<!doctype html public "-//w3c//dtd html 4.0 transitional//en"> <html> <head>    <meta http-equiv="Content-Type" content="text/html; charset=iso-8859-1">    <meta name="description" content="Colloque en l'honneur de Frdric Pham, Nice">    <meta name="keywords" content="Singularities, differential equations                   and  mathematical aspects of quantum physics">    <meta name="rating" content="general">    <meta name="revisit-after" content="31 days">    <meta name="robots" content="all">    <meta name="GENERATOR" content="Mozilla/4.78 [en] (X11; U; Linux 2.4.7-10smp i686) [Netscape]">    <title> Colloque Pham (Nice)</title> <!-- Changed by: Michel.MERLE, 21-Jan-2002 --> <!-- Changed by: Admin.CollPham, 05-Jun-2002 --> </head> <body bgcolor="#FFFFFF"> &nbsp; <center><table BORDER=0 > <tr> <td ALIGN=CENTER COLSPAN="3"> <center><b><font color="#E6241E"><font size=+3>Singularit&eacute;s, &eacute;quations diff&eacute;rentielles&nbsp;</font></font></b> <br><b><font color="#E6241E"><font size=+3>et aspects math&eacute;matiques de la physique quantique</font></font></b> <br>&nbsp; <p><b><font color="#E6241E"><font size=+3>Colloque en l'honneur de Fr&eacute;d&eacute;ric PHAM</font></font></b></center> </td> </tr> </table></center>  <blockquote> <blockquote> <blockquote> <blockquote>&nbsp; <center> <p><font color="#3333FF"><font size=+1>Titres et r&eacute;sum&eacute;s des conf&eacute;rences</font></font></center> </blockquote> </blockquote> </blockquote> </blockquote>  <br>&nbsp; <p><b>Norbert A'Campo </b>(B&acirc;le, Suisse) <blockquote><b><i>L'ombre r&eacute;elle de la fibre de Milnor</i></b></blockquote>  <blockquote>On examine la projection de la fibre de Milnor sur l'espace r&eacute;el. Comme application on prouve la nullit&eacute; des invariants d'enlacement de Milnor pour les entrelacs des singularit&eacute;s de courbes planes. On propose aussi un r&eacute;tract pour la fibre de Milnor, qui g&eacute;n&eacute;ralise le r&eacute;tract de Pham.</blockquote> <b>Kazuhiko Aomoto </b>(Nagoya, Japon) <blockquote><b><i>Gauss-Manin connections of Schlafli type for hypersphere arrangements</i></b></blockquote>  <blockquote>In the n dimensional unit sphere we consider an arrangement of hyperspheres which are hyperplane sections.&nbsp; These are parameterized by the inner products of coefficient vectors and constant coefficients which are basic invariants of the group <i>O(n+1)</i>. <br>The corresponding hyper geometric integrals satisfy a Gauss-Manin connection.&nbsp; These can be <br>described explicitly in terms of the basic invariants and their differetial 1-forms. This is an extension of well-known Schlafli formula for the volume of a geodesic simplex. <br>I shall give the explicit forms and&nbsp; raise some problems related to them.</blockquote>  <p><br><b>Carl M. Bender </b>(Saint Louis, USA) <blockquote><b><i>Properties of Non-Hermitian Quantum Field Theories</i></b> <p>Abstract: In this talk I discuss quantum systems whose Hamiltonians are <br>non-Hermitian but whose energy levels are all real and positive. Such theories <br>are required to be symmetric under CPT, but not symmetric under P and T <br>separately. Recently, quantum mechanical systems having such properties have <br>been investigated in detail. In this talk I extend the results to quantum field <br>theories. Among the systems that I discuss are $-\phi^4$ and $i\phi^3$ theories. <br>These theories all have unexpected and remarkable properties. I discuss the <br>Green's functions for these theories and present new results regarding bound <br>states, renormalization, and nonperturbative calculations. <br>&nbsp; <br>&nbsp;</blockquote> <b>Philip Boalch </b>(Strasbourg, France) <blockquote><b><i>Braiding of Stokes Multipliers</i></b> <p>I will explain how one obtains actions of (generalised) braid groups by considering isomonodromic deformations of irregular connections on G-bundles over a disc. <br>The relationship with quantum Weyl groups and with algebraic solutions of <br>Painlev&eacute; equations will then be discussed.