Calendrier des conférences CRM/ISM - Année 2008-2009
Responsables:Alexander Shnirelman (Concordia) (shnirel@mathstat.concordia.ca) 514-848-2424 #5222
Abraham Broer (Montréal) (broera@DMS.UMontreal.ca) 514-343-2053
Trimestre d'automne 2008 - hiver 2009 / Fall 2008 and Winter 2009 Semester
Le vendredi 24 avril 2009 / Friday, April 24, 2009 16 h / 4:00 p.m.
Gang Tian (Princeton)
Ricci Flow, Monge-Ampere Equation and Algebraic Spaces
Salle SH-3420, UQÀM, Pavillon Sherbrooke, 200, rue Sherbrooke Ouest
RESUME / ABSTRACT :This is an expository talk. I will discuss recent progresses on the KŠhler-Ricci flow. I will show how complex Monge-Ampere equations can be applied to studying the KŠhler-Ricci flow and how singularity formation of the KŠhler-Ricci flow interacts with the classification of algebraic manifolds. Some open problems will be also discussed.
Le vendredi 17 avril 2009 / Friday, April 17, 2009 16 h / 4:00 p.m.
Alexandru Buium (University of New Mexico)
Arithmetic Laplacians
Salle 6214, Pavillon André Aisenstadt, 2920 ch. de la Tour, Université de Montréal
RESUME / ABSTRACT :We develop an arithmetic analogue of elliptic partial differential equations. The role of the space coordinates is played by a family of primes, and that of the space derivatives along the various primes are played by corresponding Fermat quotient operators subjected to certain commutation relations. This leads to arithmetic linear partial differential equations on algebraic groups that are analogues of certain operators in analysis constructed from Laplacians. We classify all such equations on one dimensional groups, in particular on elliptic curves, and analyze their spaces of solutions.
Le vendredi 3 avril 2009 / Friday, April 3, 2009 16 h / 4:00 p.m.
Olivier Schiffmann (CNRS ENS Ulm)
Problème de Riemann-Hilbert sur la sphère et combinatoire des systèmes de racines
UQAM, Pav. Sherbrooke, 200, rue Sherbrooke O., salle SH-3420
RESUME / ABSTRACT :Soient C_1, ...C_l des classes de conjugaison dans GL(n,C). Quand peut-on choisir des éléments c_1, c_2, ... appartenant respectivement a C_1,C_2, ... tels que c_1c_2...c_l=1 ? Ce problème, qui vient de la géometrie des fibres vectoriels plats sur la sphère de Riemann privée de l points, a récemment été résolu par B. Crawley-Boevey et fait intervenir de manière surprenante la combinatoire de systèmes de racines exotiques. Nous exposerons le problème, sa solution, et nous en présenterons diverses ramifications.
Le vendredi 27 mars 2009 / Friday, March 27, 2009 16 h / 4:00 p.m.
Bjorn Poonen (MIT)
Undecidability in Number Theory
UQAM, Pav. Sherbrooke, 200, rue Sherbrooke O., salle SH-3420
RESUME / ABSTRACT :Hilbert's Tenth Problem asked for an algorithm that, given a multivariable polynomial equation with integer coefficients, would decide whether there exists a solution in integers. Around 1970, Matiyasevich, building on earlier work of Davis, Putnam, and Robinson, showed that no such algorithm exists. But the answer to the analogous question with integers replaced by rational numbers is still unknown, and there is not even agreement among experts as to what the answer should be.
Le vendredi 20 mars 2009 / Friday, March 20, 2009
16 h / 4:00 p.m. Conférence du lauréat du Prix André-Aisenstadt 2009 Lecture by André-Aisenstadt 2009 Prize Recipient Valentin Blomer (University of Toronto)
Le vendredi 13 mars 2009 / Friday, March 13, 2009 16 h / 4:00 p.m.
Sergei Yakovenko (Weizmann Institute, Rehovot, Israel, and Fields Institute, Toronto, Canada)
Infinitesimal Hilbert 16th Problem
UQAM, Pav. Sherbrooke, 200, rue Sherbrooke O., salle SH-3420
RESUME / ABSTRACT :The Hilbert 16th problem (2nd part) is about the maximal possible number of isolated ovals on the phase portraits of planar polynomial vector fields (these ovals are called limit cycles). Despite the progress of analysis, geometry and algebra in the 20th century, the general question remains open as it was hundred years before. Only various local or semilocal versions of this problem seem to be amenable, every time with great efforts. In this talk I will describe the recent progress in another direction of research going back to Petrovskii and Landis. This approach deals with limit cycles born from continuous families of (nonisolated) ovals. The corresponding infinitesimal Hilbert problem was intensely studied for the last 40 years or so. I will describe the first explicit uniform global upper bound for the number of limit cycles of near-Hamiltonian polynomial vector fields. The talk (based on works by G. Binyamini, D. Novikov and the speaker) is aimed for a general audience.
Le vendredi 6 mars 2009 / Friday, March 6, 2009
16 h / 4:00 p.m. Conférence de la Chaire Aisenstadt Lecture by Aisenstadt Chair 2008-2009 Craig A. Tracy (UC Davis)Le vendredi 27 février 2009 / Friday, February 27, 2009
16 h / 4:00 p.m. Conférence du lauréat du Prix ACP-CRM 2008 Lecture by the 2008 CAP-CRM Prize Recipient Richard Cleve (University of Waterloo)Le vendredi 20 février 2009 / Friday, February 20, 2009 16 h / 4:00 p.m.
Louigi Addario-Berry (Univ. de MontrŽal)
Branching Random Walk and Searching in Trees
UQAM-Pavillon Sherbrooke- 200 Sherbooke o., salle SH 3420
RESUME / ABSTRACT :The problem is related to searching in trees. Suppose we are given a complete binary tree (a rooted tree in which the root has degree 3 and every other vertex has degree 2) with independent, identically distributed random edge weights (say copies of some random variable X). The depth d(v) of a vertex v is the number of edges on the path from v to the root. We give each vertex v the label S_v which is the sum of the edge weights on the path from v to the root. For positive integers n, we let M_n be the maximum label of any vertex at depth n, and let M^* = sup {M_n: n =0,1,...}. It is of course possible that M^* is infinity. Under suitable moment assumptions on X, it is known that there is a constant A such that M_n/n --> A almost surely and in expectation. We call the cases A>0, A=0, and A< 0 supercritical, critical, and subcritical, respectively. We derive more precise information about the expected value E(M_n) than is captured by the above "law of large numbers"-style result, and derive exponential tail bounds for M_n-E(M_n). These results are "branching random walk" analogues of results Kesten (1987) proved for branching Brownian motion. Our techniques also allow us to derive information about branching random walks in higher dimensions (with steps in R^d, d > 1). When A <= 0 it makes sense to try to find the vertex of maximum weight M* in the whole tree. One possible strategy is to only explore the subtree T_0 containing the root and consisting only of vertices of non-negative weight. With probability bounded away from zero this strategy finds the vertex of maximum weight. We derive precise information about the expected running time of this strategy. Equivalently, we derive precise information about the random variable |T_0|. In the process, we also derive rather precise information about M*. This answers two questions posed by Aldous (1997). Parts of this work are joint with Nicolas Broutin and with Bruce Reed.
Le vendredi 13 février 2009 / Friday, February 13, 2009 16 h / 4:00 p.m.
William Byers (Concordia)
Mathematics in the Light of Metaphor and Ambiguity.
UQAM-Pavillon Sherbrooke- 200 Sherbooke o., salle SH 3420
RESUME / ABSTRACT :Mathematics is often taught and discussed as though the only thing that is going on is the logical structure. From the point of view of formal logic, ambiguity is something that must be avoided at all costs. However I shall show that a kind of metaphoric ambiguity is not only very common in mathematics but also is often the essential thing that is gong on. The conventional, formal approach to math misses what is most important, the creative essence of math, which are the mathematical ideas. Ideas are where the action is but ideas do not have to be logical. This talk will explore another way to think about mathematics and point to a totally different perspective on the philosophy of math. It will be based on my recent book, How Mathematicians Think: Using Ambiguity, Contradiction, and Paradox to Create Mathematics (Princeton University Press 2007). This is a non-technical talk that will be accessible to everyone who loves mathematics and will especially interest those who love to talk and think about mathematics..
Le vendredi 6 février 2009 / Friday, February 6, 2009 16 h / 4:00 p.m.
André D. Bandrauk (Université de Sherbrooke)
Nonlinear High Dimensional PDE's in High Intensity Laser-matter Interactions-New Mathematics for a New Science.
CRM, Pav. André-Aisenstadt, 2920, ch. de la Tour, salle 6214
RESUME / ABSTRACT :Interaction of ultrashort intense laser pulses with molecular media leads to highly nonlinear nonperturbative effects which can only be treated by large scale computation on massively parallel computers. Single molecule response to such pulses leads to Molecular High Order Harmonic Generation, MHOHG, (1), from which one can synthesize new "attosecond" pulses necessary to control electron dynamics at the natural time scale of the electron, the attoseocond (10**-18 s), (2).The single molecular response can be obtained from high level quantum Time-Dependent Schršdinger, TDSE, simulations. The collective macroscopic response of a molecular medium requires solving many TDSE,s (>10**5) coupled to the classical laser (photon) Maxwell equations (3). We will present the numerical methods necessary to achieve this goal, especially the problem of transparent and artificial boundary condition techniques in view of the different time scales, photon vs electron. Results will be shown for attosecond pulse generation and pulse filamentation in an aligned molecular medium, the one electron H2+ system(4).
(1) A.D. Bandrauk et al, "Molecular Harmonic Generation", in Progress in Ultrafast Intense Laser Science, vol III, edit K. Yamanouchi (Springer V, NY, 2008), chapt 9.
(2) A.D. Bandrauk, F. Krausz, A. Starace, "Focus on Attosecond Physics", New J. Phys, 10, 025004 (2008).
(3) E. Lorin, S. Chelkowski, A.D. Bandrauk, "Maxwell-Schršdinger Equations for Nonlinear Laser Propagation in Molecular Media", Comput. Phys. Commun. 177, 908 (2007).
(4) E. Lorin, S. Chelkowski, A.D. bandrauk, "Attosecond Pulse Generation for Aligned Molecules", New J. Phys, 10, 025033 (2008).
Le vendredi 30 janvier 2009 / Friday, January 30, 2009 16 h / 4:00 p.m.
Alexei Miasnikov (McGill)
Around Tarski's Problems
UQAM-Pavillon Sherbrooke- 200 Sherbooke o., salle SH 3420
RESUME / ABSTRACT :In this talk I am going to discuss recent solutions of Tarski's problems on elementary theory of free groups and their amazing relations to group theory, symbolic dynamics, algebraic topology, and computer science.
Le vendredi 23 janvier 2009 / Friday, January 23, 2009 16 h / 4:00 p.m.
Chantal David (Concordia University)
Statistics for the Zeroes and Traces of Zeta Functions over Finite Fields
UdeM, Pav. André-Aisenstadt, 2920, ch. de la Tour, salle 6214
RESUME / ABSTRACT :In recent years, much research has been devoted to the understanding of the distribution of zeroes of zeta functions and L-functions. The seminal work of Katz and Sarnak showed that the zeroes of zeta functions of curves over the finite fields $\F_q$ in various families are distributed as eigenvalues of random symplectic matrices as $q$ tends to infinity. Very recently, Kurlberg and Rudnick studied distributions of zeroes in similar families of curves over finite fields, but from a different perspective, fixing the finite field $\F_q$ and varying the genus of the curve. For the family of zeta functions of the hyperelliptic curves $y^2=F(x)$, they showed that the limiting distribution of the trace is that of a sum of $q$ independent random variables. We will explain in this talk how to build zeta functions of curves over finite fields, and what is the significance of their zeroes. We will also present a natural generalisation of the trace distribution from hyperelliptic curves to cyclic trigonal curves $y^3=F(x)$. In this case, the limiting distribution of the trace is no longer given by independent random variables, but involves bias and mixed probabilities.
Le vendredi 19 décembre 2008 / Friday, December 19, 2008 16 h / 4:00 p.m.
Jie Shen (Purdue University)
Spectral-Galerkin Methods for High-Dimensional PDEs
UdeM, Pav. André-Aisenstadt, 2920, ch. de la Tour, salle 6214
RESUME / ABSTRACT :Many scientific, engineering and financial applications require solving high-dimensional PDEs. However, traditional tensor product based algorithms suffer from the so called "curse of dimensionality". We shall present a new spectral-Galerkin method for non-periodic problems and/or in the whole space. The method is based on two basic ingredients: (i) Choosing the frequencies of the trial functions from the "hyperbolic cross"; (ii) Using a sparse grid or a lattice rule to perform the numerical interpolation/integration. We will present rigorous estimates as well as efficient numerical algorithms for elliptic equations. We will also present some preliminary numerical results for the six-dimensional BGK model.
Le vendredi 12 décembre 2008 / Friday, December 12, 2008 16 h / 4:00 p.m.
Alexander TURBINER (CRM and National University of Mexico, Mexico)
Solvable Schroedinger Equations and Representation theory
UQAM, Pavillon Sherbrooke, 200 Sherbrooke W., salle SH 3420
RESUME / ABSTRACT :A notion of solvability of the eigenvalue problem for a Schroedinger operator is introduced. It is shown that a classification of known exact solutions is related to a classification of spaces of polynomials which are finite-dimensional representation spaces of certain Lie algebras of differential operators. As a result one obtains a Lie-algebraic theory of exact solutions of differential and difference equations. As a surprising byproduct a new procedure for calculating Selberg integrals emerges.
Le vendredi 5 décembre 2008 / Friday, December 5, 2008 16 h / 4:00 p.m.
Robert Coquereaux (CPT, Luminy-Marseille)
Fundamental Interactions and Classical or Quantum Geometries
UdeM, Pav. André-Aisenstadt, 2920, ch. de la Tour, salle 6214
RESUME / ABSTRACT :As we know from General Relativity, gravitational forces can be seen as a manifestation of the curvature of Space-Time, but the other fundamental forces of Nature -- electroweak or strong interactions -- can also be geometrically interpreted in terms of curvatures stemming from connections on fibered spaces (gauge theories). The corresponding mathematical tools belong to the realm of classical geometry, but a quantum description of fundamental interactions requires the replacement of "spaces'' (actually commutative algebras of functions on spaces) by more general algebras, as well as the replacement of classical geometry by a non-commutative geometry. The purpose of the seminar is to give an elementary presentation of the standard mathematical structures underlying the notion of curvature (theory of connections) and to show how such structures can be generalized to the realm of non-commutative spaces.
Le vendredi 28 novembre 2008 / Friday, November 28, 2008 16 h / 4:00 p.m.
