phys. 56(2006), no. 11,2203–2229
[2] N. Roy, Regular deformations of completely integrable systems. J. Symplectic Geom. 3
(2005), no. 1, 1–16
[1]N.Roy,Thegeometryofnondegeneracyconditionsincompletelyintegrablesystems. Ann.
Fac. Sci. Toulouse Math. (6)14(2005), no. 4, 705–719
Doctoral Thesis
[0] N. Roy, Sur les déformations des systèmes complètement intégrables classiques et semiclassiques.
Grenoble (France), September2003
Submited articles
N. Roy, A Weinstein’stubular neighbourhood for Lagrangian fibrations. ArXiv 0905.0594
Articlesin preparation
N. Royand V. Humilière,The geometryof the spaceof completelyintegrablesystems
1 pdffiles availableatØØÔÛÛÛÑØÙÖÐÒÖÓÝ
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Summaries of published articles
[0] Sur lesdéformationsdessystèmescomplètementintégrablesclassiquesetsemi-classiques
Inthisthesis,westudyseveralaspectsofperturbationsofcompletelyintegrablesystems.
The first part deals with completely integrable Hamiltonian systems (classical mechanics)
and their deformations. This work as developped further and gave rise to the articles [1,2]
described below. The second part concerns perturbations of semiclassical completely integrable
systems. We introduce the theory of Pseudodifferential operators with small parameter
on the torus. The particular algebraic structure of the torus provides a weel-defined
notion of total symbol for these operators. The main results of this part were improved and
are presented in[5].
[1] Thegeometryofnondegeneracy conditions incompletelyintegrablesystems
In K.A.M-like theorems one needs to impose nondegeneracy conditions on the completely
integrable Hamiltonian H0 in order to insure that the set of Diophantine tori (which
persist after a small perturbation H0 + εK) has large measure. These conditions are usualy
given in action-angle coordinates, but it is possible to formulate them geometrically, by
considering completely integrable systems defined on a symplectic manifold by a fibration
in Lagrangian tori and a Hamiltonian function constant on the fibers. In this article, we
give a geometric definition for several nondegeneracy conditions, we explain the different
implication relation that exist between them, and we show the unicity of the fibration for
non-degenerate Hamiltonians.
[2] Regular deformationsofcompletelyintegrable systems
Given a symplectic manifold (M, ω) and a fibration M → B in lagrangian tori, we show
thatanyfamily φ ε ofsymplectomorphims canbedecomposed inauniquewayas φ ε = Φ ε ◦
ϕ ε ,with Φ ε afamilyoffiber-preservingsymplectomorphimsand ϕ ε afamilyofHamiltonian
symplectomorphims. This normal form is used in a second part to study deformations of
non-degenerate regular completely integrable hamiltonian. Namely, we make explicit the
spaceofperturbationsKofacompletelyintegrableHamiltonian H0,suchthattheperturbed
Hamiltonian H0 + εK is“completelyintegrable up toO ε 2 ”.
[3] Intersectionsof Lagrangian submanifoldsand the Mel’nikov1-form
Mel’nikov’s theory/method deals with Hamiltonian systems H0 ∈ C∞ R2 with a hyperbolic
point x0 and their perturbations Hε := H0 + εK, where H1 (t) ∈ C∞ R2 × R depends
on the time in a periodic way. We assume that the stable and the unstable manifolds
of x0 coincide, thus forming together a manifold of homoclinic points. Meln’ikov’s method
provides a way to detect homoclinic points in the perturbed system Hε, in which the stable
andunstable manifolddonolongercoincide. Namely,themethodpredictthatthesehomoclinic
points are given by the zeros of a function (called “Mel’nikov function”) and that this
function admitsan integral form, in which appearsonlythe flow of H0.
This method extends to perturbations of Hamiltonian systems having two transversally
hyperbolic periodic orbits linked by a heteroclinic manifold. Unfortunately, the most of
the works on this topic deal always with models, written in a coordinate system and with
an explicit splitting between periodic and transversal varaibles. Besides the fact that these
models are non-generic, they alsohide the geometrical characterof the method.
In this article, we develop the general theory of intersections of pairs of familiies of La-
grangian submanifolds N ± ε , with N+ 0 = N− 0
, and constrained to live in an auxiliary family
of submanifolds N ± ε ⊂ Pε ⊂ M. We show that the intersections N + ε ∩ N − ε which survive
when ε = 0 are detected by a 1-form on N0, called Mel’nikov 1-form. We show that it is
the appropriate tool for studying heteroclinic points between two transversally hyperbolic
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periodicorbits. This1-formadmitsanintegralexpressioninvolvingonlytheflowof H0,but
which is in general divergent. We give several ways to treat this divergence problem and
we explain whyit doesapparentlynot appearin the above-mentionned particular models.
[5] Asemi-classicalK.A.M. theorem
We consider a semiclassical completely integrable system defined by a pseudodifferential
operator ˆH, with small parameter ¯h, on the torus T d , whose total symbol is a classicaly
completely integrable Hamiltonian H ∈ C ∞ T ∗ T d . We study perturbed operators of the
form ˆH + ¯h κ ˆK, where ˆK is any pseudodifferential operator and κ > 0, and we show the
existence ofsemiclassical normal forms forthese operators. Thisisused toconstruct a large
number of quasimodes, in analogy with the K.A.M theorem in classical mechanics which
proves the persistency of a large number of invariant tori after perturbation. Moreover, the
first correction to the eigenvalues is related to the average of the symbol of the perturbation
ˆK. In order to prove these result, one needs to perform a very precise analysis of the
resonances and the diophantine tori with parameters dependingon ¯h.
[4] Ruelle-Pollicottresonancesfor realanalytic hyperbolicmaps
Weconsider two simplemodelsof uniformly hyperbolic dynamical systems : expansive
maponthecircleS 1 andhyperbolicmaponthetorus T 2 ,andwestudythedecayoftimecorrelation
functions, which isfundamental to establish other chaotic properties of the system.
To achieve this, we study the Ruelle transfert operator ˆM defined as the pull-back operator
ˆM (ϕ) = ϕ ◦ M, where M is a chaotic real analytic map on S 1 (resp. T 2 ) and ϕ ∈ L 2 S 1
(resp. ϕ ∈ L 2 T 2 ). We show that the Ruelle-Pollicott resonances, which describe the time
correlation functions Cϕ,φ (n) = ϕ. ˆM n (φ) dx, with n ∈ N, can be obtained as the eigenvalues
of a trace class operator on L 2 S 1 (resp. L 2 T 2 ). We show moreover that these
resonances accumulate exponentiallyfast on 0.
[6] ASemi-classicalapproachfor Anosov diffeomorphismsand Ruelle resonances
If f is an Anosov diffeomorphism on a compact manifold, the decay of the dynamical
correlation functions is governed by the so-called Ruelle resonances. It follows from the
works of Baladi & al and Liverani & al, that these resonances can be obtained by a suitable
spectral analysisofthecomposition operator (oranotherone relatedto it)calledthe "Transfer
operator". In this paper, we show how these results can be obtained by a systematic
microlocal analysis, extending the approach of the previous work [4].
[7] Thegeometryofthe spaceof fibrations
We study geometrical aspects of the space of fibrations between two given manifolds
M and B, from the point of view of Frechet geometry. As a first result, we show that any
connected component of this space is the base space of a Frechet-smooth principal bundle
with the identity component of the group of diffeomorphisms of M as total space. Second,
weprovethatthespaceoffibrationsisalsoitselfthetotalspaceofasmoothFrechetprincipal
bundle with structure group the group of diffeomorphisms of the base B.
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