</blockquote> <b>Louis Boutet de Monvel </b>(Paris, France) <blockquote><b><i>Quantification g&eacute;om&eacute;trique et op&eacute;rateurs de Toeplitz.</i></b></blockquote> <b>Yves Colin de Verdi&egrave;re </b>(Grenoble, France) <blockquote>&nbsp;<b><i>La g&eacute;om&eacute;trie et la dynamique semi-classique des croisements de modes.</i></b> <p>De nombreuses &eacute;quations d'ondes de la physique sont des syst&egrave;mes sym&eacute;triques (&eacute;quation de Maxwell, propagation d'ondes &eacute;lastiques, dynamique mol&eacute;culaire &agrave; la Born-Oppenheimer). <br>Le symbole principal d'un tel syst&egrave;me est donc une application <i>x \rightarrow A(x) </i>de l'espace <br>des phases &agrave; valeurs matrices sym&eacute;triques. <br>La vari&eacute;t&eacute; caract&eacute;ristique <i>Z={ x |&nbsp; det (A(x))=0 } </i>d&eacute;crit la microlocalisation des solutions. <br>&nbsp;En un point g&eacute;n&eacute;rique <i>x_0</i>&nbsp; de <i>Z</i>,&nbsp; le noyau de <i>A(x_0) </i>est de dimension 1 et <i>Z </i>est lisse de codimension 1 : l'onde est "polaris&eacute;e". Le syst&egrave;me semi-classique peut se r&eacute;duire en ces points &agrave; une &eacute;quation scalaire. On en d&eacute;duit des Ansatz naturels de type BKW pour les solutions. <br>D'apr&egrave;s un th&eacute;or&egrave;me de Von Neumann et Wigner, on s'attend &agrave; ce que,&nbsp; le long de sous-vari&eacute;t&eacute;s de codimension 3, le noyau de <i>A(x)</i> soit de dimension 2. L'exemple le plus simple en est la surface de Fresnel. <br>Le but de cet expos&eacute; est de d&eacute;crire&nbsp; les solutions du syst&egrave;me pr&egrave;s de ces points. Parmi les nombreuses contributions &agrave; ce probl&egrave;me, on peut citer les travaux fondateurs de Landau, Zener, <br>puis ceux de&nbsp;&nbsp; Hagedorn, Joye, Littlejohn et Flynn, C. Fermanian-Kammerer et P. G&eacute;rard. <br>Arnold, puis Duistermaat ont propos&eacute; une forme normale pour un tel syst&egrave;me dans la situation g&eacute;n&eacute;rique. <br>Apr&egrave;s avoir introduit les hypoth&egrave;ses g&eacute;om&eacute;triques de g&eacute;n&eacute;ricit&eacute;, <br>je vais d&eacute;crire cette forme normale et montrer comment on peut en d&eacute;duire <br>des r&eacute;sultats g&eacute;n&eacute;raux sur le probl&egrave;me des conversions de modes. <br>&nbsp;</blockquote>  <p><br><b>Lucia Di Vizio </b>(Toulouse, France) <blockquote><b>p<i>-adic </i>q<i>-difference equations</i></b> <p>Let <i>q </i>be a nonzero complex number.&nbsp; One usually calls <i>q</i>-difference operator the operator $<i>\varphi_q(f)(x)=f(qx)</i>$ or the associated <i>q</i>-derivation <center> <p><i>d_q(f)(x) = f(qx)-f(x) \over&nbsp; (q-1)x</i></center>  <p>both acting on a convenient ring of functions. <br>The relation&nbsp; <i>d_q x^n=(1+q+\dots+q^{n-1})x^{n-1}&nbsp;</i> shows intuitively that <br><i>d_q\rightarrow {d\over dx}</i> when <i>q </i>tends to 1: this phenomenon is known as confluence. <br>In a recent paper J. Sauloy proves the confluence of the so-called <i>Birkhoff matrix</i> of a fuchsian <i>q</i>-difference equation to the monodromy of the limit differential system. <br>We will introduce <i>p</i>-adic <i>q</i>-difference equations and their first properties:&nbsp; namely their weak Frobenius structure and transfer theorems.&nbsp; Our motivation is the study of the confluence <i>p</i>-adic monodromy.</blockquote> <b>Patrick E. Dorey </b>(Durham, UK) <blockquote><b><i>On reality properties of some problems in PT-symmetric quantum mechanics.