Alexandre Girouard (Cardiff University)
Lauréat du Prix Carl Herz (2007-2008) 2007-2008 Carl Herz Prize Recipient
Shape optimization for low eigenvalues of the Laplace operator
UdeM, Pav. André-Aisenstadt, 2920, ch. de la Tour, salle 6214
RESUME / ABSTRACT :This talk will be a survey of some classical problems on the spectrum of bounded planar domains and closed surfaces. The question to know which domain (if it exists!) minimizes --or maximizes-- a given eigenvalue of the Laplace operator is an old one. It can be traced back to Lord Rayleigh's book --The theory of sound-- published in 1877, where it was conjectured that the fundamental frequency of a drum is minimal when the drumhead is circular. A lot of progress has been made in the field of shape optimization since then, but many surprisingly simple questions still remain unsolved to this day. By the end of this talk, I hope to convince (at least some of you that these are rather interesting problems.
Le vendredi 21 novembre 2008 / Friday, November 21, 2008 16 h / 4:00 p.m.
Claude Bardos (Université Paris-Diderot (Paris 7))
Turbulence from Statistical Theory to Wigner Measure
UQAM, Pav. Sherbrooke, 200, rue Sherbrooke O., salle SH-3420
RESUME / ABSTRACT :In this talk I want to do the following things: 1. Recall a certain number of examples of pathological behavior of solutions of Euler or Navier-Stokes-like equations. 2. Relate these behaviors to the intuitive or statistical notion of turbulence. 3. Compare the principles of statistical turbulence with the description of weak limit in terms of Wigner measures. 4. Finally compare with some results on the complex singularities.
Le vendredi 14 novembre 2008 / Friday, November 14, 2008 16 h / 4:00 p.m.
Bernard Shiffman (Johns Hopkins University)
Overcrowding and Undercrowding of Random Zeros on Complex Manifolds
UQAM, Pav. Sherbrooke, 200, rue Sherbrooke O., salle SH-3420
RESUME / ABSTRACT :We discuss the distribution of zeros of random polynomials in m complex variables and, more generally, of random holomorphic sections of ample line bundles over compact Kaehler manifolds. We show that the zeros are highly likely to be uniformly distributed if the degree of the polynomial or line bundle is large. For example, the overcrowding and undercrowding probabilities for the volume of the zero set in a fixed domain decay like exp(-CN^{m+1}) as the degree N increases. We use off-diagonal asymptotics of the Bergman-Szego kernels to analyze coherent states centered at lattice points and to obtain large deviation estimates for the maximum modulus, and then we apply methods from Nevanlinna theory to obtain estimates for the zeros. This talk involves joint work with Steve Zelditch and Scott Zrebiec.
Le vendredi 7 novembre 2008 / Friday, November 7, 2008 16 h / 4:00 p.m.
Jean-Louis Loday (CNRS, Strasbourg)
Combinatorial Hopf Algebras
UQAM, Pav. Sherbrooke, 200, rue Sherbrooke O., salle SH-3420
RESUME / ABSTRACT :Many recent papers are devoted to some infinite dimensional Hopf algebras called collectively "combinatorial Hopf algebras". Among the examples we find the Faa di Bruno algebra, the Connes-Kreimer algebra and the Malvenuto-Reutenauer algebra. We give a precise definition of such an object and we provide a classification. We show that the notion of preLie algebra and of brace algebra play a key role.
Le vendredi 31 octobre 2008 / Friday, October 31, 2008 16 h / 4:00 p.m.
Robert Seiringer (Princeton University)
Dilute Quantum Gases
UdeM, Pav. AndrŽ-Aisenstadt, 2920, ch. de la Tour, salle 6214
RESUME / ABSTRACT :We present an overview of mathematical results on the low temperature properties of dilute quantum gases, which have been obtained in the past few years. The discussion includes, for instance, results on the free energy in the thermodynamic limit, and on Bose-Einstein condensation, Superfluidity and quantized vortices in trapped gases. All these properties are intensely being studied in current experiments on cold atomic gases. We will give a brief description of the mathematics involved in understanding these phenomena, starting from the underlying many-body Schroedinger equation.
Le vendredi 24 octobre 2008 / Friday, October 24, 2008 16 h / 4:00 p.m.
David Ruelle (IHES)
Nonequilibrium Statistical Mechanics and Smooth Dynamical Systems
UQAM, Pav. Sherbrooke, 200, rue Sherbrooke O., salle SH-3420
RESUME / ABSTRACT :One idealization of nonequilibrium statistical mechanics simply gives general smooth dynamics on a compact manifold, with new interpretations and new questions, which we shall review. In particular we shall introduce the natural physical (SRB) measure associated with a diffeomorphism f, and ask if it depends smoothly on f. We shall also introduce an analytic susceptibility function and study its singularities.
Le vendredi 17 octobre 2008 / Friday, October 17, 2008 16 h / 4:00 p.m.
Svante JANSON (Uppsala University)
Random Graphs: New models and the Internet
Pavillon André-Aisenstadt, Université de Montréal, 2920, ch. de la Tour, Salle / Room 1360
RESUME / ABSTRACT :Random graphs have been more or less successfully applied to many real-life problems. One important example is the Internet, which can be regarded as a very large graph. This graph can in practise not be described exactly, and to study various properties, such as vulnerability to intentional or accidental disruptions, it is natural to study random models However, the classical random graph models are often too homogeneous to be good approximations. In particular, in the Internet and many other real-life examples, it is observed that the vertex degrees (number of adjacent edges) vary a lot, often with a power-law distribution of the high degrees. This has served as a source of inspiration for random graph theorists, and during the last 10 years, a number of new random graph models have been introduced and studied in order to mimic the Internet or other similar graphs. This illustrates that not only can mathematics be useful for applications; conversely, applications can stimulate new theoretical developments. I will give some examples from the Internet and describe some different random graph models that have been proposed.
Le vendredi 10 octobre 2008 / Friday, October 10, 2008 16 h / 4:00 p.m.
Leonid Bunimovich (Georgia Institute of Technology)
Visual Chaos: Dispersing, Defocusing, Absolute Focusing and Astigmatism
UQAM, Pav. Sherbrooke, 200, rue Sherbrooke O., salle SH-3420
RESUME / ABSTRACT :The mechanisms generating chaotic (hyperbolic) behavior in billiards will be discussed. It turned out that all focusing components of the boundary of chaotic billiards should be absolutely focusing. Absolute focusing seems to be a new notion in the geometric optics. The astigmatism comes into play in dimensions greater than two which forces to reduce the sizes of focusing components of chaotic billiard tables. We conclude with the simple visual examples of billiards with divided phase space where any number of chaotic ergodic components coexist with any number of integrable islands.
Le vendredi 3 octobre 2008 / Friday, October 3, 2008 16 h / 4:00 p.m.
Elliott Lieb (Princeton University)
Some Calculus of Variations Problems in Quantum Mechanics
deM, Pav. André-Aisenstadt, 2920, ch. de la Tour, salle 6214
RESUME / ABSTRACT :Three examples are given, in order of historical development, of minimization problems in quantum mechanics arising from attempts to model the N--body Schroedinger equation by simpler energy functionals involving only densities. These simpler models are Thomas-Fermi theory, Hartree-Fock theory and the Mueller density matrix functional theory. This talk is for non-specialists: no knowledge of quantum mechanics is needed.
Le vendredi 26 septembre 2008 / Friday, September 26, 2008 16 h / 4:00 p.m.
Vladimir Sverak (University of Minnesota)
PDE aspects of the Navier-Stokes Equations
UQAM, Pav. Sherbrooke, 200, rue Sherbrooke O., salle SH-3420
RESUME / ABSTRACT :We will explain the main difficulties arising in the mathematical analysis of the Navier-Stokes equations and we mention some recent results which are related to these problems.
Le vendredi 19 septembre 2008 / Friday, September 19, 2008 16 h / 4:00 p.m.
Kenneth McLaughlin (The University of Arizona)
Some classes of random Hermitian matrices: F(Tr(V(M)) Instead of Tr(V(M))
UdeM, Pav. André-Aisenstadt, 2920, ch. de la Tour, salle 6214
RESUME / ABSTRACT :Three examples: we'll start with the usual random Hermitian matrices as an introduction. Then we'll consider two examples of probability measures on random Hermitian matrices which are invariant ensembles but which have peculiar, and different behavior. In all three cases, the primary goal will be to summarize explicit formulae for eigenvalue statistics, and with remaining time, discuss the subsequent asymptotic analysis.
Le vendredi 12 septembre 2008 / Friday, September 12, 2008 (Cette conférence s'adresse à un large auditoire / Suitable for a general audience) 16 h / 4:00 p.m.
Andrei Okounkov (Princeton University)
The Algebra and Geometry of Random Surfaces
UdeM, Pav. André-Aisenstadt, 2920, ch. de la Tour, salle 6214
RESUME / ABSTRACT :These lectures will be about one of the simplest models of random surfaces, the so-called stepped surfaces, that arise as the zero-temperature interfaces in the 3D Ising model and, also, for example, as the height function representations of hexagonal dimers. A typical question about such surfaces is to describe the behavior of a random surface spanning given boundary and the mesh size of the surfaces goes to zero. Our interest will be in both the global, macroscopic shapes that these surfaces develop, as well as in their local, microscopic properties. These are interlinked in a remarkable fashion that seem to require a certain input from several distant fields, from analysis to noncommutative algebraic geometry.
Le vendredi 5 septembre 2008 / Friday, September 5, 2008 16 h / 4:00 p.m.
Iku Nakamura (Hokkaido University)Trimestre d'hiver 2008 / Winter Semester 2008
Stability and Compactification of the Moduli of Abelian Varieties
UdeM, Pav. André-Aisenstadt, 2920, ch. de la Tour, salle 6214
RESUME / ABSTRACT :In arbitrary dimension, we canonically compactify the moduli space of abelian varieties by GIT-stability of Mumford. In the smallest possible case, dimension one and $N=3$, the problem is reduced to the study of planar cubic curves, more specifically Hesse cubic curves. Every GIT-stable cubic curve is isomorphic to a Hesse cubic curve.
Le vendredi 25 avril 2008 / Friday, April 25, 2008 16 h / 4:00 p.m.
Burkhard Wilking (Mathematisches Institut der Uni Munster)
New Ricci flow invariant curvature conditions and applications
UdeM, Pav. André-Aisenstadt, 2920, ch. de la Tour, salle 6214
RESUME / ABSTRACT :We consider a very simple curvature condition: Given constant c and a dimension n we say that a manifold (M; g) satises the condition (c; n) is if the scalar curvature is bounded below by c times the norm of the Weyl curvature. We show that in each large even dimensions there is precisely one constant c = c(n) > 0 such that this condition is invariant under the Ricci ow. The condition behaves very similar to scalar curvature under conformal transformations and we indicate how this can be utilized to get a large source of examples. Finally we speculate what kind singularities should develop under the Ricci ow.
Le vendredi 18 avril 2008 / Friday, April 18, 2008 14:30 h / 2:30 p.m. Notez le changement d'heure / Note: Time change. Pause-café : 15h30 (Salle 6245) / Coffee Break: 3:30 (Room 6245)
R. Mazzeo (Stanford University)
Flexibility of singular Einstein metrics
UdeM, Pav. André-Aisenstadt, 2920, ch. de la Tour, salle 6214
RESUME / ABSTRACT :The study of canonical metrics on stratified spaces (at least in the non-complex setting) is relatively undeveloped. I will survey some of what is known, focusing mostly on questions of moduli rather than existence. There are already interesting results in low dimensions. In dimension three, this is closely related to an old conjecture by Stoker concerning convex polyhedra.
Le vendredi 11 avril 2008 / Friday, April 11, 2008 16 h / 4:00 p.m.
Steve Zelditch (John Hopkins University)
Nodal lines of eigenfunctions, geodesics and complex analysis
UdeM, Pav. André-Aisenstadt, 2920, ch. de la Tour, salle 6214
RESUME / ABSTRACT :Nodal line patterns have intrigued mathematicians and physicists since Chladni in 1800. They are the zero sets of the eigenfunctions representing modes of vibration. My talk is about two relatively recent results describing nodal patterns on real analytic Riemannian manifolds. The main idea of the results is to analytically continue eigenfunctions to the complexification of the manifold and to study the complex zeros. When the geodesic flow is ergodic, we obtain an exact limit formula for the distribution of complex zeros for `almost all eigenfunctions'. It is also possible to determine limit formulae when the geodesic flow is integrable. For a real analytic plane domain with boundary, we obtain an upper bound on the number of nodal components which touch the boundary (joint work with John Toth).
Le vendredi 4 avril 2008 / Friday, April 4, 2008 16 h / 4:00 p.m.
Vladimir Maz'ya (Ohio State University, University of Liverpool, and Linkšping University)
Unsolved mysteries of solutions to PDEs near the boundary
UQAM, Pav. Sherbrooke, 200, rue Sherbrooke O., salle SH-3420
RESUME / ABSTRACT :Throughout its long history, specialists in the theory of partial differential equations gained a deep insight into the boundary behavior of solutions.Yet despite the apparent progress in this area achieved during the last century, there are fundamental unsolved problems and surprising paradoxes related to solvability, spectral, and asymptotic properties of boundary value problems in domains with irregular boundaries. I shall formulate some challenging questions arising naturally when one deals with unrestricted, polyhedral, Lipschitz graph, fractal and convex domains.
Le vendredi 14 mars 2008 / Friday, March 14, 2008 16 h / 4:00 p.m.
Stephen Vavasis (University of Waterloo)
Greedy algorithms and complexity for nonnegative matrix factorization
UQAM, Pav. Sherbrooke, 200, rue Sherbrooke O., salle SH-3420
RESUME / ABSTRACT :Nonnegative Matrix Factorization (NMF) has emerged in the past decade as an important tool for information retrieval and clustering. NMF was originally considered as early as 1983 in the mathematical literature, but it did not become popular for information retrieval until work by Lee and Seung in 1999. It has been successfully applied to text and image databases, biochemical laboratory data, and even music. The most popular class of algorithms uses local improvement heuristics, but another class of algorithms based on greedy rank-one downdating has also shown promise. We describe our development of a greedy downdating algorithm motivated in part by the singular value decomposition. We prove that it can successfully classify text databases in a model of text and find features in images in an image model. The new algorithm performs well in practice on standard databases. We also consider the complexity of the NMF problem, showing that it is NP-hard and relating it to a problem in polyhedral combinatorics. Parts of this talk are joint with with M. Biggs and A. Ghodsi of Waterloo.
Le vendredi 7 mars 2008 / Friday, March 7, 2008 16 h / 4:00 p.m.
