Theme Year 2003-2004

and Spectral Analysis


We wish to acknowledge The National Science Foundation (NSF) for their contribution (NSF grant DMS-0339017).

Organizing Committee

E. Bierstone (Toronto), W. Craig (McMaster), F. Finster (Regensburg), D. Jakobson (McGill), V. Jaksic (McGill), N. Kamran (McGill), Y. Last (Hebrew), R. Melrose (MIT), P. Milman (Toronto), C. Pillet (CPT-Toulon), D.H. Phong (Columbia), I. Polterovich (Montreal), J. Toth (McGill), S. Zelditch (Johns Hopkins).



Analysis has traditionally stood at the center of a wide spectrum of research activities in mathematics. In particular, the fields of geometric and spectral analysis have played a fundamental role in shaping the major themes of current research in differential geometry and mathematical physics, and now touch in an important way onto areas such as number theory and algebraic geometry. They are at the core of several of the deepest and most spectacular advances in these fields.


The thematic year in geometric and spectral analysis will focus on a number of themes in which this interaction has been particularly fruitful. The year is organized around two interconnected themes: the first, whose different subthemes cover the whole year, is principally centered on various questions in spectral analysis; it comprises a short programme on analysis on singular spaces, and a more extended period on spectral analysis in geometry, mathematical physics and number theory. The second theme relates to the analysis of the Einstein equations, a subject on which there has been spectacular progress in recent years. It is concentrated in the fall of 2003.


These themes have been chosen for a balance between the geometric and spectral components of the scientific programme, and also with the objective of highlighting some of the most interesting current applications of analytic ideas to physics.


There will be a strong emphasis on training through the short courses which will precede the proposed workshops, as well as through the coordination of the graduate course offerings in analysis and geometry in the Montreal universities.



Aisenstadt Chair Lecture Series


There will be two chairholders for the year : P. Sarnak (Princeton) and S. T. Yau (Harvard).



Short programme on analysis and resolution of singularities

August 18 - September 5, 2003


Organizers: E. Bierstone (Toronto), R. Melrose (MIT), P. Milman (Toronto), D.H. Phong (Columbia)


Effective methods in resolution of singularities are becoming central to a modern generation of problems from analysis and geometry — for example, spectral theory and Hodge theorem for algebraic varieties, stability of oscillating integrals, existence of Kähler-Einstein metrics, sharp forms of Moser-Trudinger inequalities. The diversity of the problems and their very different origins and aims have led to a lack of communication among researchers on these and related topics. This programme, bringing together leading experts in resolution of singularities, complex differential geometry, and real analysis and partial differential equations.


Week 1. Workshop on oscillatory integrals and critical integrability exponents

Topics include degeneracy of holomorphic functions in several variables, Legendre distributions and multiplier ideal sheaves.


Week 2. Short courses

Three short courses to be accessible to graduate students in analysis, given by the organizers or other participants.

  • Effective methods in resolution of singularities — ideas involved in desingularization algorithms, concrete examples with a view to applications in analysis and geometry.
  • Stability questions in real and complex analysis; for example, stable forms of the method of stationary phase, stability of critical integrability exponents, ascending chain conditions, stability problems for degenerate Fourier integral operators.
  • Real and complex blow up, resolution of metrics, configuration spaces and Lie algebras of vector fields — leading to a description of harmonic forms and L2 cohomology of various singular spaces


Week 3. Workshop on resolution of singularities, metrics and the Laplacian


The Hodge theorem, describing the harmonic forms on a smooth algebraic variety and relating them to its cohomology, has had wide impact on differential and algebraic geometry, and differential analysis. In the more general case of a singular projective variety, a description of the harmonic forms remains largely open, although there are substantial conjectures. An approach through resolution of singularities depends on understanding the structure of the Fubini-Study metric lifted to a resolution. The workshop will bring together researchers in geometric, algebraic and analytic areas related to these questions.



Workshop on the Cauchy problem for the Einstein equations

September 24 - 28, 2003


Organizers: F. Finster (Regensburg), N. Kamran (McGill)


A number of major advances have been achieved over the past few years in the analysis of the Cauchy problem in general relativity. These include the proof of the non-linear stability of Minkowski space, the proof of the Riemannian Penrose conjecture and the rigorous description of the asymptotic behavior at infinity of the admissible Cauchy data. This workshop will bring together some of the key players who have been involved in these developments, and will provide an opportunity for exploring some of the remaining open problems.


The workshop will be preceded by two short courses given by A. Ashtekar (Penn State) and G. Huisken (MPI Golm). 


Workshop on the interaction of gravity with classical fields

October 1 - 5 , 2003


Organizers: F. Finster (Regensburg), N. Kamran (McGill)


The interaction of gravity with external fields is governed by highly coupled systems of partial differential equations on manifolds. The analysis of these systems leads to rigorous analytical results on fundamental questions such as the scattering of waves by black holes and the role of external fields in the dynamics of gravitational collapse and black hole formation.