</i></b> <p>Many years ago Bessis and Zinn-Justin conjectured that the spectrum of the cubic quantum-mechanical oscillator with purely imaginary coupling should be entirely real. <br>Bender and Boettcher subsequently generalised&nbsp; this conjecture to cover a whole class of problems, all of them sharing&nbsp; a property known as PT symmetry. <br>In this talk I will sketch how an unexpected link with the theory of integrable models, itself inspired by the work of Sibuya and Voros, has led to a proof of these conjectures. <br>(Joint work with Clare Dunning [York, UK] and Roberto Tateo [Durham, UK]). <br>&nbsp; <br>&nbsp;</blockquote> <b>Jean Ecalle </b>(Orsay, France) <blockquote><b><i>R&eacute;surgence double, perturbations singuli&egrave;res, dimorphie.</i></b></blockquote>  <blockquote>Nous pr&eacute;senterons, puis nous utiliserons, une famille de ``<i>mon&ocirc;mes de r&eacute;surgence</i>" sp&eacute;cialement adapt&eacute;s &agrave; l'&eacute;tude des perturbations&nbsp; singuli&egrave;res et de leurs ph&eacute;nom&egrave;nes de Stokes.&nbsp; Ceci nous conduira tout naturellement, dans un second&nbsp; temps et par le biais que nous verrons, &agrave; aborder un sujet fascinant, &agrave; la richesse apparemment&nbsp; in&eacute;puisable, et aujourd'hui en plein essor : celui de la <i>dimorphie</i>. Curieusement, le sujet&nbsp; lui-m&ecirc;me est &agrave; la fois <i>un</i> et&nbsp; <i>dimorphe</i>, avec d'une part la&nbsp; <i>dimorphie fonctionnelle </i>(stabilit&eacute; d'une famille de <i>fonctions </i>relativement &agrave; deux produits distincts et &agrave; deux syst&egrave;mes distincts de&nbsp; ``<i>d&eacute;rivations exotiques</i>") et d'autre part la <i>dimorphie num&eacute;rique </i>(existence, sur un <b>Q</b>-module de <i>constantes </i>transcendantes, tel l'anneau des multiz&ecirc;tas,&nbsp; des hyperlogs, et bien d'autres encore, d'un double codage et d'une double table de multiplication - d'o&ugrave; une structure arithm&eacute;tique et alg&eacute;brique tr&egrave;s riche mais aussi tr&egrave;s complexe).</blockquote>  <p><br><b>Sabir Gusein Zade </b>(Moscou, Russie) <blockquote><b><i>On Poincar&eacute; series of some filtrations.</i></b> <p>There will be described a method of calculation of (generalized) Poincar&eacute; series of some multi-index filtrations elaborated jointly with A. Campillo and F. Delgado. The method can be applied, <i>e.g.</i>, for computing Poincar&eacute; series of the filtration defined by a plane curve singularity on the ring of germs of functions of two variables, for the (divisorial) filtration on the ring of functions on a rational surface singularity, ...</blockquote>  <p><br><b>Heisuke Hironaka </b>(Yamaguchi,&nbsp; Japon) <br>&nbsp; <p><b>Chris Howls </b>(Southampton, UK) <blockquote>&nbsp; <br><b><i>Hyperasymptotics for PDEs: Baby's first steps</i></b> <br>&nbsp; <p>We give a description of some recent work in the development of exponentially accurate <br>asymptotic expansions for (linear) PDEs. After an introduction to hyperasymptotic <br>techniques, we illustrate how they might be used in different types of problems, including <br>boundary layer theory and large-time behaviour. We also demonstrate the key role that can <br>be played in a time evolution equation by a higher order Stokes phenomenon where a series <br>of three singularities are colinear in an arbitrary direction in the Borel plane. We conclude <br>with a discussion of the viability of such methods.</blockquote>  <p><br><b>L&ecirc; Dung Trang </b>(Marseille, France) <blockquote><b><i>Carrousels et topologie des courbes planes.</i></b> <p>Un carrousel d&eacute;crit le diffeomorphisme de monodromie locale associ&eacute;e &agrave; une forme lin&eacute;aire du plan complexe et adapt&eacute;e &agrave; une courbe plane. On peut associer &agrave; ce carrousel un feuilletage <br>en cercles qui caract&eacute;rise la topologie de la courbe.</blockquote> <b>Fran&ccedil;ois Loeser </b>(Paris, France) <blockquote><b><i>Oscillating integrals, stationary phase and Fermat hypersurfaces.