Michael Berry (Bristol University)
Tsunami asymptotics
UdeM, Pav. André-Aisenstadt, 2920, ch. de la Tour, salle 6214
RESUME / ABSTRACT :For most of their propagation, tsunamis are linear dispersive waves whose speed is limited by the depth of the ocean and which can be regarded as a diffraction-decorated caustic in spacetime. For constant depth, uniform asymptotics gives a very accurate compact description of the tsunami profile generated by an arbitrary initial disturbance. Variations in depth can focus tsunamis onto cusped caustics, and this "singularity on a singularity" constitutes an unusual diffraction problem, whose solution indicates that focusing can amplify the tsunami energy by an order of magnitude.
Le vendredi 29 février 2008 / Friday, February 29, 2008 16 h / 4:00 p.m.
John Harnad (Concordia, CRM)
What is a tau function?
UdeM, Pav. André-Aisenstadt, 2920, ch. de la Tour, salle 6214
RESUME / ABSTRACT :What do the following have in common? - Irreducible characters of Lie groups (e.g., Schur functions) - Riemann's theta function on the Jacobian of a genus g Riemann surface - Deformation classes of random matrix integrals - Weights on path spaces of partitions, generating "integrable" random processes random tilings, and growth processes - Generating functions for Gromov-Witten invariants - Generating functions for classical and quantum integrable systems, such as the KP hierarchy (What have we left out? L-functions? Take their Mellin transforms.) In this talk, I will show how all the above may be seen as special cases of one common object: the "Tau function". This is a family of functions introduced by Sato, Hirota and others, originally in the context of integrable systems. They are parametrized by the points of an infinite dimensional Grassmann manifold, and depend on an infinite sequence of variables (t_1, t_2, ...), real or complex, continuous or discrete. They satisfy an infinite set of bilinear differential (or difference) relations, which can be interpreted as the Plucker relations defining the embedding of this "universal" Grassmann manifold into an exterior product space (called the "Fermi Fock space" by physicists) as a projective variety. This involves the "Bose-Fermi equivalence", which follows from interpreting the t-variables as linear exponential parameters of an infinite abelian group that acts on the Grassmannian and Fock space. A basic tool, which is part and parcel of the Plucker embedding, is the use of fermionic "creation" and "annihilation" operators. The tau function is obtained as a "vacuum state matrix element" along orbits of the abelian group. This is language that is familiar to all physicists, but little used by mathematicians, except for those, like Kontsevich, Witten, Okounkov (or, in earlier times, Cartan, Chevalley, Weyl), who know how to get good use out of it.
Le vendredi 22 février 2008 / Friday, February 22, 2008 16 h / 4:00 p.m.
Fernando Rodriguez Villegas (University of Texas at Austin)
Combinatorics as geometry
UdeM, Pav. André-Aisenstadt, 2920, ch. de la Tour, salle 6214
RESUME / ABSTRACT :Ever since Weil we know that counting points of varieties over finite fields yields topological information about them. In this talk I will discuss several examples of this two-way street between combinatorics and geometry involving, in decreasing order of complexity: character varieties parameterizing representations of the fundamental group of a Riemann surface into GL_n, representations of quivers and counting graphs of various sorts.
Le vendredi 15 février 2008 / Friday, February 15, 2008 16 h / 4:00 p.m.
Michael F. Singer (North Carolina State University)
Differential Groups and Differential Relations
UQAM, Pav. Sherbrooke, 200, rue Sherbrooke O., salle SH-3420
RESUME / ABSTRACT :I will present a leisurely introduction to a Galois theory of linear difference equations where the Galois groups are linear differential groups that is, groups of matrices whose entries satisfy a fixed set of polynomial differential equations. These groups measure the differential dependence among solutions of linear difference equations. I will show how this theory can be used to reprove Hšlder's Theorem of 1887 that the Gamma function satisfies no differential polynomial equation as well as new results concerning differential dependence of solutions of higher order difference equations. This is joint work with Charlotte Hardouin.
Le vendredi 8 février 2008 / Friday, February 8, 2008 16 h / 4:00 p.m. Nous vous avisons que le Colloque prévu le vendredi 8 février a été annulé. Merci de votre attention. Please note that the Colloquium on February 8, has been cancelled. Sincerely.
Alexandru Buium (New Mexico)
Arithmetic partial differential equations
UdeM, Pav. André-Aisenstadt, 2920, ch. de la Tour, salle 6214
RESUME / ABSTRACT :We develop an arithmetic analogue of linear partial differential equations in two independent "space-time" variables. The spatial derivative is a Fermat quotient operator, while the time derivative is a usual derivation. This allows us to "flow'' integers or, more generally, points on algebraic groups with coordinates in rings with arithmetic flavor. In particular, we show that elliptic curves have certain canonical "arithmetic flows'' on them that are arithmetic analogues of the convection, heat, and wave equations. The same is true for the additive and the multiplicative group and also for modular curves. This is joint work with Charlotte Hardouin.
Le vendredi 1er février 2008 / Friday, February 1, 2008 16 h / 4:00 p.m.
Victor Kac (MIT)
Quantization and chiralization
UQAM, Pav. Sherbrooke, 200, rue Sherbrooke O., salle SH-3420
RESUME / ABSTRACT :The four fundamental frameworks of physical theories are classical mechanics, quantum mechanics, classical field theory and quantum field theory. The related algebraic structures are Poisson algebras, associative algebras, Poisson vertex algebras and vertex algebras. I will discuss connections between these structures and explain the general picture on the example of W-algebras.
Le vendredi 18 janvier 2008 / Friday, January 18, 2008 16 h / 4:00 p.m.
Ivar Ekeland (PIMS, UBC)
From Elie Cartan to Gerard Debreu: some applications of exterior differential calculus to economic theory
UdeM, Pav. André-Aisenstadt, 2920, ch. de la Tour, salle 6214
RESUME / ABSTRACT :A venir / TBA
Le vendredi 11 janvier 2008 / Friday, January 11, 2008 16 h / 4:00 p.m.
Gérard Letac (Université Paul Sabetier)
L'invariance de Thomae de 3F2 par le groupe symétrique S5 et les produits de matrices (2,2) aléatoires
UdeM, Pav. André-Aisenstadt, 2920, ch. de la Tour, salle 6214
RESUME / ABSTRACT :En 1879, Thomae a découvert une étonnante propriété d'invariance par le groupe $\mathcal{S}_5$ des permutations de 5 objets de la fonction de cinq variables $$(a,b,c,d,e)\mapsto \, _3F_2(a,b,c;d,e;1)=\sum_{n=0}^{\infty}\frac{(a)_n (b)_n (c)_n }{n!(d)_n (e)_n }$$ Nous commencons par en donner une déémonstration. La formule de Thomae fournit la clé pour montrer que la loi marginale de $X$ quand $$(X,X')\sim C(1-xx')^{b-a-a'}\beta_{a,a'}(dx)\beta_{a',a}(dx')$$ est la loi de la fraction continue aléatoire $$\frac{1}{1+\frac{W_1}{1+\frac{W'_1}{1+\frac{W_2}{1+...}}}}$$ lorsque $(W_n)_{n\geq 1}$ et $(W'_n)_{n\geq 1}$ sont deux suites indépendantes et chacune \textit{i.i.d.} de lois $\beta^{(2)}_{b,a}$ et $\beta^{(2)}_{b,a'}.$ Ces compositions d'homographies aléatoires se traduisent facilement par des produits de matrices aléatoires indépendantes. Mais les calculs explicites comme celui ci sont mystérieux et rarissimes.
Le vendredi 4 janvier 2008 / Friday, January 4, 2008 16 h / 4:00 p.m.
Michael Jakobson (University of Maryland)
Attractors and invariant measures in low-dimensional dynamical systems
UdeM, Pav. André-Aisenstadt, 2920, ch. de la Tour, salle 6214
RESUME / ABSTRACT :We consider typical behavior of trajectories in dynamical systems from topological and measure theoretical prospectives. They coincide for uniformly hyperbolic systems, but can be different for non-uniformly hyperbolic ones. Sinai-Ruelle-Bowen (SRB) measures are crucial for understanding ergodic properties of dynamical systems. In the case when a system has an absolutely continuous SRB measure its typical trajectories exhibit random behavior and at the same time their statistical properties can be described quantitatively. We discuss several types of dynamics which arise in one-parameter families of low-dimensional dynamical systems.
Trimestre d'automne 2007 / Fall Semester 2007
Le vendredi 14 décembre 2007 / Friday, December 14, 2007 16 h / 4:00 p.m.
Michael Shub (U. Toronto)
(Workshop on chaos and ergodicity of dynamical systems) Smale's 17th Problem
UdeM, Pav. André-Aisenstadt, 2920, ch. de la Tour, salle 6214
RESUME / ABSTRACT :In a series of papers written in the first half of the 1990's Steve Smale and I studied the complexity of solving systems of n polynomial equations in n complex variables. We studied path following techniques. A system with known solution is connected by a path to the system we want to solve and the solution is "continued" along the path. The path we chose was the straight line connecting the systems. We proved that "on average" systems can be solved with polynomial cost but we did not prove the existence of a uniform algorithm. The question of the existence of a uniform algorithm is Smale's 17th problem. Recently, Beltran and Pardo have made significant progress on this problem. Moreover, Jointly with Beltran I have linked the complexity to the length of the (problem,solution) path in the condition number Riemannian structure. Surprisingly short paths exist!
Le vendredi 30 novembre 2007 / Friday, November 30, 2007 16 h / 4:00 p.m.
Pavel Bleher (Indiana University-Purdue University Indianapolis)
Exact Solution of the Six-Vertex Model of Statistical Physics UdeM, Pav. André-Aisenstadt, 2920, ch. de la Tour, salle 6214
RESUME / ABSTRACT :
We prove the conjecture of Zinn-Justin that the partition function of the six-vertex model on the $N\times N$ lattice with domain wall boundary conditions has the asymptotics, $Z_N\sim CN^\kappa e^{N^2}f$ as $N\to \infty$, and we find the exact value of the exponent $\kappa$. The solution is based on the Riemann-Hilbert approach and the Deift-Zhou nonlinear steepest descent method.
Le vendredi 23 novembre 2007 / Friday, November 23, 2007 16 h / 4:00 p.m.
Ben Green (Cambridge)
Nilsequences in Additive Combinatorics UQAM, Pav. Sherbrooke, 200, rue Sherbrooke O., salle SH-3420
RESUME / ABSTRACT :Harmonic analysis has proved to be a very effective tool for studying additive questions in number theory. There are many problems, however, for which harmonic analysis does not provide enough information. For some of these problems success has been possible by considering, instead of simply the linear exponentials e^{2 \pi i theta n}, more exotic "harmonics" called nilsequences. The goal of this talk will be to explain (to a general mathematical audience) what nilsequences are, why they are needed, and how they have helped to solve concrete problems concerning primes. In particular we will outline a proof of an asymptotic for the number of quadruples p_1 < p_2 < p_3 < p_4 <= N of primes which lie in arithmetic progression. Joint work with T. Tao.
Le vendredi 16 novembre 2007 / Friday, November 16, 2007 15 h 45 / 3:45 p.m.