The workshop will be preceded by two short courses given by J. Smoller (Michigan). It will be simultaneous with the first series of Aisenstadt lectures for the year, to be delivered by S.T. Yau.



Workshop on large N limits of U(N) gauge theory in physics and mathematics

January 5 - 9, 2004


Organizers: P. Bleher (IUPUI), V. Kazakov (Ecole Normale) and S. Zelditch (Johns Hopkins)


This workshop is devoted to the large N expansion in quantum Yang-Mills theory, particularly in the explicitly solvable 2D setting. During the 90's a series of articles by such physicists as D. J. Gross, W. Taylor, G. Matytsin, M. Douglas, V. Kazakov, and G. Moore produced a series of conjectured expansions for objects of 2D Yang-Mills with gauge group U(N), such as the partition function of a closed surface of genus g, the partition function of a cylinder, the expected value of the Wilson loop functional, as well as certain characters χR(U). These quantities are related to traces and other invariants of heat kernels, as well as to volumes and traces over moduli spaces of flat connections. The asymptotics of the partition functions are governed by statistics of branched covers of surfaces. Among the topics of the conference:

  • The Matytsin asymptotics for the characters χR(U), recently proven by A. Guionnet and O. Zeitouni
  • The Kazakov-Douglas phase transition in g = 0, recently proven by A. Boutet de Monvel and M. Shcherbina
  • Zelditch's limit formula for the partition function on the cylinder Statistics of branched covers (integrals over Hurwitz spaces)
  • Volumes and trace integrals over moduli spaces of flat bundles
  • The large N limit of objects of SN Relations between large N theory of YM2 and random matrix models
  • Relations with free probability
  • The new, very fast developing work of Dijkgraaf-Vafa.


Workshop on Spectral Geometry

March 4 - 6, 2004


Organizer: I. Polterovich (Montreal)


Relations between the geometric properties of manifolds and the spectrum of the Laplacian have been actively studied for decades. It is well known that many important geometric invariants are determined by the spectrum, and, vice-versa, the behavior of eigenvalues is strongly dependent on the underlying geometry and topology. Still, our understanding of the interplay between geometry and the spectrum is very far from being complete. In the recent years some major developments have occurred in various areas of spectral geometry, such as spectral asymptotics, eigenvalue estimates, isospectrality, and others. These problems and their applications will be in the focus of the workshop.


AARMS-CRM - Workshop on singular integrals and analysis on CR manifolds

May 3 - 8, 2004

Organizers: Galia Dafni (Concordia), Andrea Fraser (Dalhousie)

The theory of singular integral operators in the context of analysis on CR submanifolds of Cn, in particular the Heisenberg group, has been studied and proven fruitful over the last 30 years. In recent years, the emphasis has shifted to singular integral operators which do not fall under the standard Calderon-Zygmund theory. These include operators arising from product kernels on nilpotent Lie groups, which in turn lead to the study of flag kernels. The workshop combines the areas of harmonic analysis, several complex variables, symmetric spaces and Lie groups. It will include two series of lectures, to be delivered by Alexander Nagel (Wisconsin) and Elias M. Stein (Princeton). The workshop will be held in Halifax (Nova Scotia).


Workshop on spectral theory and automorphic forms

May 3 - 8, 2004


Organizers: D. Jakobson (McGill), Y. Petridis (CUNY)


In the last 40 years it has been understood that there is a close connection between the spectral theory of hyperbolic manifolds and the theory of L-functions attached to automorphic forms. Trace formulas of Selberg and Kuznetsov-Bruggeman are extremely useful in studying the spectrum and eigenfunctions of the hyperbolic Laplacian. Surprising connections have also been discovered between subconvexity estimates for L-functions and the equidistribution results for Eisenstein series and cusp forms.


Analytical questions about families of L-functions include questions about the distributions of zeros and GRH, value-distribution, special values and applications, as well as connections with arithmetical questions (such as distribution of primes, size of class groups, analytic ranks of elliptic curves). One of the most fruitful approaches to the study of statistical properties of zeros of L-functions involves establishing connections with random matrix theory.


The goal of this workshop is to bring together leading researchers in those fields, to introduce young researchers and graduate students to the state of the art results and to give an account of applications of techniques from analytic number theory to problems in analysis.


The workshop will coincide with the second series of Aisenstadt lectures for the year, to be given by Professor Peter Sarnak.


Workshop on Hamiltonian Dynamical systems
(jointly with the Fields Institute)

May 24 - 28, 2004

Organizing Committee: D. Bambusi (Milano), W. Craig (McMaster), S. Kuksin (Edinburgh), C.E. Wayne (Boston), E. Zehnder (ETH-Zentrum)

A conference on analytic techniques of dynamical systems, including perturbation theory, variational methods and stability theory. The workshop will cover both finite dimensional Hamiltonian systems such as in celestial mechanics, and infinite dimensional Hamiltonian systems, such as those arising from PDE or from other dynamical systems with infinitely many degrees of freedom. Part of the Fields Institute thematic programme, it follows a workshop on Integrable and Near-integrable Hamiltonian PDE, held the previous week in Toronto.