</i></b> <p>We shall present some results obtained in collaboration with Jan Denef. <br>We shall explain motivic analogues of the stationary phase <br>and of the Thom-Sebastiani Theorem and describe a new, geometric, convolution <br>for which Fermat curves (a special instance of Pham-Brieskorn hypersurfaces) <br>play a central role. <br>&nbsp;</blockquote>  <p><br><b>Bernard Malgrange </b>(Grenoble, France) <b>et Andr&eacute; Voros</b> (Saclay, France) <blockquote><b><i>Travaux de Fr&eacute;d&eacute;ric Pham</i></b></blockquote> <b>Jean-Fran&ccedil;ois Mattei </b>(Toulouse, France) <blockquote><b><i>Invariants topologiques locaux et d&eacute;termination finie topologique d'un</i></b> <br><b><i>germe de feuilletage holomorphe singulier en dimension 2</i></b> <p>Apr&egrave;s avoir d&eacute;crit une famille compl&egrave;te d'invariants topologiques locaux pour une &eacute;quation diff&eacute;rentielle <i>$\omega = a(x,y) dx + b(x, y) dy = 0$</i>, les coefficients <i>a(x,y) </i>et <i>b(x,y)</i> <br>&eacute;tant des germes de fonctions holomorphes, nous mettons en &eacute;vidence les invariants qui ne d&eacute;pendent que d'un jet d'ordre fini de la forme diff&eacute;rentielle <i>$\omega$.</i></blockquote> <b>Zoghman Mebkhout </b>(Paris, France) <blockquote><b><i>Exposants de la monodromie </i>p<i>-adique et structures de Frobenius</i></b></blockquote>  <p><br><b>Olivier Piltant </b>(Palaiseau, France) <blockquote><b><i>Sur la methode de Jung en caracteristique positive</i></b> <p>La m&eacute;thode de Jung donne une strat&eacute;gie simple de r&eacute;solution&nbsp; des singularit&eacute;s pour les surfaces complexes. L'ingr&eacute;dient essentiel est qu'on r&eacute;duit le probl&egrave;me aux singularit&eacute;s toriques apr&egrave;s avoir effectu&eacute; la r&eacute;solution plong&eacute;e du discriminant d'une projection plane. En caract\'eristique p>0, la situation est beaucoup plus compliqu&eacute;e quand la projection est&nbsp; sauvagement ramifi&eacute;e au-dessus du support du discriminant. Je montrerai dans cet expos&eacute; que l'obstruction &agrave; r&eacute;soudre les singularit&eacute;s des surfaces &agrave; la Jung en caract&eacute;ristique p>0 est&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; pr&eacute;cisement ce qu'on appelle le d&eacute;faut en th&eacute;orie des valuations. <br>(travail commun avec Dale Cutkosky) <br>&nbsp; <br>&nbsp;</blockquote> <b>Jean-Claude Risset </b>(Marseille, France) <blockquote><b><i>Math&eacute;matiques et musique</i></b></blockquote>  <blockquote>Les liens des math&eacute;matiques avec la musique sont anciens et profonds : la num&eacute;rologie des intervalles musicaux tient une grande place dans la th&eacute;orie de la musique - elle a jou&eacute; &eacute;galement un r&ocirc;le scientifique significatif. La notation musicale anticipe sur l'usage des coordonn&eacute;es cart&eacute;siennes. <p>Mais l'incarnation des nombres dans le sensible ne va pas de soi. D&eacute;j&agrave;, &agrave; la conception pythagoricienne selon laquelle les nombres gouvernent l'harmonie, Aristox&egrave;ne objectait que la justification de la musique &eacute;tait dans l'oreille de l'auditeur et non dans la raison math&eacute;matique. <p>Avec l'av&egrave;nement de l'ordinateur, il est devenu possible de produire des sons en calculant des nombres. En 1957, Max Mathews a pu enregistrer des sons sous forme d'une suite de nombres, et aussi synth&eacute;tiser des sons musicaux &agrave; l'aide d'un ordinateur se chargeant de calculer les nombres repr&eacute;sentant le son. La synth&egrave;se permet ainsi de composer le son lui-m&ecirc;me: elle a suscit&eacute; l'int&eacute;r&ecirc;t des musiciens. Le "son num&eacute;rique" &eacute;t&eacute; popularis&eacute; par les disques compacts, les synth&eacute;tiseurs et les &eacute;chantillonneurs et aussi par l'activit&eacute; d'institutions comme l'IRCAM. Les math&eacute;matiques sont l'outil omnipr&eacute;sent de ce nouvel artisanat du son musical, qui permet d'imiter les instruments