Charles Radin (Texas, Austin)
UdeM, Pav. André-Aisenstadt, 2920, ch. de la Tour, salle 6214 RESUME / ABSTRACT :The densest possible packings of congruent spheres in space are highly symmetric, for reasons still unknown. We will review an approach to understand this phenomenon through packings which are merely dense rather than densest, and conclude with recent attempts to model the properties of sand. |
UdeM, Pav. André-Aisenstadt, 2920, ch. de la Tour, salle 6214 |
UQAM, Pav. Sherbrooke, 200, rue Sherbrooke O., salle SH-3420 RESUME / ABSTRACT :Un réseau du plan est un sous-groupe discret de R^2 isomorphe à Z^2 ; il est unimodulaire si l'aire du quotient est 1. L'espace de ces réseaux unimodulaires est un objet vénérable en mathématiques qui a de nombreux aspects, topologiques, dynamiques et arithmétiques. Dans cet exposé, je proposerai une visite guidée de cet espace, dans laquelle j'insisterai sur l'aspect topologique. Je décrirai en particulier les orbites périodiques du "flot modulaire" qui donnent de jolis "noeuds modulaires". L'approche sera très visuelle, et je montrerai quelques petits films... |
UdeM, Pav. André-Aisenstadt, 2920, ch. de la Tour, salle 6214 RESUME / ABSTRACT :It was understood by Minkowski that one could prove interesting results in number theory by considering the geometry of lattices in R^n. (A lattice is simply a grid of points.) I will begin by explaining some examples of this technique, called "the geometry of numbers." More recently, dynamics and analysis on the space of all lattices in R^n have been studied intensively, and this is, in some ways, a modern successor to the geometry of numbers. I will discuss this picture and how it has enhanced our understanding of some classical questions. |
UQAM, Pav. Sherbrooke, 200, rue Sherbrooke O., salle SH-3420 RESUME / ABSTRACT :In pseudo-Riemannian geometry the spaces of space-like and time-like geodesics on a pseudo-Riemannian manifold have natural symplectic structures (just like in the Riemannian case), while the space of light-like geodesics has a natural contact structure. Furthermore, the space of all geodesics has a structure of a Jacobi manifold. In the talk I will describe the geometry of these structures, define pseudo-Euclidean billiards and discuss their properties. In particular, I will outline complete integrability of the billiard in the ellipsoid and the geodesic flow on the ellipsoid in pseudo-Euclidean space; these results are pseudo-Euclidean counterparts to the classical theorems of Euclidean geometry that go back to Jacobi and Chasles. |
UQAM, Pav. Sherbrooke, 200, rue Sherbrooke O., salle SH-3420 RESUME / ABSTRACT :The fundamental philosophy of the lattice Boltzmann method (LBM) is to construct simplified kinetic type models that preserve the conservation laws and necessary symmetries so that the macroscopically averaged properties obey the desired continuum equations. The LBM and its historical development, including the derivation of the Navier-Stokes equations, are presented. The treatment of boundary conditions within the LBM framework is described. The flow of a Newtonian fluid past a confined cylinder is investigated and comparisons made with existing results in the literature. An extension of the LBM to axisymmetric flows is presented. The resulting method is validated for the classical problem of flow past a sphere. Excellent agreement is found with analytical and empirical expressions for the drag. The LBM is then developed for immiscible binary fluids with different viscosities and densities. The model is shown to recover the Navier-Stokes equations for two-phase flow in the macroscopic limit. A theoretical expression for surface tension is derived and the validity of the analysis is confirmed by comparing numerical and theoretical predictions of surface tension as a function of density. Finally, a formal perturbation analysis of the generalized LBE is presented. The generalized LBE overcomes some of the shortcomings of the single relaxation parameter LBM such as its ability to model complex fluids. |
UQAM, Pav. Sherbrooke, 200, rue Sherbrooke O., salle SH-3420 RESUME / ABSTRACT :The topology of Lagrangian submanifolds is still mysterious today even after more than twenty five years of intensive investigation. Understanding this type of submanifold is motivated by many problems some originating in physics some others in real algebraic geometry as well as in other areas of mathematics. I will first review some of the standard results in the subject starting with Gromov and Floer and further pursued by Hofer, Salamon, Oh and many others. They all depend on detecting certain types of pseudoholomorphic curves with Lagrangian boundary conditions. Managing algebraically this type of curves is a hot topic of current research and I will describe some results in this direction obtained in joint work with Francois Lalonde as well as some obtained together with Paul Biran. I will also describe the way this is related to work of Fukaya-Oh-Ohta-Ono. |
UdeM, Pav. André-Aisenstadt, 2920, ch. de la Tour, salle 6214 RESUME / ABSTRACT :The space of representations of the fundamental group of a surface in a Lie group is a rich geometric object. Examples include symplectic vector spaces, Jacobi varieties and Teichmueller spaces. The topological symmetries of the surface acts on this space preserving a natural Poisson geometry. This action of the mapping class group closely relates to Hamiltonian flows on these moduli spaces. When the Lie group is compact, the action is chaotic. For uniformization representations corresponding to geometric structures, the action is properly discontinuous. In general the dynamics falls between these two extremes. |
UdeM, Pav. André-Aisenstadt, 2920, ch. de la Tour, salle 6214 RESUME / ABSTRACT :In this talk, I will give a brief survey of my series of joint papers with Jiaping Wang and also other subsequent results in the subject. Objects of our investigation are complete manifolds whose Ricci curvature has a lower bound. If the manifold also admits a certain Poincaré inequality, then one can conclude certain topological and metric properties of the manifold. |
Calendrier des conférences CRM/ISM - Année 2006-2007
Responsables: Jacques Hurtubise (McGill) (hurtubise@math.mcgill.ca) & Sasha Shnirelman (shnirel@mathstat.concordia.ca)
UdeM, Pav. André-Aisenstadt, 2920, ch. de la Tour, salle 6214 RESUME / ABSTRACT :Cette conférence présente un survol des processus aléatoires. Nous débuterons avec une introduction des marches aléatoires comme jeux répétés, et la solution du problème de la ruine du joueur. Ensuite, nous considérerons les limites des distributions des chaînes de Markov, liées aux algorithmes Monte Carlo par chaînes de Markov (MCMC), tout particulièrement les algorithmes de marches aléatoires Metropolis. Nous discuterons les chaînes de Markov couplées et l'inégalité de couplage pour borner les temps de convergence. Finalement, nous examinerons le potentiel et les difficultés des algorithmes MCMC adaptatifs. Toutes ces notions seront illustrées par des exemples très simples, à l'aide de simulations graphiques avec des applets java. Une réception suivra la conférence au Salon Maurice l'Abbé (salle 6245). There will be a reception after the lecture in Salon Maurice-l'Abbé (Room 6245). |
UQAM, Pav. Sherbrooke, 200, rue Sherbrooke O., salle SH-3420 RESUME / ABSTRACT :We will give a survey of some results about eigenfunctions of Laplacians, from 1787 to 2007. We shall also discuss some recent estimates for the spectral function and the remainder in Weyl's law. |
UQAM, Pav. Sherbrooke, 200, rue Sherbrooke O., salle SH-3420 RESUME / ABSTRACT :I will discuss, both verbally and through actual toy demonstrations, a number of nonholonomic problems. They exhibit signatures of integrability - conserved quantities, quasi-periodicity, etc. - but they do not fit into the current models of integrable systems. Other signatures are chirality and finite-time singularity, and I argue that these should be part of integrability in a generic nonholonomic problem. |
UdeM, Pav. André-Aisenstadt, 2920, ch. de la Tour, salle 6214 RESUME / ABSTRACT :En 1925 un étudiant au doctorat, Ernst Ising, proposait un modèle simple ayant pour but de décrire le ferromagnétisme. Il réussit à montrer qu'en une dimension, son modèle ne possédait pas de phase ferromagnétique, mais ne parvint pas à trancher la question en deux dimensions. Nous savons maintenant que le modèle d'Ising décrit correctement une transition de phase en deux dimensions. Mais la liste de questions ouvertes reliées à ce modèle demeure fort longue. Cette liste attire de plus en plus de mathématiciens à cause des liens inattendus qui ont été tissés depuis vingt ans entre ce modèle physique et plusieurs champs des mathématiques (l'algèbre, la géométrie, l'analyse, les probabilités). |
UQAM, Pav. Sherbrooke, 200, rue Sherbrooke O., salle SH-3420 RESUME / ABSTRACT :I hope my talk will be mostly an introductory survey about the complexity of search algorithms, about second Hamiltonian paths (traveling salesman routes) in graphs, and about bimatrix games (2-person Nash equilibria), with a bit about the connection between them. Biography: Jack Edmonds is widely regarded as one of the most important and influential contributors to the fields of computational complexity and combinatorial optimization. He was the recipient of the 1985 John von Neumann Theory Prize, and his seminal papers on matroids and matchings are regarded as amongst the most important papers in all of discrete mathematics. |
UQAM, Pav. Sherbrooke, 200, rue Sherbrooke O., salle SH-3420 RESUME / ABSTRACT :Borrowing examples from statistical physics, we address various enumerative and algebraic combinatorial problems, such as counting maps with marked points at fixed geodesic distances, or exploring relations between alternating sign matrices, plane partitions and the geometry of orbital varieties, via a connection to lattice loop gas. We will show how classical or quantum integrability inherited from the structure of the physical theories at hand can be used to derive exact combinatorial results. |
UQAM, Pav. Sherbrooke, 200, rue Sherbrooke O., salle SH-3420 RESUME / ABSTRACT :I will discuss the problem of nonlinear wave motion of the free surface of a body of fluid with a varying bottom. The ob ject is to describe the character of wave propagation in a long-wave asymptotic regime. The case in which the bottom topography is periodic is shown to homogenize completely, and we compute how the bottom irregularities affects the effective Boussinesq equations and in the appropriate unidirectional limit, the Korteweg de Vries (KdV) equation. We also consider the case of a bottom described by a random, stationary ergodic process with sufficiently strong mixing conditions. We show that in the long wave limit, the random effects are governed by a canonical limit process which is equivalent to a white noise through Donsker’s invariance principle, with one free parameter the variance. The coherent wave motions of the KdV limit are shown to be preserved, while at the same time the random effects are described, and the degree of scattering due to the variable bottom is quantified. Our analysis is performed from the point of view of perturbation theory for Hamilto- nian PDEs with a small parameter. |
UQAM, Pav. Sherbrooke, 200, rue Sherbrooke O., salle SH-3420 RESUME / ABSTRACT :In the past 25 years there has been a rapid evolution in our understanding of the differential topology of smooth four dimensional manifolds. I will try to give an overview of this evolution and how it has affected the mathematics and physics. |
UQAM, Pav. Sherbrooke, 200, rue Sherbrooke O., salle SH-3420 RESUME / ABSTRACT :Self assembly refers to the dream of being able to mix small components in a jar and have them spontaneously assemble into a functional device. The primary obstacle with making this work in practice is that in general a set of N interacting objects have a large number of metastable states, which grows rapidly (exponentially) with N. In principle this can be dealt with by either designing the energy function so that there is only a unique equilibrium state or by tuning the dynamics so that the desired equilibrium is accessed from a specified initial conditions. Both of these methods require a close interplay between mathematics and experimentation. This talk will summarize opportunities in this field, and also discuss two examples of our recent research in this direction: First we discuss a method for assembling uniquely specified packings of spheres where the non-uniqueness problem does not exist; secondly we discuss recent efforts to discover whether and how interaction specificity between spheres can lead to selection of desired structures, avoiding the nonuniqueness problem. |
UQAM, Pav. Sherbrooke, 200, rue Sherbrooke O., salle SH-3420 RESUME / ABSTRACT :The temperature of the sun's atmosphere is about three magnitudes higher than that of the surface of the sun. A classical explanation of this remarkable phenomenon involves the notion that intense electrical currents are produced when tangled magnetic field lines try to move to lower energy configurations. A simple model problem is used to demonstrate how imposition of topological constraints can produce singularities in a solution to an energy minimization problem which would not arise in the absence of such constraints. More precisely a topological decomposition of W^{1,2} Sobolev functions in two dimensions is used to establish existence of magnetic fields in cylindrical symmetry with prescribed field line topology. This is applied to a classical example related to existence of current sheets in the solar corona to illustrate a method of establishing existence of discontinuities in magnetic fields. |
CRM, UdeM, Pav. André-Aisenstadt, 2920, ch. de la Tour, salle 6214 RESUME / ABSTRACT :In 1977 Szemeredi proved that any subset of the integers of positive density contains arbitrarily long arithmetic progression. A couple of years later Furstenberg gave an ergodic theoretic proof of Szemeredi's theorem. At around the same time Furstenberg and Sarkozy independently proved that any subset of the integers of positive density contains a perfect square difference, namely elements x,y with x-y=n^2 for some positive integer n. In 1995, Bergelson and Leibman proved, using ergodic theoretic methods, a vast generalization of both Szemeredi's theorem and the Furstenberg-Sarkozy theorem, establishing the existence of arbitrarily long polynomial progression in subsets of the integers of positive density. The ergodic theoretic methods are limited, to this day, to handling sets of positive density. However, in 2004 Green and Tao proved that the question of finding arithmetic progressions in some special subsets of the integers of zero density - for example the prime numbers - can be reduced to that of finding arithmetic progressions in subsets of positive density. In recent work with T. Tao we show that one can make a similar reduction for polynomial progressions, thus establishing, through the Bergelson- Leibman theorem, the existence of arbitrarily long polynomial progressions in the prime numbers. |
CRM, UdeM, Pav. André-Aisenstadt, 2920, ch. de la Tour, salle 6214 RESUME / ABSTRACT :Siegel introduced local representation densities in his quantitative study of the number of representations of one integral quadratic form by another. After reviewing this classical theory, we will introduce secondary invariants, the derivatives of representation densities which seem to contain new arithmetic information. We will discuss some examples from joint work with M. Rapoport in which these derivatives of representation densities are related to intersection numbers in arithmetic geometry. |
CRM, UdeM, Pav. André-Aisenstadt, 2920, ch. de la Tour, salle 6214 RESUME / ABSTRACT :When should we say that a singularity on one complex algebraic variety is the "same" as that on another? If the two varieties are plane curves, then we have known the answer for eighty years, but in other cases, the notion of equisingularity has been "elusive," as Zariski put it. Yet, if one variety is a special member of an algebraic family and the other a general member, then it is easier to say when; namely, the total space should satisfy the Thom--Whitney conditions. Thus we seek numerical invariants whose constancy across the members of a family is necessary and sufficient for the the Thom--Whitney conditions to hold. For isolated complete-intersection singularities (ICIS), such numerical invariants arise from the Buchsbaum--Rim multiplicity of the column space of the Jacobian matrix; the proof involves the theory of integral dependence. This talk will review equisingularity theory, and describe some recent efforts to extend it from ICIS to more general singularities. |
CRM, UdeM, Pav. André-Aisenstadt, 2920, ch. de la Tour, salle 6214 RESUME / ABSTRACT :I will explain some results, constructions, and conjectures of the last two decades, motivated by number theory and quantum string theory respectively, whose common theme is: counting rational points/rational curves on algebraic varieties. |
CRM, UdeM, Pav. André-Aisenstadt, 2920, ch. de la Tour, salle 1140 |
Calendrier des conférences CRM/ISM - Année 2006-2007
Responsables: Marco Bertola (Concordia) (bertola@mathstat.concordia.ca) & Vojkan Jaksic (jaksic@math.mcgill.ca)
UQAM, Pav. Sherbrooke, 200, rue Sherbrooke O., Salle / Room SH-3420 RESUME / ABSTRACT :A fixed elliptic curve over the rational numbers is known to have approximately p points modulo p for any prime number p. In about 1960 Sato and Tate gave a conjectural distribution for the error term. Laurent Clozel, Michael Harris, Nick Shepherd-Barron and I recently proved this conjecture in the case that the elliptic curve has somewhere multiplicative reduction. In this talk I will describe the Sato-Tate conjecture and the ideas Tate and Serre had for proving it. I will also sketch how we were able to prove sufficient higher dimensional modularity results to complete the proof. |
CRM, UdeM, Pav. André-Aisenstadt, 2920, ch. de la Tour, Salle / Room 6214 RESUME / ABSTRACT :In various scientific fields from astro- and high energy physics to neuroimaging, researchers observe entire images or functions rather than single observations. The integral geometric properties, notably the Euler characteristic of the level/excursion sets of these functions, typically modelled as Gaussian random fields, have found some interesting applications in these domains. In this talk, I will describe some of the statistical applications of the (average) integral geometric properties of these random sets, particularly their Lipschitz-Killing curvature measures. Two aspects I will focus on our: i) using the Euler characterstic of the excursion at high level to approximate excursion probabilities and ii) a class of non-Gaussian random fields (but built up of Gaussians) and its relation to the classical Kinematic Fundamental Formulae of integral geometry. |
UQAM, Pav. Sherbrooke, 200, rue Sherbrooke O., Salle / Room SH-3420 RESUME / ABSTRACT :In computational physics, molecular dynamics refers to the computer simulation of a material at the atomic level. In principle the only inputs to a molecular dynamics simulation are a set of particles and a description of the forces between them. An initial condition is chosen, and the evolution in time of the system is simulated by applying a simple numerical technique to Newton's equations. Given its scientific importance there is very little rigorous justification of molecular dynamics. From the viewpoint of numerical analysis it is surprising that it works at all. The problem is that individual trajectories computed by molecular dynamics are accurate for only small time intervals, whereas researchers trust the results over very long time intervals. It has been conjectured that molecular dynamics trajectories are accurate over long time intervals in some weak statistical sense. However, no one has been able to rigourously establish this for any systems of interest. I will present an overview of different approaches to the problem and some recent work which clarifies some of the issues. |
CRM, UdeM,
Pav. Andre-Aisenstadt, |
UQAM, Pav. Sherbrooke, 200, rue Sherbrooke O., Salle / Room SH-3420 RESUME / ABSTRACT :There is a great variety of physical systems outside thermal equilibrium. At this time a big effort is underway to understand nonequilibrium steady states for infinite systems. But it is difficult to identify what are the natural physical quantities for such systems. As an example we shall discuss the heat flow in a system of coupled classical rotators and compare this problem with that of the heat flow in a quantum spin system. |
Le vendredi 20 octobre 2006 / Friday, October 20, 2006 Exceptionnellement à 16 h 15 / Exceptionally at 4:15 p.m.