Workshop on semi-classical theory of eigenfunctions and PDEs

June 1 - 11, 2004


Organizers: D. Jakobson (McGill), J. Toth (McGill)


Many questions in quantum chaos are motivated by the correspondence principle in quantum mechanics. It asserts that certain features of the classical system manifest themselves in the semiclassical (as Planck's constant h → 0) limit of a quantization of the classical system. The exact relationship between classical dynamics and asymptotic properties of high energy eigenstates of a quantized system is still not completely understood, despite exciting developments in the last 20 years. Important issues related to the correspondence principle include asymptotic L (Lp) bounds for the eigenfunctions, integrated (and pointwise) Weyl errors and scarring. Another fundamental question concerns the local and global statistical properties of eigenfunctions (eg. the random wave model), their nodal sets and critical points. These problems draw extensively on the theory of partial differential equations and so we propose to bring together experts in these areas.


The workshop will include several short courses. Harold Donnelly (Purdue) (*), Nikolai Nadirashvili (Chicago) and David Jerison (M.I.T.) (*) have been invited.



Workshop on spectral theory of Schrödinger operators

July 26-30, 2004


Organizers: V. Jaksic (McGill), Y. Last (Hebrew)


This workshop will focus on the spectral theory of random and quasiperiodic Schrödinger operators. In solid state physics random and almost periodic Schrödinger operators serve as models of disordered systems, such as alloys, glasses and amorphous materials. The disorder of the system is reflected by the dependence of the potential on some random parameters.


From a mathematical point of view, random Schrödinger operators show quite

" unusual " spectral behavior. If the disorder is large enough then these operators have dense point spectrum with exponentially decaying eigenfunctions (Anderson localization). The appearance of dense point spectra is a reflection of the physical fact that the strongly disordered systems are bad conductors. It is believed that in the weak disorder regime and for dimensions larger then 2 these operators have some absolutely continuous spectrum which corresponds to non-zero conductivity of the weakly disordered systems. The mathematical proof of this expected spectral phase transition (Anderson delocalization) is a fundamental open problem in mathematical physics.


This workshop will bring together the world leaders in spectral theory of random and quasiperiodic Schrödinger operators. Its goal is to review the state of the art of the field and to map new directions of the research.


The programme includes short courses to be given by M. Aizenman (Princeton), B. Simon (Caltech) (*), and S. Jitomirskaya (Irvine). The workshop is being held in conjunction with the following one, and many participants will be attending both.



Workshop on dynamics in statistical mechanics

August 2-6, 2004


Organizers: V. Jaksic (McGill), C.-A. Pillet (Toulon)


During the last years, significant efforts have been devoted to the study of dynamical properties of (classical and quantum) open systems. In particular, through the study of noisy or forced dissipative systems, or Hamiltonian systems with a large number of degrees of freedom, our understanding of the mathematical structure of nonequilibrium statistical mechanics has greatly improved. The aim of this meeting is to present the latest results and discuss the possible future directions of research in this area. The following topics will be discussed:


Axiomatic approaches: Under appropriate hypotheses on the ergodic properties of the underlying dynamical system (chaotic hypothesis, asymptotic abelianness, etc), it is possible to prove various predictions of nonequilibrium thermodynamics (linear response, Kubo formula, Onsager's relations, etc.). This approach also lead to unexpected results, like the Gallavotti-Cohen fluctuation theorem.


Specific models: Modern techniques (quantum field theory, algebraic quantum dynamical systems, spectral analysis, renormalization group, etc.) have been successfully applied to the study of various models (spin-boson, spin-fermion, Pauli-Fierz, Lorentz-gas, etc.). Elementary physical properties like return to equilibrium or existence and structural properties of nonequilibrium steady states, have been obtained in this way. More difficult questions, like the emergence of the Fourier law, are currently under investigation.


Markovian Dynamics: It gives a natural mathematical framework to study the dynamics of various nonequilibrium processes — Hamiltonian systems coupled to reservoirs, exclusion processes on the lattice, noisy extended systems.


The program includes short courses to be given by H. Araki (Kyoto), B. Derrida (École Normale), J. Froehlich (ETH), J.-P. Eckmann (Geneva) (*). The workshop is being held in conjunction with the preceding one, and many participants will be attending both.

Advanced Courses

Several advanced graduate courses are being given as part of the thematic programme, including:

Those wishing to participate in the above activities are invited to write to:

Louis Pelletier
Centre de recherches mathématiques (CRM)
Université de Montréal
C.P. 6128, Succ. Centre-ville
Montréal (Québec)

*Credits: «Trichaotic» image is reproduced by courtesy of Eric J. Heller. Prof. Heller's artistic renditions of physical phenomena will be featured in an exposition in Montreal during the thematic programme. Those interested in his work can also consult www.ericjhellergallery.com.

January 27, 2004