acoustiques; de cr&eacute;er des sons synth&eacute;tiques "illusoires" ou "paradoxaux"; de composer des textures sonores in&eacute;dites; d'interpr&eacute;ter des musiques en "temps r&eacute;el", gr&acirc;ce &agrave; la norme MIDI de description num&eacute;rique des &eacute;v&egrave;nements musicaux. Cependant, dans la ligne d'Aristox&egrave;ne le musicien, il faut tenir compte des sp&eacute;cificit&eacute;s de la perception. <br>&nbsp; <br>&nbsp;</blockquote>  <p><br><b>Claude Sabbah </b>(Palaiseau, France) <blockquote><b><i>Gauss-Manin systems, Brieskorn lattices and Frobenius structures</i></b> <p>I will give some examples of Frobenius structures arising from <br>Laurent polynomials. This is a joint work with Antoine Douai (Nice). <br>&nbsp;</blockquote>  <p><br><b>David Sauzin </b>(Paris, France) <blockquote><b><i>Two examples of resurgence</i></b> <p>The second-order difference equation <p>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; <i>u(z+1)-2u(z)+u(z-1) = - u&sup2;(z)</i></blockquote>  <blockquote>admits a divergent formal solution <i>u(z)=\sum a_n z^{-2n}</i>, which is the common asymptotic expansion of two distinct analytic solutions. This difference equation corresponds to the parametrization of invariant curves of the Henon map :&nbsp; <i>(u,v) --> (u + v - u^2, v - u^2) </i>and <i>{ (u(z),v(z)=u(z)-u(z-1)) } </i>is the "formal separatrix" of this <br>discrete dynamical system. We study carefully the Borel transform of <i>u(z)</i>, compare its two natural sums and describe the whole resurgent structure of the problem (joint work with Vassili Gelfreich, Warwick). <p>An analogous analysis is performed on a PDE, which is an auxiliary equation in a study of&nbsp; "separaratrix splitting" <br>via the Hamilton-Jacobi method, and which admits a unique formal solution <i>u(z,t)</i> expanded in negative powers of <i>z </i>with <i>t </i>periodic coefficients. This second example corresponds to a work in progress in collaboration with Carme <br>Oliv&eacute; (Tarragona) and Tere Seara (Barcelona).</blockquote>  <p><br><b>Laurent Stolovitch </b>(Toulouse, France) <blockquote><b><i>Compl&egrave;te int&eacute;grabilit&eacute; singuli&egrave;re et ph&eacute;nom&egrave;ne de type KAM</i></b> <p>Nous montrons qu'un champ de vecteurs holomorphe au voisinage de son point <br>singulier&nbsp; 0 de <b>C</b>^<i>n</i><b> </b>est analytiquement normalisable pourvu qu'il ait un commutant holomorphe suffisamment grand, suffisamment d'int&eacute;grales premi&egrave;res formelles, et qu'une condition de petits <br>diviseurs associ&eacute;e &agrave; sa partie lin&eacute;aire soit satisfaite. C'est un ph&eacute;nom&egrave;ne de compl&egrave;te int&eacute;grabilit&eacute; singuli&egrave;re. La situtation perturb&eacute;e donnera lieu &agrave; un ph&eacute;nom&egrave;ne de type KAM non-symplectique.</blockquote> <b>Yoshitsugu Takei </b>(Kyoto, Japon) <blockquote><b><i>Exact WKB analysis of non-adiabatic transition probabilities for three levels</i></b> <p>We discuss the Schr&ouml;dinger equation $\sqrt{-1}d\psi/dt=\eta H\psi$ <br>with a large parameter $\eta$ and an appropriate $3\times 3$ matrix <br>$H$ by using the exact WKB analysis, i.e., WKB analysis based on the <br>Borel resummation. In particular, transition probabilities of <br>solutions of the equation are explicitly calculated with the help of <br>the connection formula for WKB solutions.</blockquote> <b>Jean Zinn-Justin</b> (Saclay, France) <blockquote><b><i>Generalized Bohr-Sommerfeld formula</i></b> <p>We will recall how, in quantum mechanics in the case of potentials with degenerate classical minima, semi-classical instanton calculations have led to a series of conjectures, generalizing <br>Bohr-Sommerfeld formula to such a situation. Some of these conjectures have eventually proven by Pham's group, but interesting problems remain. <br>&nbsp;</blockquote>  <br>&nbsp; </body> </html> 