Anatole Katok (Penn State University)
UQAM, Pav. Sherbrooke, 200, rue Sherbrooke O., Salle / Room SH-3420 RESUME / ABSTRACT :In the classical theory of dynamical systems which deals with diffeomorphisms and smooth flows on compact manifolds hyperbolic behavior is known to imply stability of the global topological orbit structure under small perturbations of the system, called structural stability. However, differentiable orbit structure is never stable. Furthermore, even topological stability has to be qualified in the continuous time case where time change must be allowed. And full hyperbolic structure is necessary for structural stability. It is quite remarkable that for actions of higher rank abelian groups much stronger rigidity phenomena appear. First, there is global rigidity of differentiable orbit structure for standard examples of actions with global hyperbolic behavior, such as commuting hyperbolic automorphisms of a torus, or Weyl chamber flows (which are higher rank counterparts of geodesic flows on symmetric spaces of negative curvature). Notice that for R^k actions with k>1 only linear time changes are allowed. Rigidity for these actions was established in the mid-1990ies in a series of papers joint with M. Guysinsky and R. Spatzier. The central idea of the method is proving regularity of the structural stability maps by building invariant geometric structures for perturbed actions and showing that the topological conjugacy must intertwine the invariant structures for perturbed and unperturbed actions. More recently jointly with D. Damjanovic we showed that similar differentiable rigidity takes place for several classes of partially hyperbolic actions where there is no structural stability in the rank one case and hence the previous method is totally useless. I Instead we developed two mutually complementary methods which may even be more interesting that the results they produce. One method is based on linearization of the conjugacy equation, solving the linearized problem with tame estimates (based on vanishing of the obstructions due to higher rank) and using KAM (Kolmogorov-Arnold-Moser) type iteration scheme to construct a converging sequence of approximate conjugacies. The other method is based on translating the conjugacy problem to a cohomology problem over the perturbed action and using description of generators and relations in classical split Lie groups to solve those equations. |
CRM, UdeM,
Pav. André-Aisenstadt, It took several centuries to mathematicians to realize that a space can be intrinsically "curved". This lead to the discovery of non-Euclidean geometries. Bernhard Riemann introduced the mathematical tool that still now lies at the heart of our understanding of the concept of curvature, namely the "curvature tensor". Since it is a complicated object, both analytically and geometrically, many attempts have been made to cut it into pieces and to get at it from more geometrical points of view. Getting the curvature connected to more global properties of a space is still an active part of present day research in Geometry, as the spectacular developments concerning the PoincarŽ conjecture show. The purpose of the lecture is to try and give an overview of progress in our understanding of the curvature through a step by step analysis of the tools introduced in spite of the non-linear advancement of our knowledge. |
UQAM, Pav. Sherbrooke, 200, rue Sherbrooke O., Salle / Room SH-3420 RESUME / ABSTRACT :Randomness is a known source of deterministic laws. A manifestation of that is found in the spectral theory of linear operators which incorporate extensive disorder. Among the relevant issues is the distinction between regimes of pure-point and of continuous spectra, and in the finite-volume case questions concerning the local level statistics. Observations made in different contexts have led to the somewhat vaguely expressed expectation that the distribution of the level statistics in regimes of continuous spectrum generically resembles the distribution found in the Wigner random matrix ensembles. A different law, the Poisson distribution, is known to describe the level statistics in regimes of spectral localization. We shall comment on interesting deviations from this dichotomy, and also outline some recent works concerning related topics in the context of Schroedinger operators with random potential, and random quantum graph operators. |
CRM, UdeM,
Pav. André-Aisenstadt, I will give a historical account of the study of CR structures. I will discuss some relations with the theory of several complex variables and with partial differential equations. I will indicate how the analysis of CR structures involves techniques of microlocalization and multiplier ideals. The emphasis will be on on existence and regularity theorems which lead to the E. E. Levi problem in several complex variables and to Hans Lewy's example of a non-solvable PDE and its generalizations. |
UQAM,
Pav. Sherbrooke, Ordinary matter is held together with electromagnetic forces, and the dynamical laws governing the constituents (electrons and nuclei) are those of quantum mechanics. These laws, found in the beginning of this century, were able to account for the fact that electrons do not fall into the nuclei and thus atoms are quite robust. It was only in 1967 that Dyson and Lenard were able to show that matter in bulk was also stable and that two stones had a volume twice that of one stone. Simple as this may sound, the conclusion is not at all obvious and hangs by a thread-- namely Pauli's "exclusion principle" (which requires that for N electrons one must consider only the N-fold antisymmetric product of L^2(R^3) instead of the full product). In the ensuing 3 decades much was accomplished to clarify, simplify and extend this result. We now understand that matter can, indeed, be unstable when relativistic effects and magnetic fields are taken into account -- unless the electron's charge is small enough (which it is, fortunately). These delicate and non-intuitive conclusions will be summarized. The requisite mathematical apparatus needed for these results is itself interesting. Finally, we can now hope to begin an analysis of the half-century old question about the ultimate theory of ordinary matter, called quantum electrodynamics (QED). This is an experimentally successful theory, but one without a decent mathematical foundation. Some recent, preliminary steps in resolving the infinities of QED will be presented. |
Applications of an asymptotic expansion for the one-point function of random matrix theory: Loop equations, partition function, large deviation principles
CRM, UdeM,
Pav. André-Aisenstadt, We will first define all the terms in the title and their interplay. Time remaining, we will explain a workhorse result concerning integrals against the RMT mean density of eigenvalues, and how it is used in the applications mentioned in the title. |
On a Riemannian manifold (M^n, g) equipped with a smooth measure m, one can associate geometric quantities such as Ricci tensor, scalar curvature to the Riemannian measure space (M^n,g, m). An example of such an association was introduced by Bakry- ƒmery from the point of view of probability theory and has played interesting roles in the recent works of Perelman, Lott, Lott-Villani, Sturm in the study of problems in the Ricci-flow, collaping of Riemannian metrics and optimal transport. In this talk, I will discuss properties of Bakry-ƒmery Ricci tensor. I will report some recent joint work with Matt Gursky and Paul Yang, in which we introduce some conformal ly invariant notion of Ricci and scalar curvatures associated to a Riemannian measure space (M^n,g, m) which as dimension tends to infinity can be identified with the Bakry-ƒmery Ricci tensor and scalar curvature. I will also discuss construction of covariant operators, some geometric variational problems and some cohomology vanishing theorem in this setting.
Le vendredi 7 avril 2006 / Friday, April 7, 2006 16 h / 4:00 p.m.
Helmut Hofer Courant Institute Quantitative Symplectic Geometry CRM, UdeM, Pav. André-Aisenstadt, 2920, ch. de la Tour, Salle / Room 6214 RESUME/ABSTRACT:
This will be a very elementary talk about a very non-elementary subject. The beautiful fact is that problems and facts are easy to understand, while the proofs usually have to rely on heavy duty machinery like symplectic field theory. Quantitative symplectic geometry is concerned with the "size" of a symplectic space. A good starting point is already the consideration of open sets in a linear symplectic space. Since about 20 years, through the work of Gromov and others, we know that there is a fascinating world of invariants going beyond volume measurents. The resulting theory is the theory of symplectic capacities. It is the main goal of these lectures to show that this theory is rather at its beginnings and far from being a mature theory.
IMPORTANT: ASSEMBLEE GENERALE DES MEMBRES DU CRM DE 17H00 A 18H00 La conférence de Helmut Hofer sera suivie, dans la même salle, de l'Assemblée générale des membres du CRM (membres réguliers, associés, visiteurs et invités), qui ne s'est pas tenue depuis 2002. Tous les membres du CRM de toutes les universités partenaires sont vivement invitŽs à y participer. Le CRM ne peut fonctionner sans tenir cette assemblée. Le directeur du CRM présentera le Centre, ses labos, les six semestres thématiques à venir jusqu'en 2009. L'ordre du jour comprendra Žgalement l'adoption des nouveaux statuts du CRM et l'élection de deux membres au Bureau de direction. Le tout ne prendra pas plus d'une heure.
IMPORTANT: GENERAL MEETING of CRM MEMBERS FROM 5 to 6 PM. Helmut Hofer's lecture will be followed in the same room by the General Meeting of theCRM's Members (regular, associate, invited and visiting members). The last general meeting was held in 2002. All CRM members of all partner universities are strongly encouraged to participate. CRM cannot function without this general meeting. CRM Director will present the activities of the Centre, its research laboratories, and the 6 upcoming thematic semesters until 2009. The agenda also includes the adoption of new statutes and the election of two representatives from the membership to the CRM's Bureau de direction. The meeting should last about one hour.
Le vendredi 24 mars 2006 / Friday, March 24, 2006 16 h / 4:00 p.m.
John Mather Princeton University "Arnold Diffusion" CRM, UdeM, Pav. Andre-Aisenstadt, 2920, ch. de la Tour Salle / Room 6214 RESUME/ABSTRACT:
The problem of Arnold Diffusion concerns small perturbations of integrable Hamiltonian systems. Arnold Diffusion is said to take place when there exist orbits that wander over phase space, in a suitable sense, to be defined. The problem is to prove that such orbits exist (or otherwise). I will discuss a result concerning the existence of Arnold Diffusion in two and one half and three degrees of freedom.
Le vendredi 17 mars 2006 / Friday, March 17, 2006 16 h / 4:00 p.m.
Vladimir Ratakh Rutgers University "Algebras associated to directed graphs and related to factorizations of noncommutative polynomials" UQAM, Pav. Sherbrooke, 200, rue Sherbrooke O. Salle SH-3420 RESUME/ABSTRACT:
Polynomials with noncommutative coefficients such as matrices, operators, quaternions,... appear in many areas of mathematics. One of the main problems in the theory of such polynomials is to find their possible factorizations into a product of linear factors. This problem is not trivial even for polynomials in one variable: a polynomial with noncommutative coefficients may have a lot of different factorizations into a product of linear factors. The structure of such factorizations can be rather complicated and the associated subalgebra of pseudo-roots seems to be very interesting. One way to describe the situation is to study the directed graph of the right divisors of a polynomial. In this talk we present a "universal" approach to this problem. We introduce and study a wide class of algebras associated to directed graphs and universal polynomials over these algebras. For many "natural" graphs such algebras are Koszul and and they have nice Hilbert series. The talk is based on joint papers with I. Gelfand, S. Serconek and R. Wilson.
Le vendredi 10 mars 2006 / Friday, March 10, 2006 16 h / 4:00 p.m.
Peter Zograf Steklov Mathematical Institute, St. Petersburg "Witten-Kontsevich theory and Weil-Petersson volumes of moduli spaces of algebraic curves" CRM, UdeM, Pav. Andre-Aisenstadt, 2920, ch. de la Tour Salle / Room 6214 RESUME/ABSTRACT:
This talk is an introduction to the intersection theory of moduli spaces of algebraic curves. We discuss the famous Witten's conjecture (originally proven by Kontsevich) that the tautological intersection numbers on moduli spaces are determined by a solution to the Korteveg-deVries hierarchy. We describe the relationship betweeen these numbers and Weil-Petersson volumes of moduli spaces and outline a recent proof of Witten's conjecture by Mirzakhani that utilizes Weil-Petersson volumes.
Le vendredi 3 mars 2006 / Friday, March 3, 2006 16 h / 4:00 p.m.
Fedor Bogomolov Courant Institute "Geometry of algebraic varieties over small fields ($\bar F_p, \bar Q$)" CRM, UdeM, UQAM, Pav. Sherbrooke, 200, rue Sherbrooke O., Salle/Room: SH-3420 RESUME/ABSTRACT:
Modern algebraic geometry considers algebraic varieties defined over different fields( usually algebraically closed) Though there is a similarity between classification results over diffrent fields, there are also drastic differences. In the talk I will describe several new results about special properties of algebraic varieties over fields $\bar F_p, \bar Q$.
Le vendredi 24 février 2006 / Friday, February 24, 2006 16 h / 4:00 p.m.
Des Higham University of Strathclyde "A New Model for Protein-Protein Interaction Networks" CRM, UdeM, Pav. Andre-Aisenstadt, 2920, ch. de la Tour Salle / Room 1140 RESUME/ABSTRACT:
Advances in experimental biology are creating challenging modelling and data analysis problems for researchers in bioinformatics. In particular, protein-protein interaction data sets can be viewed as large unweighted, undirected graphs that, when analyzed appropriately, may reveal important biological information. Researchers have considered high-level questions, such as ``can we describe these networks in terms of a few parameters?'' and low-level questions such as ``can we identify interesting groups of proteins?''. I will describe a new graph model that aims to contribute at both levels. Results for real biological data sets will be given.
Le vendredi 17 février 2006 / Friday, February 17, 2006 16 h / 4:00 p.m.
Roman Schubert University of Bristol "Universality in wave propagation for large times" UQAM Pav. Sherbrooke, 200, rue Sherbrooke O. Salle SH-3420 RESUME/ABSTRACT:
We look at wave propagation on compact Riemannian manifolds, and are interested in the fate of a propagating wavepacket for large times and small wave-length. By classical geometric optics constructions for small wavelength the wavefronts are propagated along geodesics perpendicular to them. This means that for large times the ergodic properties of the geodesic flow determine what happens to a wavepacket. For manifolds of negative curvature the geodesic flow is hyperbolic and shows universal features like mixing and the validity of a central limit theorem. We discuss how these features lead to similar universal behavior for propagation of wavepackets.
Cafe et biscuits seront servis avant le colloque et un aperitif suivra. Coffee and cookies will be served before the colloquium and a cocktail will follow.
Le vendredi 10 février 2006 / Friday, February 10, 2006 16 h / 4:00 p.m.
Yousef Saad University of Minnesota "Solution of sparse matrix problems by domain decomposition-type methods" CRM, UdeM, Pav. Andre-Aisenstadt, 2920, ch. de la Tour Salle / Room 1140 RESUME/ABSTRACT:
Domain
decomposition is a powerful and pervasive method used for solving
partial differential equations, principally in parallel computing
environments. This method can be extended to the solution of sparse
linear systems of equations or eigenvalue problems. One such extension
is the Algebraic Recursive Multilevel Solver (ARMS). ARMS is a purely
algebraic technique and it is general purpose in the same way that
ILUT (for example) is. The main idea on which ARMS is based, is to
recursively form the Schur complement system associated with a set of
"coarse points" which are determined algebraically. The Schur
complement is sparsified and the same process of reduction can then be
applied recursively until a last level is reached, where a standard
ILU factorization is invoked. ARMS has been developed by combining
tools from sparse matrix computations and graph theory, and insight
from Domain Decomposition-type methods. Probably the most important
ingredient of ARMS is the divide and conquer approach made possible by
exploiting Schur complement ideas combined with dropping. This
viewpoint has proven useful in many related methods and the goal of
this talk is to stress its power and its versatility in an algebraic
framework.
In addition to ARMS, and its parallel implementation (pARMS) we will
present a highly parallel multilevel Schur complement method termed
PHIDAL, which can be viewed as an extension of wirebasket techniques,
to general sparse linear systems. We will also briefly discuss how
Schur complements can be effectively used for eigenvalue problems. The
talk will end with the description of a new method which exploits two
sided permutations in conjunction with Schur complements and a
multilevel framework.
Cafe et biscuits seront servis avant le colloque et un aperitif suivra. Coffee and cookies will be served before the colloquium and a cocktail will follow.
Le vendredi 3 février 2006 / Friday, February 3, 2006 16 h / 4:00 p.m.
George Zaslavsky Courant Institute UQAM, Pav. Sherbrooke, 200, rue Sherbrooke O. Salle SH-3420 RESUME/ABSTRACT:
Evidence of absence of ergodicity and mixing will be presented for few typical physical models. A notion of dynamical quasi-traps will be introduced and the related examples will be provided. An application of these features will be extended for pseudochaos i.e. for a kind of randomness of trajectories with zero Lyapunov exponent. As the main application we consider the field lines of a divergence free vector field. Its field lines can be interpreted as trajectories of tracers (Lagrangian particles). Their randomness in the coordinate space can be considered as a specific type of turbulence. We will review the problem and discuss some new discoveries in particle dynamics and pseudochaotic behavior of fieldlines.
Cafe et biscuits seront servis avant le colloque et un aperitif suivra. Coffee and cookies will be served before the colloquium and a cocktail will follow.
Le vendredi 27 janvier 2006 / Friday, January 27, 2006 16h / 4:00 p.m.
Adrian Iovita
Concordia University
"On the arithmetic of elliptic curves"
CRM, UdeM, Pav. Andre-Aisenstadt, 2920, ch. de la Tour Salle / Room 6214 RESUME/ABSTRACT:
An elliptic curve over the rationals is a smooth projective curve of genus one together with a fixed rational point. The set of rational points of such a curve has a natural structure of finitely generated abelian group. This talk will discuss analytic methods complex and p-adic) for computing the ranks of these groups.
Café et biscuits seront servis avant le colloque et un aperitif suivra. Coffee and cookies will be served before the colloquium and a cocktail will follow.
Le vendredi 20 janvier 2006 / Friday, January 20, 2006 16 h / 4:00 p.m.
Vitali Bergelson Ohio State University "Ergodic theory and the properties of large sets"
UQAM, Pav. Sherbrooke, 200, rue Sherbrooke O. Salle SH-3420 RESUME/ABSTRACT:
(i) If A is a
set of reals having positive Lebesgue measure, then there exists a
positive real a, so that A-A contains the interval (-a,a). (ii) If A
is a set of natural numbers having positive upper density, then for
any polynomial p(n) having integer coefficients and zero constant
term, the set A-A contains infinitely many integers of the form
p(n). (iii) If F is an infinite algebraic field and G is a subgroup of
finite index in the multiplicative group F*, then G-G = F.
In this talk we shall discuss these and other similar results from the
perspective of Ergodic Ramsey Theory. This discussion will lead us to
new interesting results and conjectures. In particular we will see the
foregoing as a special case of the appearance of rather arbitrary
finite configurations inside sufficiently large sets.
The talk is intended for a general audience.
Le vendredi 13 janvier 2006 / Friday, January 13, 2006 16 h / 4:00 p.m.
Michael Goldstein University of Toronto "Anderson localization for shifted and skew-shifted potentials : some recent developments" CRM, UdeM, Pav. Andre-Aisenstadt, 2920, ch. de la Tour Salle / Room 1140RESUME/ABSTRACT:
Eigen-functions and spectrum of Schroedinger equation with potentials exhibiting random behavior were studied extensively in the last fourty years starting from the famous works of Anderson and Harper. Properties of this type of equations are fundamental in understanding phase transitions in quantum mechanical disordered systems of solid state physics. Besides their relevance to physics these equations suggest a reach mathematical program. The central part of this program consists of the study of the structure of the so-called set of resonances and intersections of the different shifts of this set under the translations in the space of potentials. The most important questions regarding the properties of the eigen-functions, in particular their exponential decay, know as Anderson localization, are closely related to this set. The answers to these questions are expected to depend on the dimension of the problem and also on the stochastic properties of the translations in the space of the potentials (regular stationary processes, hyperbolic dynamical systems and i.i.d. random values like in the Anderson model or quasi-periodic dynamics like in HarperÕs model). These questions were studied first in perturbative regimes with use of ideas of KAM theory. In last five years new methods of the analysis of resonances for quasi-periodic and skew-shifted dynamics were developed in the works of Bourgain, Goldstein and Schlag. In this talk, we will describe the status of the main problems in thie field, some recents results in particular we will outline the connection between the fine properties of the integrated density of states and the equidistribution of zeros trigonometric polynomials which arrise in these problems.
Le vendredi 6 janvier 2006 / Friday, January 6, 2006 16 h / 4:00 p.m.
Vašek Chvátal Concordia University "Recent advances in solving the Travelling Salesman Problem" UQAM, Pav. Sherbrooke, 200, rue Sherbrooke O. Salle SH-3420
Trimestre d'automne / Fall Semester 2005
2005-2006 Le vendredi 16 decembre 2005 / Friday, December 16, 2005 16 h / 4:00 p.m.
James W. Cogdell Ohio State University
"L-functions, modularity, and functoriality"
CRM, UdeM, Pav. Andre-Aisenstadt, 2920, ch. de la Tour Salle / Room 6214RESUME/ABSTRACT:
There is a very interesting, and still very mysterious, complex analytic invariant attached to an arithmetic object -- its L-function. (The Riemann zeta function is an example.) There is also a family of more analytic objects that have similar complex analytic invariants -- modular forms or automorphic forms. In this talk I would like to discuss both arithmetic and automorphic L-functions. I will pay particular attention to the the nature of the ``Converse Theorem for GL(n)'', which morally says: any object with a nice L-function should be modular. I will explain how this leads naturally to both Langlands' conjectures on the modularity of Galois representations and Langlands' Functoriality conjecture. Finally I will discuss the Converse Theorem as a practical tool for establishing functoriality, concentrating on the cases of the lifting of automorphic forms from the classical groups to GL(n). The hope is that the talk will be expository, self contained, and understandable to a general mathematical audience.
Le vendredi 9 decembre 2005 / Friday, December 9, 2005 16 h / 4:00 p.m.
Tamas Erdelyi A&M University
"Excursions in Unimodular Polynomials"
UQAM, Pav. Sherbrooke, 200, rue Sherbrooke O., salle SH-3420RESUME/ABSTRACT:
Unimodular polynomials are polynomials with complex coefficients of modulus one. In 1957 Erdšs asked two natural questions about unimodular polynomials. The easier one was disproved by Kahane some twenty five years later. The correct answer to the other one is still unknown today. Kahane's result gave rise to the study of ultraflat unimodular polynomials. The talk discusses some conjectures about ultraflat unimodular polynomials, and presents the resolution of some conjectures of Saffari closely related to each other. A few other related problems will also be touched. In particular, at least one question of Peter Borwein on Littlewood polynomials will be answered.
Le vendredi 2 decembre 2005 / Friday, December 2, 2005 16 h / 4:00 p.m.
Alexei Kokotov Concordia University
"Extremal properties of some functionals on the moduli space of genus two Riemann surfaces"
CRM, UdeM, Pav. Andre-Aisenstadt, 2920, ch. de la Tour Salle / Room 6214RESUME/ABSTRACT:
The regularized determinant of the Laplacian in the Bergman metric is shown to be a Morse function on the moduli space $M_2$ of Riemann surfaces of genus two. The critical points of this functional are identified. The results are used to study the orbifold geometry of $M_2$ and Bolza's strata of symmetric surfaces of genus two.
Le vendredi 25 novembre 2005 / Friday, November 25, 2005 16 h / 4:00 p.m.
Edward Nelson Princeton University
"The Mystery of Stochastic Mechanics"
UQAM, Pav. Sherbrooke, 200, rue Sherbrooke O., salle SH-3420RESUME/ABSTRACT:
We develop in strict analogy with the classical case a conservative mechanics for Markov processes. This leads in a natural but surprising way to an alternative interpretation of quantum mechanics, called stochastic mechanics. In this picture particles have continuous trajectories. But there are two major difficulties with this interpretation; one has been known for some years and one is newly discovered. The mystery of stochastic mechanics is how a theory can be so right and yet so wrong.
Le vendredi 18 novembre 2005 / Friday, November 18, 2005 16 h / 4:00 p.m.
Gregory Margulis Yale University
"Quantitative Oppenheim conjecture"
UQAM, Pav. Sherbrooke, 200, rue Sherbrooke O., salle SH-3420RESUME/ABSTRACT:
The Oppenheim conjecture proved in 1986 asserts that for a nondegenerate indefinite quadratic form Q in n>2 variables the set Q(Z^n) is dense in R. In the talk I will discuss the quantitative version of this conjecture. In particular I will describe results in the case of signature (2,2) and some corollaries about eigenvalue spacing on flat 2-tori.
Le vendredi 11 novembre 2005 / Friday, November 11, 2005 16 h / 4:00 p.m.
Walter Craig McMaster University
"On the Boltzmann equation: global solutions in one spatial dimension"
CRM, UdeM, Pav. Andre-Aisenstadt, 2920, ch. de la Tour Salle / Room 6214RESUME/ABSTRACT:
The Boltzmann equation is a description of the statistical mechanics of gases which lies between the many-body description of individual particles and the macroscopic description of a fluid. It is the first place in continuum mechanics where the arrow of time is introduced. I will discuss solutions of the Boltzmann equation which have bounded initial density, for which the total energy and entropy are controlled. In one spatial dimensional geometries (in which solutions depend upon three velocity variables but only one space variable) our result is that the density remains bounded for all finite times. Among several corollaries, the result has implications on the uniqueness of dissipative weak solutions in the sense of P.-L. Lions. The main theorem depends upon a newly described averaging property of the Boltzmann collision kernel. This is joint work with A. Biryuk and V. Panferov.
Le vendredi 4 novembre 2005 / Friday, November 4, 2005 16 h / 4:00 p.m.
A. Shnirelman Concordia University
"The mystery of 2-dimensional fluid"
UQAM, Pav. Sherbrooke, 200, rue Sherbrooke O., salle SH-3420RESUME/ABSTRACT:
This talk is devoted to the properties of an ideal incompressible fluid moving in a 2-dimensional domain. This is an infinite-dimensional dynamical system with peculiar properties whose visible manifestation is the inverse energy cascade: the kinetic energy of the flow is transferred from small to large scales, and eventually accumulates at the global scale, so that (almost) any flow, whatever the initial velocity field, ends up as a steady and stable large scale flow. Thus, steady and stable flows form an infinite-dimensional attractor in the space of velocity fields, which is in apparent contradiction with the time reversibility of the Euler equations. The inverse cascade is observed in experiments and computer simulations, but has not been established rigorously. Moreover, the existence of even one solution of Euler equations displaying the inverse cascade is not proved. In the present talk we discuss some results giving a partial justification of the inverse cascade. In particular, we show the existence of weak solutions of the Euler equations displaying the extreme form of the inverse cascade, when the scale of the initial velocity is zero.
Le vendredi 28 octobre 2005 / Friday, October 28, 2005 16 h / 4:00 p.m.
Emmanuel Letellier Universite Paris VI
"From Kazhdan-Springer to the topological properties of the Riemann-Hilbert monodromy map"
CRM, UdeM, Pav. Andre-Aisenstadt, 2920, ch. de la Tour Salle / Room 6214RESUME/ABSTRACT:
In the seventies, D. Kazhdan and T. A. Springer established a connection between the irreducible character of finite reductive groups and the trigonometric sums on finite reductive Lie algebras via the theory of Green functions. This connection has been generalized into a commutation formula between Fourier transforms and Deligne-Lusztig induction. Recently, T. Hausel conjectured that the Riemann-Hilbert map which goes from Nakajima's star-shaped quiver varieties to their corresponding character varieties preserves mixed Hodge structures and induces an isomorphism on the pure parts. Via the theory of e-polynomials we can connect the generalization of Kazhdan-Springer theory with Hausel's conjecture. As a result, we get a conjectural formula for the mixed Hodge polynomials of these character varieties in terms of the (q,t)-Macdonald polynomials (this work is partly joint with Tamas Hausel and Fernando Rodriguez-Villegas).
Le vendredi 21 octobre 2005 / Friday, October 21, 2005
Fadil Santosa University of Minnesota
**** 14h30 ***** / **** 2:30 p.m. ****
"Seeing better with Mathematics -- A mathematical problem arising in design of ophthalmic lenses"
CRM, UdeM, Pav. Andre-Aisenstadt, 2920, ch. de la Tour Salle / Room 6214RESUME/ABSTRACT:
Progressive addition lenses are prescribed to patients who need correction for both far and near visions. A progressive lens needs to have power that gradually changes from the far vision zone, used for example in driving, and the near vision zone, used for example in reading a map. The basics of optics and lens design will be described. In particular, it will be shown that the problem can be reduced to one of surface design. The surface design problem itself is solved by a variational approach, which can be further simplified by linearization, leading to a fourth order elliptic PDE. Numerical results are presented to illustrate the process of lens design.
RESUME/ABSTRACT:
A multiplier is a function such that local a priori estimates for partial differential equations hold only after the test function is multiplied by it. The ideal sheaf consisting of multipliers identifies the location and the jet orders where local a priori estimates fail to hold. Solvability of a partial differential equation is reduced to algebraic conditions which force the multiplier ideal sheaf to be the structure sheaf. On the side of analysis, the method of multiplier ideal sheaves has been applied to problems such as the global regularity problem of the complex Neumann equation on pseudoconvex domains and the existence of Kaehler-Einstein metrics of Fano manifolds. On the side of algebraic geometry, the method of multiplier ideal sheaves has been successfully applied to solving, or making substantial progress towards solving, a number of long outstanding problems in algebraic geometry such as the Fujita conjecture, the effective Matsusaka big theorem, the deformational invariance of plurigenera, and the finite generation of canonical rings."
RESUME/ABSTRACT:
In this lecture we will give a description of the Yamabe problem (concerning constant scalar curvature Riemannian metrics) aimed at a general audience. We will give an account of the progress which has been made on the problem, and describe an old conjecture of ours concerning the full set of solutions of the problem on higher dimensional manifolds. We will state a recent joint theorem of ours with Marcus Khuri which largely solves this conjecture. Typically there are many high energy solutions with high Morse index for this variational problem. The problem is intimately connected with positive energy theorems of General Relativity, and we will describe this connection and the recent progress on these theorems in the generality needed for the Yamabe application. We will also give an account of recent work on the convergence of the Yamabe flow which has parallels with the variational theory.
RESUME/ABSTRACT:
Consider a closed compact hypersurface M embedded in R'n. Assume that for any pair of points (x',a) and (x',b) on M,with a<b, The mean curvature of M at the first is not greater than that at the second. Then under some conditions,it is shon that M is symmetric about a hyperplane : x'n = constant. The problem leds to new forms of the classical Hopf Lemma. The talk will be expositury.
RESUME/ABSTRACT:
In this talk I will describe some models of chaotic scattering on
classical and quantum levels and introduce the notion of resonances
which replace eigenvalues, familiar from basic quantum mechanics,
and omnipresent in mathematics, as quantum states.
I will then describe recent mathematical, numerical, and experimental
advances in counting these quantum states, stressing the emergence of
fractal Weyl laws.
RESUME/ABSTRACT:
Ramanujan dealt with the arithmetic properties of the
Fourier coefficients of some classical modular forms, and
unearthed striking congruences that contain important number
theoretic information. For example, fix an even, positive
definite, unimodular lattice in $24$-dimensional space; the
number of vectors of given length in it is an arithmetic
function difficult to understand, but nevertheless has a
simple expression modulo $691$. This same arithmetic function,
now taken modulo $11$, also has a simple expression once you
know the arithmetic function mod $11$ that counts the number
of solutions modulo $p$, for all $p$, of the equation $y^2+y = x^3+x$.
Ever since such discoveries, the search for {\it congruences}
that unify otherwise seemingly remote problems has broadened
to guide much number-theoretic research. For example, it played
its role in the dramatic proof of modularity of elliptic curves
over {\bf Q}, a decade ago. The standard way of organizing this
web of congruences is to view modular forms, and the representations that are associated to them, as members of a continuous family
(with $p$-adic parameters).
This leads to one of the major themes of the workshop (to be) held
Sept 12-16 in Montreal. The aim of this colloquium is to describe
this theme to a nonspecialized audience.
RESUME/ABSTRACT:
After reviewing several definitions of K(X), I will explain how they can be 'twisted' to give K-Theories classified by H3(X,Z). One motivation for these theories is the B-field in string theory. Another is projective vector bundles.
I'll give examples and describe some new developments.
RESUME/ABSTRACT:
In this talk we discuss a very simple type of integral performed on a
vector field defined as the gradient of the Euclidean distance function
to the bounding curve (or surface) of a binary object. The limiting behavior
of this integral as the enclosed area (or volume) shrinks to zero reveals
a very useful invariant which can be used to compute the Blum skeleton
as well as to reveal the geometry of the object that it describes. We shall
discuss some recent applications of this technique to problems in computer
vision, medical imaging and computer graphics.
RESUME/ABSTRACT:
After a brief review of some aspects of the fractional quantum
Hall effect, we shall discuss the role of a multi-resolution analysis (MRA) of L^2(R), in connection with the problem of finding the ground state for the Hamiltonian of the quantum system. This analysis suggests a general strategy to produce examples of MRA, starting from a given seed function h(s). We shall discuss the role of a canonical map between certain canonically conjugate operators, showing that our procedure is really model-independent.
RESUME/ABSTRACT:
The classical Weyl character formula has 'q-analogs' which
provide links between representation theory, geometry and the
combinatorics of symmetric functions. In 1988, Macdonald discovered
'q,t-analogs' involving an extra parameter. Macdonald's functions
appear in geometric and representation-theoretic contexts not
visible from the one-parameter q-theory. The talk will be an
introduction to some of these developments, with hints about
tantalizing combinatorial aspects of Macdonald theory that we are
just beginning to understand.
RESUME/ABSTRACT:
We will discuss some aspects of the use of integrable
techniques in the theory of Toeplitz determinants and
its applications to random matrices and statistical mechanics.
RESUME/ABSTRACT:
A trajectory (orbit) of a flow on a 3-manifold is wild
if the closure of at least one of the semi-trajectories
is a wild arc. A trajectory is 2-wild if the closure of
each semi-trajectory is a wild arc.
We describe a method of embedding wild trajectories in
flows on 3-manifolds. This methods yields interesting
examples of dynamical systems. In particular, every
orientable boundaryless 3-manifold admits a flow with
a discrete set of fixed points and such that the closure
of every non-trivial trajectory is 2-wild.
RESUME/ABSTRACT:
In this talk we will describe some equations of Monge-Ampere
type which naturally arise in conformal geometry. We will explain
their origins, some applications of these equations to problems
in global Riemannian geometry, and finally some existence results.
Abel, Galois, Liouville, Picard, Vessiot, Kolchin and others found
a lot of results about solvability and insolvability of equations
in finite terms. According to them, algebraic equations are
usually not solvable by means of radicals. Ordinary linear
differential equations and holonomic systems of linear
differential equations in partial derivatives are not usually
solvable by quadratures. Galois theory belongs to algebra. In fact
results about insolvability of differential equations belong to
differential algebra (and are also purely algebraic).
About 30 years ago I constructed a topological version of Galois
theory for functions in one complex variable. According to it,
there are topological restrictions on the way the Riemann surface
of a function representable by quadratures covers the complex
plane. If the function does not satisfy these restrictions, then
it is not representable by quadratures. Beside its geometric
clarity the topological results on nonrepresentability of
functions by quadratures are stronger than the algebraic results.
Recently I have constructed a multi-dimensional topological version
of Galois theory.
No preliminary knowledge is required.
A brief look at Ueno's book on the classification
theory of compact complex manifolds will convince
the reader that there are many conjectures and very
few general results in this setting aside from the
ones we may generalize directly from Kahler manifolds.
In this talk, I will look at the positivity theorems
of Kawamata-Viehweg and of Miyaoka in a fresh light and
purely within complex geometry so as to give some
applications to the geometry and classification of compact
complex manifolds. Many results obtained this way are
complementary to results in Mori's Program but we will
focus on the structure theorems for various canonical
fibrations of compact complex manifolds, resolving and
generalizing the conjectures of Ueno concerning some of
these fibrations.
Le chaos quantique est l'étude semi-classique de
systèmes (spectraux) quantiques dont la limite
classique est un système dynamique hamiltonien
chaotique. Une question centrale (parmi beaucoup
d'autres) est la compréhension du comportement
des fonctions propres de ses systèmes dans la limite
semi-classique. J'expliquerai cette problématique
à l'aide d'exemples, puis je passerai en revue
quelques re&ecute;sultats recents plus sp&ecute;cifiques pour
les systèmes dynamiques discrets, les "applications
quantiques" comme le chat d'Arnold et ses perturbations
non-linéaires.
Phylogenetics is the art of reconstructing evolutionary history,
usually represented as an evolutionary tree (a phylogeny). The
reconstruction of history is an inherently uncertain endeavour.
There is the uncertainty we know (sampling error) and
the uncertainty we know we don't know (modelling error).
My approach has been to generalise phylogenetic trees and their
associated stochastic models, giving phylogenetic networks. These
networks can be used to represent error or uncertainty in our trees in
a manner reminiscent of mixture models. I will describe applications of
networks to the study of two billion year old divergences in the tree
of life, and to more recent divergences in the United Nations. I also
hope to touch on links with numerical mathematics, probability,
statistics, combinatorics and algebraic geometry.
The main example is the discrete Laplacian but even the study of the discrete Laplacian leads to considering more general operators of this kind. I will discuss how techniques and ideas from PDE can be used to study these difference operators and point out some examples where analogies between difference and differential operators break down.
Homotopy theory offers the prospect of providing geometric enrichments of algebraic structures. In characteristic zero this enrichments is often the well known habit of trying to see an integer as an Euler characteristic. In this talk I will sketch the homotopy theory which has the theory of families of elliptic curves as its surface manifestation. One result is the contruction of a new object, the spectrum of Òtopological modular forms,Ó which seems every bit as natural as the ring of modular forms, but which has geometric significance. This is project led by Mike Hoplins, with contributions form Paul Goerss, Charles Rezk, and me.
Café et biscuits seront servis avant le colloque et un aperitif suivra. Coffee and cookies will be served before the colloquium and a cocktail will follow.
Le vendredi 7 janvier 2005 / Friday January 7, 2005 16 h / 4:00 p.m.
Vladimir Remeslennikov (UQAM) & Ilya Kazatchkov (Omsk State University) "Free partially commutative groups" UQAM, Pav. Sherbrooke, 200, rue Sherbrooke O., salle SH-3420 RESUME/ABSTRACT:
Let $G$ be an arbitrary partially commutative group.
- We develop divisibility theory for $G$ (and not just for semigroups of positive elements, as in the classical paper on Artin groups). It closely resembles the classical divisibility theory of integers. In particular, we have reasonable concepts of the least common multiple and greatest common divisor.
- We prove the existence of polynomial algorithms for solving main algorithmic problems and estimate the boundaries for their complexity.
- With the help of the algorithms of divisibility theory in each class $w^{G}$ of all elements conjugate to the element $w$ we compute the canonical representative. Its normal formal is called in the paper the \emph{minimal exhausted form} of $w^G$. This form is easy from the computational point of view.
- These results allow us to present a solution of time complexity $O(n^{3})$ for the conjugacy problem for partially commutative groups.
Café et biscuits seront servis avant le colloque et un aperitif suivra. Coffee and cookies will be served before the colloquium and a cocktail will follow.
Le vendredi 10 décembre 2004 / Friday December 10, 2004 16 h / 4:00 p.m.
Vladimir Korepin (SUNY Stony Brook) "Quantum Correlations and Number Theory" CRM, UdeM, Pav. Andre-Aisenstadt, 2920, ch. de la Tour, salle 6214 RESUME/ABSTRACT:
The most famous quantum integrable model is XXX Heisenberg spin chain. It was solved by Hans Bethe in 1931. I will argue that correlation functions in the model can be represented as polynomials [with rational coefficients] of values of Riemann zeta function at odd arguments. These values are celebrated objects of number theory. In 1979 Roger Apery proved that zeta(3) is an irrational number. Later P. Cartier conjectured that the values of Riemann zeta function at odd arguments are algebraically independent transcendental numbers. ******************************************************************* Responsables : O. Cornea (cornea@dms.umontreal.ca) D. Jakobson (jakobson@math.mcgill.ca)
Café et biscuits seront servis avant le colloque et un aperitif suivra. Coffee and cookies will be served before the colloquium and a cocktail will follow.
Le vendredi 3 décembre 2004 / Friday December 3, 2004 16 h / 4:00 p.m.
YURI GUREVICH (Microsoft Research) "What is an Algorithm?" UQAM, Pav. Sherbrooke, 200, rue Sherbrooke O., salle SH-3420 RESUME/ABSTRACT:
One may think that the title problem was solved long ago by Church and
Turing but it wasn't; there is more to an algorithm than the function
it computes. (Besides, what function does an operating system
compute?) The interest to the problem is not only theoretical;
applications include specification and verification of software and
hardware.
In the main part of the talk, we formalize the notion of sequential
algorithm, recall the definition of sequential abstract state machines
(ASMs), and then state precisely the Sequential Characterization
Theorem according to which, for every sequential algorithm A, there
exists a sequential ASM B indistinguishable from A as far as behavior
is concerned; in particular B simulates A step-for-step. The full
proof of the theorem is found in the ACM Transactions on Computational
Logic 1:1 (2000) or on the author's webpage (article 141).
If time allows, we will also discuss
(a) generalizations of the Sequential Characterization Theorem to
parallel and distributed algorithms, and
(b) the applications of the ASM approach at Microsoft.
Bio:
Yuri Gurevich is Sr. Researcher at Microsoft Research in Redmond, WA.
He is also Professor Emeritus of the University of Michigan, ACM
Fellow, Guggenheim Fellow, and Dr. Honoris Causa of Limburg University
in Belgium.
Café et biscuits seront servis avant le colloque et un aperitif suivra. Coffee and cookies will be served before the colloquium and a cocktail will follow.
Le vendredi 26 novembre 2004 / Friday November 26, 2004 16 h / 4:00 p.m.
BRUCE REED (McGillUniversity) "Routed Routing and Graph Minors" CRM, Université de Montréal, Pavillon André-Aisenstadt, 2920, ch. de la Tour Salle / Room 6214 RESUME/ABSTRACT:
Graphs are the simplest and most ubiquitous discrete models of connectivity. A graph is a simply a set of vertices together with a set of edges, each of which joins two pairs of vertices. Thus graphs can represent highway networks, telecommunications networks, or social networks. We consider the following fundamental connectivity question in graph theory: Given sets $S={s_1,...,s_k}$ and $T={t_1,...,t_k}$ of vertices in a graph $G$ can we find $k$ vertex disjoint paths $P_1,...,P_k$ such that $P_i$ joins $s_i$ and $t_i$ (a path is a sequence of vertices every consecutive pair of which is joined by an edge). We give a polynomial time algorithm to solve this problem and discuss its links with the Robertson-Seymour graph minor theory. This talk is directed at a general mathematical audience. Much of the work we discuss is due to Robertson and Seymour.
Café et biscuits seront servis avant le colloque et un aperitif suivra. Coffee and cookies will be served before the colloquium and a cocktail will follow.
Le vendredi 19 novembre 2004 / Friday November 19, 2004 16 h / 4:00 p.m.
ALEJANDRO ADEM (University of British Columbia) "Periodic Complexes and Group Actions" UQAM, Pav. Sherbrooke, 200, rue Sherbrooke O., salle SH-3420 RESUME/ABSTRACT:
A classical problem in topology is that of characterizing those finite groups that act fixed-point freely on a sphere. In this talk we will review these results and describe recent work towards extending this to (i) a product of two spheres and (ii) actions of discrete groups. The methods we use are a combination of techniques in topology and group theory.
Café et biscuits seront servis avant le colloque et un aperitif suivra. Coffee and cookies will be served before the colloquium and a cocktail will follow.
Le vendredi 5 novembre 2004 / Friday November 5, 2004 16 h / 4:00 p.m.
CLAUDE Le BRIS (École nationale des ponts et chausées, France) "Flot généralisés de solutions pour des équations différentielles déterministes et stochastiques à coefficients irréguliers" CRM, Université de Montréal, Pavillon André-Aisenstadt, 2920, ch. de la Tour Salle / Room 6214 RESUME/ABSTRACT:
On utilise et on étend la théorie de DiPerna/Lions pour
les solutions renormalisées d'équations de transport
linéaires, de sorte d'obtenir des notions de flot
généralisé pour des équations différentielles à
coefficients irréguliers.
L'étude est motivée par des questions de modélisation
multiéchelle de fluides viscoélastiques.
Il s'agit d'un travail conjoint avec
Pierre Louis Lions (Collège de France, Paris)
Café et biscuits seront servis avant le colloque et un aperitif suivra. Coffee and cookies will be served before the colloquium and a cocktail will follow.
Le vendredi 22 octobre 2004 / Friday October 22, 2004 16 h / 4:00 p.m.
JEAN-PIERRE GAZEAU (Univ. Paris VII Denis Diderot) "États cohérents et quantification de systèmes simples" UQAM, Pav. Sherbrooke, 200, rue Sherbrooke O., salle SH-3420 RESUME/ABSTRACT:
On exposera une méthode générale de quantification basée sur certaines familles d'états cohérents ou \emph{repères}. On donnera quelques exemples, comme la quantification du mouvement sur le cercle ou le mouvement sur un ensemble discret de points. On présentera aussi de nouvelles inégalités quantiques portant sur les spectres respectifs des opérateurs position et impulsion et résultant d'un tel schéma de quantification pour le mouvement sur la droite.
Café et biscuits seront servis avant le colloque et un aperitif suivra. Coffee and cookies will be served before the colloquium and a cocktail will follow.
Le vendredi 15 octobre 2004 / Friday October 15, 2004 16 h / 4:00 p.m.
STEPHANOS VENAKIDES (Duke University) "The nonlinear analogues of the Fourier transform, steepest descent and eikonal analysis" CRM, Université de Montréal, Pavillon André-Aisenstadt, 2920, ch. de la Tour Salle / Room 6214 RESUME/ABSTRACT:
We will outline and develop the application of steepest descent methods in linear dispersive wave equations using the semiclassical linear Schrödinger equation as our prototype. We will then develop our understanding of basic linear/nonlinear analogies, e.g.
Café et biscuits seront servis avant le colloque et un aperitif suivra. Coffee and cookies will be served before the colloquium and a cocktail will follow.
Le vendredi 8 octobre 2004 / Friday October 8, 2004 16 h / 4:00 p.m.
RONALD FINTUSHEL (Michigan State University) "Lagrangian tori in 4-manifolds" UQAM, Pav. Sherbrooke, 200, rue Sherbrooke O., salle SH-3420 RESUME/ABSTRACT:
A Lagrangian submanifold of a symplectic manifold is a half-dimensional submanifold on which the symplectic form vanishes identically. I will talk about a simple invariant of nullhomologous Lagrangian tori which for tori in C^2 has been studied by Eliashberg and Polterovich. They showed that in C^2 this invariant vanishes on all Lagrangian tori. In contrast, I will discuss families of examples which show that in more complicated symplectic 4-manifolds the invariant need not vanish. On the one hand, it is related to the Maslov class and on the other hand to Seiberg-Witten invariants. Using Seiberg-Witten invariants I will discuss a proof (joint with Ron Stern) that the invariant in question is often a topological (rather than symplectic) invariant. Un vin d'honneur suivra le colloque. A reception, with wine and cheese, will follow the talk.
Le vendredi 1^{er} octobre 2004 / Friday October 1, 2004 16 h / 4:00 p.m.
JOHN HARNAD (Université Concordia et CRM) "Random Matrices, Orthogonal Polynomials and Integrable Systems" CRM, UdeM, Pav. André-Aisenstadt, 2920, ch. de la Tour, salle 6214 RESUME/ABSTRACT:
The spectral theory of random matrices has appeared and re-appeared
in various applications over the past few decades.
Aside from well-known applications to multivariate
statistics, it has been of importance in such
diverse and interesting physical problems
as the statistical theory of
high-lying energy levels of large atomic nuclei
(Wigner, Dyson, 1960's) and attempts at
discretization of the Feynman path integral
underlying 2D-quantum gravity and conformal models (1990's).
More recently, connections have also been made with
supersymmetric Yang-Mills theory, and also some quite
different problems amenable to similar analysis, such
as growth problems in random media, random words,
random tilingsÊ and random permutations, as well
as the seemingly unrelated domain of classical
and quantum integrable systems.
A key step in understanding these relations is to note,
first, that there is an immediate connection with
the theory of orthogonal polynomials, which dates
back to the work of Stieltjes in the 19th century,
and second, that an effective way to study the relevant
statistics of the eigenvalues is by varying the
parameters governing the measure and support of
the spectrum. The latter leads directly to the
types of deformation equations studied in the
theory of integrable systems.
Un vin d'honneur suivra le colloque. A reception, with wine and cheese, will follow the talk.
Le vendredi 24 septembre 2004 / Friday September 24, 2004 16 h / 4:00 p.m.
PAUL SCHUPP (University of Illinois, Urbana Champaign) "The Uniform Membership Problem, Foldings and Polynomial Time" UQAM, Pav. Sherbrooke, 200, rue Sherbrooke O., salle SH-3420 RESUME/ABSTRACT:
The basic question is "To what extent is a group having its uniform membership problem solvable in polynomial time evidence of "good geometry"? This talk will survey known and recent results about when the uniform membership problem is solvable in polynomial time.
Café et biscuits seront servis avant le colloque et un aperitif suivra. Coffee and cookies will be served before the colloquium and a cocktail will follow.
Le vendredi 17 septembre 2004 / Friday September 17, 2004 16 h / 4:00 p.m.
PENGFEI GUAN (University McGill) "Convexity of solutions of geometric nonlinear partial differential equations" CRM, UdeM, Pav. André-Aisenstadt, 2920, ch. de la Tour, salle 6214 RESUME/ABSTRACT:
Convexity is one of the simplest but the most important geometric properties. It is related to positivity of certain curvature tensors which are often guided by nonlinear partial differential equations. This in turn related to a fundamental issue of convexity of solutions of nonlinear PDEs. In a important development 1985, a new technique was devised to deal with the convexity issue via homotopy method of deformation in the work of Caffarelli-Friedman and Yau. We will review historic developments in this direction and describe a general principle of the convexity related to the structure of nonlinear differetial equations in a recent joint work with Caffarelli and Ma.
Archives 2003-2004
Responsable: Iosif Polterovich (iossif@dms.umontreal.ca) 2003-2004 Le vendredi 30 avril 2004, 16h JILL PIPHER (Brown University) "Multiparameter Fourier Analysis" CRM, Université de Montréal, Pavillon André-Aisenstadt, 2920, ch. de la Tour Salle / Room 6214
Le vendredi 23 avril 2004, 16h TERRY LYONS (University of Oxford) "Rough Paths" UQAM, Pav. Sherbrooke , 200, rue Sherbrooke O. Salle / Room SH-3420
Le vendredi 16 avril 2004, 16h GORDAN SAVIN (University of Utah) "Exceptional groups and the arithmetic of cubic fields" UQAM, Pav. Sherbrooke , 200, rue Sherbrooke O. Salle SH-3420
Le vendredi 26 mars 2004, 16 h ALINA STANCU "Curvature flows from the viewpoint of convex geometry" UQAM, Pav. Sherbrooke, 200, rue Sherbrooke O., Salle SH 3420
Le vendredi 19 mars 2004, 16 h CHARLES NEWMAN Courant Institute of Mathematical Sciences "Continuum Nonsimple Loops and 2D Critical Percolation" UQAM, Pav. Sherbrooke, 200, rue Sherbrooke O., salle SH-3420
Le vendredi 12 mars 2004, 16 h YAKOV SINAI (Princeton University) "Mathematical Hydrodynamics" CRM, Université de Montréal, Pavillon André-Aisenstadt, 2920, ch. de la Tour Salle 6214
Le vendredi 5 mars, 16h RAFE MAZZEO (Stanford University) "Positive scalar curvature and Poincare-Einstein fillings" CRM, Université de Montréal, Pavillon André-Aisenstadt, 2920, ch. de la Tour Salle 6214
Le vendredi 27 février 2004, 16 h ERNST HAIRER (Université de Genève) "Are linear multistep methods suitable for long-time integration ?" UQAM, Pav. Sherbrooke, 200, rue Sherbrooke O., Salle SH 3420
Le vendredi 20 février 2004, 16 h DOROTHY BUCK (Brown University & IHES) "The Topology of DNA-Protein Interactions" CRM, Université de Montréal, Pavillon André-Aisenstadt, 2920, ch. de la Tour, Salle 6214
Le vendredi 13 février 2004, 16 h FRANCOIS BERGERON (UQAM) "Invariants de groupes finis et leur applications ˆ la combinatoire" UQAM, Pav. Sherbrooke, 200, rue Sherbrooke O., Salle SH 3420
Le vendredi 30 janvier 2004, 16 h RAVI VAKIL(Stanford University) "A geometric Littlewood-Richardson rule" UQAM, Pav. Sherbrooke, 200, rue Sherbrooke O., Salle SH 3420
Le vendredi 23 janvier 2004, 16 h ECKHARD MEINRENKEN (Toronto) "Chern-Weil theory and Lie algebras" CRM, Université de Montréal, Pavillon André-Aisenstadt, 2920, ch. de la Tour, Salle 6214
Le vendredi 16 janvier 2004, 16 h VICTOR IVRII ( University of Toronto) "Sharp Spectral Asymptotics for Operators with Irregular Coefficients" CRM, Université de Montréal, Pavillon André-Aisenstadt, 2920, ch. de la Tour, Salle 6214
2003 Le vendredi 5 décembre 2003, 16 h VICTOR HAVIN, St.Petersburg University & McGill University "On the separation of singularities of analytic functions" Centre de recherches mathématiques, U de M Pavillon André-Aisenstadt, 2920, ch. de la Tour Salle 6214
Le vendredi 28 novembre 2003, 16 h OCTAV CORNEA, Université de Montréal "The Morse complex" Centre de recherches mathématiques, U de M Pavillon André-Aisenstadt, 2920, ch. de la Tour, Salle 6214
Le vendredi 21 novembre 2003, 16 h STEPHANE FISCHLER, École Normale Supérieure & University of Ottawa "Irrationality of zeta values (after Ap\'ery, Rivoal, ...)" UQAM, Pav. Sherbrooke, 200, rue Sherbrooke O. Salle SH 3420
Le vendredi 24 octobre 2003, 16 h Mikhail SODIN, Tel Aviv University "Growth, zeroes, and area estimates. Variations on the theme" Centre de recherches mathématiques, U de M Pavillon André-Aisenstadt, 2920, ch. de la Tour, Salle / Room 6214
Le vendredi 17 octobre 2003, 16 h Jonathan PILA, University McGill "Integer and rational points on curves and surfaces" Centre de recherches mathématiques, U de M Pavillon André-Aisenstadt, 2920, ch. de la Tour Salle / Room 6214
Le vendredi 10 octobre 2003, 16 h ROBERT RUSSELL, Simon Fraser University and McGill University "Numerical solution of PDEs using moving grids -- an overview" UQAM, Pav. Sherbrooke, 200, rue Sherbrooke O. Salle / Room SH 3420
Le vendredi 26 septembre 2003 IGOR PAK, MIT "The nature of partition bijections" UQAM, Pav. Sherbrooke, 200, rue Sherbrooke O., Salle SH 3420
Le vendredi 19 septembre 2003, 16h00 Felix Finster, Universität Regensburg "The Dirac Sea and the Principle of the Fermionic Projector" CRM, UdeM, Pav. André-Aisenstadt, 2920, ch. de la Tour, salle 6214
Le vendredi 12 septembre 2003, 16h00 Gianni Cassinelli, University of Genoa "Group Theoretical Quantum Tomography" CRM, UdeM, Pav. André-Aisenstadt, 2920, ch. de la Tour, salle 6214
PRINTEMPS 2003 Le vendredi 25 avril, 16h00 Michel Mendès-France, Université de Bordeaux "ZŽros réels des polyn™mes réels" UQAM, Pav. Sherbrooke, 200, rue Sherbrooke, Salle SH-3420
Le vendredi 11 avril, 16h00 (Dans le cadre des Journées Joyal) Pierre Cartier, ?Ecole Normale Supérieure "Categories, groupoides et theorie de Galois des equations différentielles" UQAM, Pav. Sherbrooke, 200, rue Sherbrooke, Salle SH-3420
Le vendredi 4 avril, 16h00 Kathryn Hare, University of Waterloo "Fractal dimensions and the uncertainty principle in harmonic analysis" UQAM, Pav. Sherbrooke, 200, rue Sherbrooke, Salle SH-3420
Le vendredi 28 mars, 16h00 Michael Monastyrsky, Institute of Theoretical and Experimental Physics "Topology and Physics, Old and New Applications" CRM, UdeM, Pav. André-Aisenstadt, 2920, chemin de la Tour, salle 6254
Le vendredi 21 mars, 16h00 Doron Zeilberger, Rutgers University "The Devious and Divine DIAGONAL" UQAM, Pav. Sherbrooke, 200, rue Sherbrooke, Salle SH-3420
Le vendredi 14 mars, 16h00 Mikhail Shubin, Northeastern University "Discreteness of spectrum for Schrädinger operators" CRM, UdeM, Pav. André-Aisenstadt, 2920, chemin de la Tour, salle 6254
Responsable: T. Tokieda (tokieda@dms.umontreal.ca) HIVER 2003 Le vendredi 21 février, 16h00 / Prix André-Aisenstadt 2002 Alexander Brudnyi, University of Calgary "Center problem for ordinary differential equations" CRM, UdeM, Pav. André-Aisenstadt, 2920, chemin de la Tour, salle 6254
Le vendredi 14 février, 16h00 Chris Skinner, University of Michigan "Eisenstein series and arithmetic" UQAM, Pav. Sherbrooke, 200, rue Sherbrooke, Salle SH-3420
Le vendredi 7 février, 16h00 Eugene Lerman, University of Illinois at Urbana-Champaign "Contact group actions and stratified spaces" CRM, UdeM, Pav. André-Aisenstadt, 2920, chemin de la Tour, salle 6254
Le vendredi 31 janvier, 16h00 Maurice Nivat, Université Denis Diderot "Pavages du plan et suites bidimensionnelles homogènes" UQAM, Pav. Sherbrooke, 200, rue Sherbrooke, Salle SH-3420
Le vendredi 24 janvier, 16h00 Alexandre Borovik, UMIST, Manchester "Groups of finite Morley rank and a strange question from number theory" UQAM, Pav. Sherbrooke, 200, rue Sherbrooke, Salle SH-3420
Le vendredi 17 janvier, 16h00 Lisa Jeffrey, University of Toronto "Symplectic quotients and their cohomology" CRM, UdeM, Pav. André-Aisenstadt, 2920, chemin de la Tour, salle 6254
Pour de plus amples informations : activites@CRM.UMontreal.CA For further information: