CMS NOTES de la SMC - Canadian Mathematical Society

CMS

NOTESde la SMC

Volume 33 No. 3 April / avril 2001

In this issue / Dans ce numero

Editorial . . . . . . . . . . . . . . . . . . . . . 2

CRM/Fields Prize Lecture . . . . 3

Book Review: Large Devia-tions . . . . . . . . . . . . . . . . . . . . . . 10

Some trends in Modern Mathe-matics and the Fields Medal–Part II . . . . . . . . . . . . . . . . . . . . . 11

Awards / Prix . . . . . . . . . . . . . . . . 13

Education Notes . . . . . . . . . . . . . 17

Millenium Prize Problems. . . . . 19

News from Departments . . . . . . 22

Du bureau du directeur admin-istratif . . . . . . . . . . . . . . . . . . . . 23

Call for Nominations / Appel deCandidatures . . . . . . . . . . . . . . 24

CMS Summer Meeting 2001Reunion d’ete de la SMC2001 . . . . . . . . . . . . . . . . . . . . . . 25

Schedule / horaire . . . . . . . . . . 26

Calendar of events / Calendrierdes evenements . . . . . . . . . . . . . 27

Rates and Deadlines / Tarifs etEcheances . . . . . . . . . . . . . . . . . 28

FROM THEEXECUTIVE

DIRECTOR’S DESK

Graham Wright

2000 Annual Report

In March 1998, under the directionof the then president Katherine Hein-rich, the Society began an extensive re-view of all of its activities and opera-tions. Richard Kane, CMS President1998 - 2000, and Jonathan Borwein,the current President, have overseenthis important strategic planning exer-cise. The final reports of seven TaskForces (Budget and Policy, Board Rep-resentation, CMS Endowment Fund,Publications, Finances and Fundrais-ing, Support of the Mathematics Com-munity and Office Strategies) and onead-hoc committee (Electronic Servicesand Camel) are available on Camel(www.cms.math.ca/Projects/).

The Executive Committee has beencharged to review all of the reportsand recommendations and to developan overall structure and strategy for theCMS for the next several years. During2000, each standing committee, as wellas members of the Society, were givenan opportunity to comment upon the re-ports and recommendations. The Ex-ecutive has already began the process ofreviewing all the submissions and willpresent its conclusions to the Board ofDirectors in June 2001. A great deal ofeffort has gone into the present strate-gic planning approach and although thefinal stages of the work are not yet com-pleted, numerous changes have alreadybeen implemented. When the processis completed, the CMS will have takenthe necessary steps to ensure our ac-tivities and services are delivered inthe most effective, efficient and cost-effective manner.

The 2000 Annual Reports fromeach standing committee chair indicatethe extensive range of research, publi-cations and educational activities sup-ported by the Society. All of these ac-tivities are possible because of the in-valuable assistance received from bothmembers and others.

Over the past few years the numberof sessions at our semi-annual meetingshave increased markedly, as has thenumber of delegates attending thesemeetings. Although the Executive

(see EXEC–page 20)

APRIL/AVRIL CMS NOTES

CMS NOTESNOTES DE LA SMC

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EDITORIAL

S. Swaminathan

One of my students approached me re-cently for advice on whether he shouldpursue graduate studies in mathemat-ics. Here is a part of our conversation:

Student: What are the prospects fora job after I graduate with a degree inmathematics?

S. S.: The prospects are expected tobe very good when you graduate with aPh. D. five or six years hence. The rateof retirement is increasing in many uni-versities opening up positions for newgraduates.

S: Which is better - mathematics orcomputer science?

S. S.: Such comparisons are diffi-cult. It depends on your aptitude andattitude. I note that you have a betteraptitude for mathematics although youare doing very well in both subjects.Moreover more students are opting forcomputer science. Hence competitionwith them for jobs will be tougher.

S: I see that computer science isvery useful to our technologically ori-ented society. We cannot say the sameabout mathematics.

S. S.: In general the usefulness ofmathematics to society is not well un-derstood. Actually mathematics has asignificant part to play in almost ev-ery aspect of society. Let me giveyou just one example. Earlier thismonth (February), almost exactly twohundred years after the first asteroidwas discovered, a spacecraft bearingthe acronym NEAR (Near Earth Aster-oid Rendezvous), launched from CapeCanaveral, Florida, in 1996, made a

smooth crash-landing on the asteroidEros which is hurtling in orbit at atremendous speed some 196 millionmiles away. This made-on-Earth con-traption jumped on the rugged rockat commands from Earth by scien-tists who based their calculations us-ing advanced mathematical techniques.Further, since Andrew Wiles’ proofof Fermat’s Last Theorem, there isgreater awareness among the publicabout mathematics. A successful playcalled Proof was staged in New Yorkin Fall 2000, in which three of thefour characters were mathematicians.It was followed by a symposium in NewYork University in which there werethree panel discussions by mathemati-cians and nonmathematicians on someof the themes and ideas raised in theplay. More can be said about the use-fulness of mathematics but I supposethis much will suffice.

S.: I hear that computer scientistscan make a lot of money. Can mathe-maticians become rich?

S. S.: It is true that computer scien-tists are paid more both in universitiesand in industry. As a mathematicianyou can become a millionaire if yousolve one or more of the Clay problems.See page 18 for these problems.

*****

Il y a quelque temps, un étudiant estvenu me demander s’il devait pour-suivre des études supérieures en math-ématiques. Voici une partie de notreconversation :

Étudiant : Quelles seraient mesperspectives d’emploi à la fin de mesétudes?

S. S. : On s’attend à ce qu’il yait beaucoup de travail pour les titu-laires d’un doctorat qui arriveront surle marché du travail dans cinq ou sixans. De plus en plus de professeursd’université prennent leur retraite, et ilfaut les remplacer.

Étudiant : Qu’est-ce qui est mieux :les mathématiques ou l’informatique?

(voir EDITORIAL–page 21)

2

NOTES de la SMC APRIL/AVRIL

CRM / Fields Prize Lecture

The Renormalization Group Approach in Spectral Analysis

and the Problem of RadiationI.M. Sigal, University of Toronto

This article includes the slides of the talk I gave at theCRM on November 10, 2000. I gave a related talk at theFields Institute on November 4, 2000. I completed sentencesindicated on the slides, added a few explanations of the no-tation and concepts presented which I gave orally during thetalks and inserted brief literature comments and a list of ref-erences. Apart from this I changed nothing. As a result thepaper retains the informal style of the talk.

SPECTRAL ANALYSISI want to address the problem of perturbation of spectra of

operators. For example, consider the problem of perturbationof a single eigenvalue. There are two possible cases:

# 1 Isolated eigenvalues

x x

EVs Cont Spec

# 2 Embedded eigenvalues

Cont SpecEVs

x xx

In physical applications the second situation is generic,while the first one arises as a crude idealization when oneconsiders a small part of a system in question.

Let us consider several examples of the second case.

HOPF BIFURCATION FROM SOLITONSConsider the nonlinear Schrodinger equation

i∂ψ

∂t= −∆ψ + g

(|ψ|2)ψ,where ψ: R

n × R → C. This equation has soliton solutions

ψsol(x, t) = eiΦ(x,t) f(x− vt)

where Φ(x, t) is some real phase depending on the veloc-ity v. The spectrum of fluctuations around ψsol, i.e. of thelinearization, Lψsol , of the r.h.s. around ψsol, is

0

rotational modestranslation and

Re z

x

Im z

x

x

oscillatory modes

Do oscillatory modes lead to the bifurcation of time-periodicsolutions?

0T T

ϕ

Following the Hopf bifurcation analysis we have to con-sider the Floquet operator

−T−1 ∂

∂t+ Lψsol on L2(Rn × S1) ,

where S1 is the unit circle and T is an unknown period of thebifurcating periodic solution we are looking for. The spec-trum of this operator is

spec(Lψsol) + iT−1

Z, (∗)

where spec(Lψsol) is shown on the figure preceding the oneabove. Spectrum (∗) consists of a continuum filling in theentire imaginary axis and translation/rotation and oscillatoryeigenvalues repeated periodically and embedded into this con-tinuum.

Thus the answer to the question of what kind of solutionbifurcates from oscillatory modes depends on an understand-ing of what happens to embedded oscillatory modes under a

3


nonlinear perturbation.

VORTEX SPECTRUMConsider the Ginzburg-Landau equation

∆ϕ+ (1 − |ϕ|2)ϕ = 0

ϕ : R3 → C

with the boundary condition that |ϕ| → 1 as |x⊥| → ∞,where x⊥ = (x1, x2) for x = (x1, x2, x3). Solutions of thisequation can be specified by smooth curves of zeros of ϕ anda topological degree of ϕ with respect to these curves.

Nullϕdeg ϕ

This equation has special–equivariant–solutions calledvortices

ϕn = fn(r)einθ .

where (r, θ) are cylindrical coordinates. The spectrum of thelinearized equation (i.e. of vortex fluctuations) is

Cont Spec

x

EV (=0) mult 3

x

(The negative eigenvalues are present for |n| > 1 and absentfor |n| = 1.)

A detailed analysis of perturbation of the zero embeddedeigenvalue is a key to understanding the dynamics of many(interacting) vortices.

QUANTUM SPECTRUM OF GEODESICSConsider a space of curves given by their parameteriza-

tions, ϕ. Let V (ϕ) be an energy of a curve ϕ. A quantizationof V (ϕ) yields the Schrodinger operator

−∆ϕ + V (ϕ) on L2(S′, dµC) , (∗)

where dµC is a Gaussian measure on the Schwartz spaceS′ = S(Rn) and the meaning of the “Laplacian”, ∆ϕ, actingon functionals of the fieldϕ ∈ S′(Rn) will be alluded at later.

Now, let ϕCP be a critical point of V (ϕ). The questionwe want to ask is: What are the quantum corrections to theenergy of ϕCP ?

Answering this question involves understanding the lowenergy spectrum of (∗) near the classical energy V (ϕCP )which in turn leads to a perturbation of embedded eigenval-ues and the nearby spectrum.

In a special situation ϕCP could be a geodesic or, moregenerally, a minimal submanifold.

An important example of the situation above is that ofquantum vortices. In this case ϕ: R

3 → C and V (ϕ) is ofthe form

V (ϕ) =∫

12|∇ϕ|2 + F (ϕ, x) (∗∗)

x)(ϕ,F

ϕ

double-well potential

The (line) vortices arise as critical points of V (ϕ), ϕ :R

3 → C, satisfying certain topological conditions (seeabove). The latter conditions imply that the null sets of thesecritical points are curves which are geodesics in a certain Rie-mannian metric (see a figure above).

One can think of the dynamics of vortices as motion oftheir centers – Null ϕ – with relatively rigid vortex riggingaround them.

Another interesting case is that of functional (∗∗) withx-independent F ≥ 0 and for ϕ : [0, 1] → R

m. In thiscase, critical points of V (ϕ) are (modulo parametrization)geodescics in the Riemannian metric ds2 = F (y) dy2 (Ja-cobi metric). The latter fact is related to Maupertuis principlein Classical Mechanics.

PROBLEM OF RADIATIONI want to present an example of a common physical situ-

ation when a small system (with finite number of degrees offreedom) is coupled to a large system (of infinite number ofdegrees of freedom) – the problem of radiation. This problemis reduced to finding the low energy spectrum of the quantumHamiltonian for the system of matter and radiation

H(e) =∑ 1

2mjp2j,eA + V (x) +Hrad

on Hmatter ⊗ Hrad (Schrodinger equation coupled to quan-tized Maxwell equations).

4


SPEC H(0)

The spectrum of the unperturbed (=uncoupled) HamiltonianH(0) contains eigenvalues sitting on the top of the thresholdsof continuous spectrum. They correspond to bound states ofan atom in a vacuum. Are these bound states stable or unstable(when e = 0)?

REFINEMENT OF NOTION OF SPECTRUMStandard notions of spectral analysis are insufficient for

treating perturbation of embedded eigenvalues. We extendthe notion of spectrum as follows. Consider a self-adjointoperatorH on a Hilbert space H. Then point and continuousspectra are poles and cuts of 〈f, (z −H)−1g〉 ∀ f, g ∈ H

branch points

cutpoles

Consider the Riemann surface of 〈f, (z −H)−1g〉 for fand g in some dense set D ⊂ H. In other words we want tocontinue this analytic function from, say, C

+ across the cut(continuous spectrum ofH) into the second Riemann sheet:

We see that non-threshold eigenvalues of H become iso-lated poles of this analytic continuation while new complexpoles, not seen before, are revealed. Clearly, real poles com-ing from embedded eigenvalues and complex poles must betreated on the same footing.

DEFORMATION OF SPECTRANow I outline a constructive tool used in the study of the

Riemann surface for a given operator H–the spectral defor-mation method. It goes as follows. Consider the orbit

H → H(θ) = U(θ)HU(θ)−1

ofH under a one-parameter group,U(θ), of unitary operators,s.t. H(θ) has an analytic continuation in θ into a neighbour-

hood of θ = 0. The spectrum of such a continuation lookstypically as on the figure below.

The resolvent(H(θ)−z)−1

provides the desired informa-tion about the Riemannian surface of the operatorH . In par-ticular, the real eigenvalues ofH(θ) coincide with the eigen-values of H , i.e. with the real poles mentioned above, whilethe complex eigenvalues of H(θ) are related to the complexpoles on the second Riemann sheet. These complex eigen-values are called the resonances of H .

Thus the problem of understanding the behaviour of em-bedded eigenvalues and the continuous spectrum ofH undera perturbation is reduced to the problem of understanding thecomplex spectrum of the operator H(θ) for complex θ’s.

MATHEMATICAL PROBLEM OF RADIATION

The goal here is to construct a mathematical theory ofemission and absorption of electro-magnetic radiation by sys-tems of non-relativistic matter s.a. atoms and molecules:

ground state

photon

excited state

Mathematically, this translates into the problem of under-standing the bound state–resonance structure of the quantumHamiltonian of a system of quantum matter coupled to quan-tum radiation.

QUANTIZED MAXWELL EQUATIONS

I review quickly a mathematical framework of quantumtheory of radiation. First, I describe the quantized Maxwellequations and then their coupling to quantum matter.

The quantized Maxwell equations can be presented as the

Schrodinger equation∂φ

∂t= Hradφwith the quantum Hamil-

5


tonian operator

Hrad :=12

∫: Eop(x)2 +

(curlAop(x)

)2 : d3x

acting on Hrad = L2(S′, dµC). Here S′ is the Schwartzspace of the transverse vector fields, A(x), divA(x) = 0, onR

3, dµC is the Gaussian measure on S′ with the mean 0 andcovariance C = (−∆)−

12 , Aop(x) and Eop(x) are the quan-

tum operators of the vector potential and electric field in theCouloub gauge,

Aop(x) = operator of multiplication by A(x)

Eop(x) = −i δ

δ A(x)+ iC− 1

2A(x) ,

and the double colons signify the Wick, or normal, ordering,i.e. some sort of deformation of the quantization procedure.

Now we explain briefly an origin of this Hamiltonian.The Maxwell equations in a vacuum is an infinitely dimen-

sional Hamiltonian system with the Hamiltonian functional

H(A,E) =12

∫E2 + (curlA)2

defined on the phase-spaceHtransv1 ×Ltransv

2 equipped with astandard symplectic structure. HereA(x) is the vector poten-tial in the Coulomb gauge and E(x) is the electric field (thefield conjugate to A(x)), which are transverse vector fieldson R

3 (i.e. divA(x) = 0 and divE(x) = 0), and Htransv1

and Ltransv2 are the Sobolev space of order 1 and L2-space of

transverse vector fields.A naive quantization of this dynamical system patterned

on the quantization of the Newton equations goes as follows.Concept CFT QFTPhase/state space Htransv

1 ⊗ Ltransv2 “L2(Htransv

1 , DA)”Symplectic structure Poisson brackets commutatorsCanonic. variables A(x) Aop(x) = operator of(w.r. to the multiplication byA(x)symplectic structure) E(x) Eop(x) = −i δ

δA(x)Observables Real functionals Self-adjoint operators

f(A,E) on A = f(Aop, Eop) onHtransv

1 ⊗ Ltransv2 “L2(Htransv

1 , DA)”Dynamics Ham.fnctnlH(A,E) Ham.oprH(A,E)

The third column does not make sense mathematically,but its natural modification leads to the formulation presentedat the beginning of this section.

MATTERQuantum non-relativistic matter is described by a

Schrodinger operator of the form (in units: = 1, c = 1,and me = 1)

Hmatter =N∑1

12mj

p2j + V (x)

on Hmatter(e.g. L2(R3N )). Here pj = −i∇xj , mj > 0 ∀jand x = (x1, . . . , xN ).

SPECTRUM OF HmatterTypically, the spectrum of Hmatter is of the form

(Hunziker-van Winter-Zhislin Theorem)

MATTER + RADIATIONTo introduce the coupling between matter and radiation,

we think about the vector potential A(x) as a quantum con-nection on R

3 and pass to the covariant derivatives

p → pA = p− eA(x) .This leads to the Hamiltonian for the system of matter andradiation

H(e) =∑ 1

2mjp2j,A + V (x) +Hrad

on Hmatter ⊗ Hrad. This operator is not well defined. Toremedy this we introduce

Ultraviolet cut-off: A(x) → χ∗A(x),∫

|χ|2d3k < ∞in the interaction terms p2j,A. The resulting operator (whichwe still denote by the same symbol H(e)) is self-adjoint andis bounded from below.

MATHEMATICAL PROBLEMProblem of Radiation: Fate of the bounded states of mat-

ter

(ϕj ⊗ Ω

)e−iEjt , ∀j

(∗)bound state of matter vacuum of

with energy Ej quantized EM field

as charges are “turned on”.

6


E.g. one would like to show that an atom in an excited statein a vacuum is unstable, that it emits a photon and descendsinto the stable ground state.

RENORMALIZATION GROUP APPROACH

Assume we want to study a part of the spectrum of theoperator H(e) near E0. We proceed as follows:

• Pass to a metric space M of operators

• Construct a flowΦτ onM s.t. - Φτ eliminates “inessential”degrees of freedom - Φτ is isospectral in |z − E0| ≤ e−τ

• Find fixed points of Φτ and their stability.

X

M

X

Observe now that

- Isospectrality of Φτ allows us to transfer the spectral in-formation we have about fixed points to the initial operatorH(e);

- Classify possible behaviour of physical systems in questionaccording to the fixed points to which they are attracted.

Since φτ eliminates “inessential” degrees of freedom (akind of partial dissipation) we expect that the HamiltoniansHτ := φτ (H) at levels τ simplify as τ → ∞ and in particularthat the fixed points are especially simple.

DECIMATION MAP

In order to define the RG-flow we first define a map, calledthe decimation map, which eliminates inessential degrees offreedom. First, we observe that the map

H → PτHPτ ,

wherePτ is the spectral projector associated with the inequal-ity |H(0) − E0| ≤ e−τ , eliminates the part of H acting on(RanPτ )⊥ but it distorts Spec H . We modify this map inorder to restore the spectral fidelity in a small neighbourhoodof z = 0:

Dτ : H → Pτ (H −HR⊥H)Pτ ,

where R⊥ = P⊥τ (P⊥

τ HP⊥τ )−1P⊥

τ and P⊥τ = 1 − Pτ . This

new map is called the decimation map.

Trade-off: Dτ is isospectral at z = 0, but is non-linear.

RESCALINGNext, we introduce the rescaling map Sτ acting on oper-

ators by a unitary conjugation, Sτ : H → UτHU−1τ , which

rescales the photon momenta as k → e−τk. As a result wehave

Sτ : |H(0) − E0| ≤ e−τ → |H(0) − E0| ≤ 1 .

RG - FLOWNow we are ready to define the RG-flow:

Φτ = Eτ Sτ Dτ ,where Eτ (A) = eτA, a normalization map.

Observe that Φτ has the following properties

- Φτ is a semi-flow

- Φτ projects out |H(0)−E0| ≥ e−τ and magnifies the result

- Φτ is isospectral in z ∈ C||z| ≤ e−τ modulo the factoreτ .

RG FLOW DIAGRAM

The figure below shows fixed point, stable, and unstablemanifolds of Φτ . The point here is that the fixed point mani-fold is very simple: Mfp = C ·Hrad, while the stable man-ifold, Ms, has the codimension 2. Hence given an operatorH there is a complex number E = E(H), s.t. H −E ∈ Ms.

Now proceed as follows. Apply the RG-flow toH−E. ThenΦτ (H −E) converges to the fixed point manifold so that forτ sufficiently large

Φτ (H − E) ≈ wHrad for some w ∈ C

and as a resultH is isospectral to wHrad −E in |z −E| ≤e−τ. This way we transfer the spectral information aboutHrad, which is available to us, to the operator H .

MATHEMATICAL RESULTS (Bach-Frohlich-IMS)Assume that |e| is sufficiently small and, in the third state-

ment below, that the particle potential, V (x), is confining, i.e.V (x) → ∞ as |x| → ∞. Then we have

7


I. Binding. H(e) has a ground state ψ. ‖eα|x|ψ‖ < ∞α > 0.

II. Instability. H(e) has no EVs near the excited EVs ofHmatter.

III. Resonances. The excited states ofHmatter bifurcate intoresonances of H(e).

X

X

1/Life-timeshiftLamb

X

Projection of Riemann surface of H(e) onto C

(EV stands for “eigenvalue”.)

REMARKSThe result on the vortex spectrum mentioned is equivalent

to the property of (linearized) stability/instability of vortices(see [G, GS, LL, M, OS1]). Recent results on dynamics ofvortices are reviewed in [OS2].

The spectrum of critical points of the potential in QuantumMechanics was found [Sim, BCD, Sj].

One can also try to understand spectra of critical points ofthe action functional

S(φ) =∫ (∫

12|φ|2 dx− V (φ)

)dt

which are periodic in time.The method of spectral deformation and the theory of

resonances based on it were proposed by Aguilar, Balslev,Combes and Simon and extended in works of Balslev, Hun-ziker, Jensen, Sigal, Simon and others. See [HisSig, HunSig]for recent reviews. A different approach was proposed byHelffer and Sjostrand (see [HeSj, HeM]).

For a text on rigorous quantum field theory see [GJ] andfor a physical discussion of the problem of radiation, [C-TD-RG].

A recent review of the quantum theory of many particlesystems can be found in [HunSig].

The renormalization group approach and the results on theradiation problem presented here were obtained in [BFS1-3],where the reader can also find many references to the earlier orsimultaneous work. Some improvements of these results aregiven in [BFS4]. The result on the ground state was improvedin [G, GLL].

REFERENCES

[BCD] P. Briet, J.M. Combes and P. Duclos, On the loca-tion of resonances for Schrodinger operators, II. Barrier topresonances, Comm. Part. Diff. Eqns, 12 (1987), 201-222.

[BFS1] V. Bach, J. Frohlich and I.M. Sigal, Quantum electro-dynamics of confined non-relativistic particles, Adv. Math.,137 (1998), 205-298.

[BFS2] V. Bach, J. Frohlich and I.M. Sigal, The Renormal-ization group approach to spectral problems in QFT, Adv. inMath. 137 (1998), 299-395.

[BFS3] V. Bach, J. Frohlich and I.M. Sigal, Mathematicaltheory of non-relativistic matter and radiation, Comm. Math.Phys. 207 (1999), 249-290.

[BFS4] V. Bach, J. Frohlich and I.M. Sigal, The renormaliza-tion group and theory of radiation (in preparation).

[C-TD-RG] C. Cohen-Tannoudji, J. Dupont-Roc, and G.Grynberg, Photons and Electrons. Introduction to QuantumElectrodynamics, John Wiley & Sons 1989.

[G] S. Gustafson, Symmetric solutions of the Ginzburg-Landau in all dimensions, IMRN 16 (1997), 801-816.

[Ger] Ch. Gerard, On the existence of ground states for mass-less Pauli-Fierz Hamiltonians, Preprint 1999.

[GJ] J. Glimm and A. Jaffe, Quantum Physics, 2nd edition,Springer 1987.

[GLL] M. Griesemer, E.M. Lieb and M. Loss, Ground statesin non-relativisitic quantum electrodynamics, Preprint 2000.

[GS] S. Gustafson and I.M. Sigal, The stability of magneticvortices, Comm. Math. Phys. 210 (2000), 257-276.

[HeM] B. Helffer and A. Martinez, Comparaison entre lesdeverses notion de resonances, Helv. Phys. Acta 60 (1987),992-1003.

[HeSj] B. Helffer and J. Sjostrand, Resonances en limite semi-classique, Bull.S.M.F., Memoire, Vol. 24/25 (1986).

[HisSig] P. Hislop and I.M. Sigal, Lectures on Spectral Theoryof Schrodinger Operators, Springer-Verlag series of mono-graphs on Applied Mathematics, 1996.

[HunSig] W. Hunziker and I.M. Sigal, Quantum many-bodyproblem, J. Math. Phys., 41 (2000), 3448-3510.

[LL] E.M. Lieb and M. Loss, Symmetry of the Ginzburg-Landau minimizers in a disc, Math. Res. Lett. 1 (1994),701-715.

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[M] P. Mironescu, On the stability of radial solutions of theGinzburg-Landau equation, J. Funct. Anal. 130 (1995), 334-344.

[OS1] Yu. N. Ovchinnikov and I.M. Sigal, The Landau-Ginzburg equation I. Static vortices. Part. Diff. Eqn andtheir Appl. CRM Proceedings 12 (1997), 199-220.

[OS2] Yu. N. Ovchinnikov and I.M. Sigal, Dynamics of lo-

calized structures. Physica A 261 (1998), 143-158.

[Sim] B. Simon, Semiclassical analysis of low lying eigen-values, Ann. Inst. H. Poincare 38 (1983), 296-307.

[Sj] J. Sjostrand, Semiclassical resonances generated by non-degenerate critical points, Lect. Notes in Math., Springer1987.

9


How Rare is Rare?Book Review by C.C.A. Sastri, Dalhousie University

Large Deviationsby Frank den Hollander

Fields Institute Monograph, 14American Mathematical Society,

Providence, 2000x + 143 pages

Large Deviations is a branch of proba-bility theory that deals with rare events.It is a very active field with applica-tions to several areas including statis-tics, communication networks, com-puting, statistical physics, and mathe-matical finance. (For an example ofa rare event, consider the problem ofspecies sampling: suppose that onewishes to determine the species of fishnative to a body of water and that, af-ter repeated sampling, one identifies acertain number of species. Supposethat n trials have been conducted. Theproblem is to estimate the probability,for large n, that one would observe anew species on the next trial.) Roughlyspeaking, one may know, in a given sit-uation, that the probabilities of certainevents tend to zero as some parame-ter goes to ∞ (or zero). The questionarises as to how fast the probabilitiesconverge. Large deviation techniquesenable one to find the rate of conver-gence. Central to the subject is the large

deviation principle (LDP), enunciatedby S.R.S. Varadhan in 1966. A naturalcontext for stating the LDP is that ofBorel probability measures on a Pol-ish (i.e., a complete, separable metric)space. The reason is that the space ofBorel probability measures on a Polishspace can be metrized in such a waythat it itself becomes a Polish space.The topology induced by the metric isthe so-called topology of weak conver-gence of measures.

A sequence Pn of Borel prob-ability measures on a Polish space Sis said to satisfy the LDP with a ratefunction I if there exists a functionI : S −→ [0,∞] such that

(i) I has compact level sets, i.e., the sets | I(s) ≤ 1 is compact in S, forevery 1

(ii) for every closed set F ⊂ S,

lim supn→∞

1nlogPn(F ) ≤ − inf

s∈FI(s),

and

(iii) for every open set G ⊂ S,

lim infn→∞

1nlogPn(G) ≥ − inf

s∈GI(s).

Prior to Varadhan’s work, large de-viation results, including such impor-tant theorems as those due to Cramerand Sanov, lay scattered in the litera-ture; the LDP brought them all underone rubric. Thus a typical theorem inlarge deviations states that, in a certainsetting, the LDP holds.

An important application of largedeviation techniques is to the asymp-totic evaluation of certain expectationsor function space integrals. This isbased on a variational formula, also dueto Varadhan, which plays a key role inlarge deviation analysis. In fact, a con-verse of Varadhan’s result, due to W.Bryc (1990), can be used as a startingpoint for the application of weak con-vergence ideas to large deviation the-ory. This is the approach followed byP. Dupuis and R.S. Ellis in their book

A Weak Convergence Approach to theTheory of Large Deviations (1997).

There exist several books onlarge deviations, written from differentpoints of view and at different levels ofdepth. Nevertheless, the book by denHollander is a welcome addition. Itis ideally suited for nonspecialists in-terested in learning the subject as wellas for graduate students who have hadsome exposure to measure-theoreticprobability. One advantage it has overthe other books is its brevity. The basicideas and principles are presented in thefirst part of the book, which takes uponly sixty pages. The topics coveredinclude the theorems of Cramer andSanov, which deal with independentidentically-distributed sequences, andthe Gartner-Ellis theorem, which dealswith weakly dependent sequences. Thesecond part of the book, which is onlyslightly longer than the first, containsapplications to a variety of areas such ashypothesis testing and interacting dif-fusions. There are exercises through-out the book. The appendix, containingsolutions to the exercises, is an attrac-tive feature, making the book suitablefor self-study. The constraint of brevityforced den Hollander to leave out im-portant topics such as Schilder’s theo-rem, but still the resulting book pro-vides a quick, relatively painless in-troduction to the subject. Indeed, thebook is user-friendly, as den Hollandersays in his preface. One can read it inconjunction with the books by Demboand Zeitouni (Large Deviations Tech-niques and Applications (1998)), Ellis(Entropy, Large Deviations, and Sta-tistical Mechanics (1985)) or Shwartzand Weiss (Large Deviations for Per-formance Analysis (1995)). One canthen go on, if one wishes, to the bookby Deuschel and Stroock (Large Devi-ations (1989)) or perhaps to the papersof Donsker and Varadhan.

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Some Trends in Modern Mathematics and the Fields Medalby Michael Monastyrsky

This is the second and concluding part of this article. Thefirst part appeared in last month’s issue.

If we very quickly review the results of the Fields medal-lists, keeping in mind the fundamental principles of J. Fields,we can observe several interesting developments:

1. Allocation of stable fields of interest.

2. Succession of mathematics.

3. Zigzags of mathematical fashion.

I will try to illustrate these theses with excerpts from theFields medallists’ results.

1. Allocation of InterestIndisputably, if we divide the mathematics of the second halfof the century into two parts, the first thirty years is mainlyconcentrated around problems of algebraic topology, alge-braic geometry, and complex analysis. Here new conceptsand methods appeared, and this evidently is reflected in thelist of Fields medallists. A definite change in this tendency,a return to the more classical topics, but of course on a newlevel, can be observed in mathematics from the end of the70’s. With some delay, this has been reflected in the Fieldsmedals awarded at the last two congresses.

It is important to note the new convergence between math-ematics and physics. The traditional contacts between math-ematics and physics are well known. If we consider the par-allel development of mathematics and fundamental physics,we are astonished that the most revolutionary theories in 20thcentury physics are based on mathematics, which was espe-cially developed for this purpose. It is enough to mentionEinstein’s special and general relativity based on the classicaldifferential geometry of Riemann spaces, quantum mechan-ics and Hilbert spaces and the theory of linear operators, theSchrodinger equation and spectral theory, and so on. Thisconnection was broken, somewhere in the 30’s, at the timeof the solution of several more concrete problems in physics,when it seemed to physicists that most of their problems couldbe solved without the application of sophisticated and abstractmodern mathematics. The development of pure mathematicsin the period between the two world wars, and especially inthe post-World War II period, was also characterized by weakconnections with applied science, in particular with physics.This association was especially true of the areas of mathemat-ics in which many Fields medalists worked. It was difficultto imagine that the concepts of sheaf, etale cohomology, J-functor, and the like would ever be applied in physics. Itwas still more difficult to imagine that physics could assistalgebraic topology and geometry.

This point of view was widespread. The French mathe-matician Jean Dieudonne, one of the founders of Bourbaki,expressed himself unambiguously on this subject in 1962. “Iwould like to stress how little recent history has been willing toconform to the pious platitudes of the prophets of doom whoregularly warn us of the dire consequences that mathemat-ics is bound to incur by cutting itself off from applicationsto other sciences. I do not intend to say that close contactwith other fields, such as theoretical physics, is not beneficialto all parties concerned; but it is perfectly clear that of allthe striking progress I have been talking about, not a singleone, with the possible exception of distribution theory, hadanything to do with physical applications.” (Quoted from anaddress delivered at the University of Wisconsin in 1962, inwhich Dieudonne gave a survey of the achievements of thepreceding decade in pure mathematics. He emphasized al-gebraic topology, algebraic geometry, complex analysis, andalgebraic number theory.) But as often happens with globallyexpressed opinions, the situation underwent a vast change tenyears later.

At the begining of the 70’s, both in mathematics andphysics, results were obtained that absolutely changed thispoint of view. Among the mathematicians who quickly un-derstood the new opportunities and challenges hidden in thenew physics were some Fields laureates. It is enough to men-tion S. Novikov, S. T. Yau, A. Connes, S. Donaldson, and E.Witten. Witten was the first physicist to be awarded a Fieldsmedal. Among the results of Fields laureates which were in-spired by physical ideas, let us mention first of all the work ofSimon Donaldson. After the work of Milnor on differentialstructures on S7, the paper of Donaldson appearing in 1983had a similar striking impact. Donaldson proved the exis-tence of different differential structures on simply-connected4-dimensional manifolds. (Unfortunately the case of S4 isnot covered by his method and is still open.) Immediatelyafter the work of Donaldson, in papers of R. Gompf and C.Taubes, the following remarkable result was proved: Thereexist an infinite number of different differential structures onR4. This result, that the “well-known” space R4 hides suchdeep structures, is absolutely astonishing. It has very deepconsequences for quantum gravity, where integrating over allmetrics and so over different differential structures is neces-sary. It is not less important than the proof, which is basedon earlier discoveries in field theory, mostly in the gauge the-ory of strong and weak interactions. Such interactions in theworld of elementary particles are described by highly nonlin-ear equations with deep topological properties—the so calledYang-Mills equations. These equations were invented by thephysicists C. N. Yang and R. Mills in 1954, but for manyyears were considered as nonphysical and attracted very little

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attention from physicists. Only the newest development ofthe theory of elementary particles—the creation of the the-ory of weak and strong interactions based on the Yang-Millsequations—led physicists to a deeper study of the structureof these equations.

In the early 70’s, the physical-mathematical union less-ened the gap in the transmission of information, leading to thefinal score in this striking mathematical achievement. Theseand other more recent results led to a new and deeper con-nection between mathematics and physics. The value of thisunion for modern mathematics is indispensible and is basedon a series of achievements of the first rank. It is enough tomention the results of V. Drinfeld, M. Kontsevich, and manyothers.

2. Mathematical SuccessionThe best confirmation of continuity and fruitfulness in thedevelopment of mathematics is the solution of deep classicalproblems left by the previous generations of mathematicians.And here the results of Fields medalists confirm this ideanicely.

The first recipient of the Fields Medal was Jesse Douglas,who solved the classical two-dimensional Plato problem. Itis necessary to say that this problem was solved simultane-ously by Tibor Rado, but Douglas’ solution was consideredas deeper and could be applied to higher dimensions.

Mathematicians of this mind-set include the famous num-ber theorists like A. Selberg, K. Roth, and A. Baker. In thelatest period, we see this tradition in the works of GregoryMargulis and Pierre Deligne.

The most important result of Margulis is his proof of Sel-berg’s conjecture that a certain class of discrete subgroupsof the group of motions of symmetric spaces of higher rankwith finite volume is arithmetic. While the conjecture can bestated rather easily, its proof required a virtuoso mastery of thetechnique of the theory of algebraic groups, use of the multi-plicative ergodic theorem, the theory of quasi-conformal map-pings, and much more. In recent years Margulis has examinedthe properties of discrete groups in different and sometimesunexpected areas. By combining ideas from the theory ofdiscrete groups and ergodic theory, he recently solved an oldproblem of the geometry of numbers: Oppenheim’s conjec-ture on the representation of numbers by indefinite quadraticforms.

Pierre Deligne received the prize for a proof of a conjec-tures of A. Weil on zeta functions over finite fields. His resultsare included as a special case of the proof of the classical Ra-manujan conjecture.

Ramanujan Conjecture: Consider the parabolic form

2π−12∆(z) = xΠ∞n=1(1 − xn)24 =

∞∑n=1

τnxn

where x = exp(2πiz). Then |τp| ≤ 2p11/2 for all primes p.

The proof of Deligne is one of the most brilliant and strik-ing examples of the unity and continuity of mathematics. It isstriking in its beauty and complexity, but required the appli-cation of the wealth of techniques accumulated in algebraicgeometry over preceding years.

The last example which I give, but only to mention inpassing to support this thesis, is the proof of the “Moonshinehypothesis” by Richard Borcherds. Here the statement re-garding the relations between the coefficients of special mod-ular forms, dimensions of the representations of the Monstergroup and some infinite-dimensional Kac-Moody algebras ledto the proof by applying methods from different fields of math-ematics. It was inspired by the recent development of stringtheory.

3. Zigzags in Mathematics

What I mean are the zigzags of mathematical fashion. Ialready talked about the domination of three mathematicaldiciplines in the list of Fields awards. Some reaction to thisbias, even beside some objective background, appeared atthe two last congresses. The awardees were mathematiciansworking in more classical fields. Let us mention here JeanBourgain and Tim Gowers–Banach spaces, harmonic analy-sis, combinatorics; P-L. Lions–partial differential equations;Jean-Cristoph Yoccoz, Curtis McMullen–dynamical systems,holomorphic dynamics; and the algebraist Efim Zelmanov,who solved the classical restricted Burnside problem. This re-sult capped off an extended period in group theory. J. Bougainand T. Gowers solved several classical problems in the the-ory of Banach spaces, discovered in very deep structures. J.C. Yoccoz and C. McMullen got important results in the so-called holomorphic dynamics. Here the study of sequencesof mappings of complex sets led to the theory of dynamicalsystems. A typical problem of holomorphic dynamics is todescribe the limiting sets of points of the mapping z → R(z),where R(z) is a rational function and z is in C or C. Eventhe the study of sequences of iterations of such a seeminglysimple map as fc(z) = z2 + c conceals highly nontrivialresults. This theory is placed at the meeting point of manybeautiful mathematical theories, such as dynamical systems,Kleinian groups, Fricke-Teichmuller spaces and many others,including computer graphics.

This theory is very remarkable and instructive if you lookat it from a historical perspective. Created in the end of the19th and the beginning of the 20th centuries in the works ofthe famous mathematicians P. Fatou, P. Montel, and G. Julia,it was seriously forgotten for more than forty years and wasrestored only in modern times. Now, besides being a very in-teresting theory, it has a wide field of applications in physics.Let us mention the famous universality law of Feigenbaumwhich has important applications in turbulance.

The unity of mathematics is shown best with these seem-ingly simple yet extraordinary complicated examples.

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To finish this very sketchy review of some of the achieve-ments of modern mathematics in the light of Fields medals,let me say that the results honored by Fields medals substan-tially determined the development of mathematics in our timeand its laureates are worthly representives of the mathematicalcommunity. Whether or not the Fields medal can be comparedwith Nobel prize, Fields’ idea of awarding it to the young hasmet with complete success.

References

1. M. Monastyrsky, Modern Mathematics in the light ofFields medals, (with a forward by F. Dyson), A. K.Peters, Wellesley, 1998.

2. M. Monastyrsky, Modern Mathematics in the light ofFields medals (in Russian), Yanus-K, Moscow, 2000(expanded and revised version of (1)).

AWARDS / PRIX

2001 CRM-Fields Prize

William T. TutteThe Centre de recherches mathématiques and the Fields In-stitute has announced the winner of the CRM-Fields prizefor 2001: Professor William T. Tutte from the University ofWaterloo.

William T. Tutte is one of the leading experts in graphtheory and matroid theory worldwide. He is responsible forsome of the most fundamental results in both of these fields; tomention just a couple, in graph theory, he established the fun-damental theorems of matching theory, an important branchof combinatorial optimization, and in matroid theory, he char-acterized regular matroids in terms of excluded minors, oneof the deepest results in the field.

Professor Tutte received his education at Cambridge Uni-versity, graduating with his Ph.D. in 1948. He joined thefaculty of the University of Toronto in 1948 and moved tothe University of Waterloo in 1962. He was for a long timethe editor of the Journal of Combinatorial Theory and is theHonorary Director of the Centre for Applied CryptographicResearch. He is a Fellow of the Royal Societies of Londonand Canada and was recently made an Officer of the Order ofCanada.

The CRM-Fields prize recognizes exceptional achieve-ment in mathematical research conducted primarily in Canadaor in affiliation with a Canadian university.

As a CRM-Fields prize winner, Professor Tuttewill present a lecture at both Institutes in Fall 2001.For further information, visit the Institute websites:www.fields.utoronto.ca or www.crm.umontreal.ca .

Three Honoured for Outstanding Research

The Canadian Mathematical Society (CMS) has selectedPriscilla Greenwood as the winner of the 2002 Krieger-NelsonPrize, Edwin Perkins as the winner of the 2002 Jeffery-Williams Prize and Kai Behrend as the winner of the 2001Coxeter-James Prize.CMS 2002 Krieger-Nelson Prize - Dr. Priscilla Greenwood(Arizona State University)

The Krieger-Nelson Prize recognizes outstanding re-search by a female mathematician.

Dr. Priscilla Greenwood was born in Kansas, USA andobtained her B. A. in mathematics from Duke University in1959, and her Ph. D from the University of Wisconsin (Madi-son) in 1963. She held positions at the University of Wiscon-sin and North Carolina College and, from 1966 to 2000, atthe University of British Columbia. In 2000 she moved toArizona State University. She was elected a Fellow of theInstitute of Mathematical Statistics (IMS) in 1985 and servedon the IMS Council from 1988-2000.

Over 35 years, Priscilla Greenwood’s research career hasspanned a broad range of topics in probability and statisticsand she has brought together ideas from different fields ina sustained research program. During the past 15 years shehas made fundamental contributions to mathematical statis-tics and, in particular, to the statistics of general stochasticprocesses. Her continued interest in the applications of prob-ability and statistics led to her leadership of a large interdisci-plinary research project at the University of British Columbia(1997 to 2000) which was funded by the Wall Institute and in-volved dozens of scientists in mathematics, geophysics, psy-chology, zoology, and physics.

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Dr. Greenwood will present the 2002 Krieger-NelsonPrize Lecture at Laval University (Québec) in June 2002.

CMS 2002 Jeffery-Williams Prize - Dr. Edwin Perkins(University of British Columbia)

The Jeffery-Williams Prize recognizes mathematicianswho have made outstanding contributions to mathematicalresearch.

Dr. Edwin Perkins was born in Toronto and obtained hisB.Sc. in mathematics from the University of Toronto in 1975,and his Ph.D. from the University of Illinois (Urbana) in 1979.He has been at the University of British Columbia since 1979.He was awarded the Rollo Davidson prize for young proba-bilists in 1983 and the CMS Coxeter-James Prize in 1986.He was elected as a Fellow of the Royal Society of Canadain 1988, and was awarded an NSERC Steacie Fellowship in1992.

Edwin Perkins has made outstanding contributions toprobability theory. Early in his career he pioneered the ap-plication of non-standard analysis to probability theory, andproved several spectacular results on the sample paths ofBrownian motion. Perkins has been a world leader in thestudy of "superprocesses", or measure valued diffusions. Us-ing non-standard analysis, he found a particle descriptionof super-Brownian motion which enabled the establishmentof very accurate estimates of the space-time behaviour ofthis process. Recently, with Richard Durrett (Cornell) andTheodore Cox (Syracuse) he has made a very exciting linkbetween particles systems and superprocesses.

Dr. Perkins will give the 2002 Jeffery-Williams PrizeLecture at Laval University (Québec) in June 2002.

CMS 2001 Coxeter-James Prize - Dr. Kai Behrend (Uni-versity of British Columbia)

The Coxeter-James Prize recognizes young mathemati-cians who have made outstanding contributions to mathemat-ical research.

Dr. Kai Behrend obtained his M.A. from the University ofOregon in 1984, his Diploma from the University of Bonn in1989 and his Ph.D. from the University of California at Berke-ley in 1991. He was a Moore Instructor at MIT from 1991to 1994 before joining the staff at the University of BritishColumbia. He has held visiting positions at the Max-Planck-Institut für Mathematik, Bonn, and at Research Institute forMathematical Sciences, Kyoto, Japan.

Kai Behrend is one of the world’s leading experts in thetheory of algebraic stacks and the geometry of moduli spacesof stable maps, which has become one of the most importantareas in algebraic geometry because of the unexpected predic-tions in enumerative algebraic geometry made by physicistsbased on string theory. Two of his publications, one with Bar-bara Fantechi (Trento, Italy) are among the most widely citedworks in algebraic geometry over the last five years. Kai’s

recent work on differential graded stacks is an extremely im-portant contribution to the construction of extended modulispaces.

Dr. Behrend will present the 2001 Coxeter-James PrizeLecture at York University (Toronto) in December 2001.

Fifth Canadian Open Challenge

The annual Canadian Open Mathematics Challenge(COMC) is organized and administered by the CanadianMathematical Society (CMS) in collaboration with the Centrefor Education in Mathematics and Computing (University ofWaterloo). Over 4,500 high school students from all acrossCanada participated in the Fifth Open on November 29, 2000.Students had to solve 12 questions during the two and one-halfhour time limit.

The top seven winners in the 5th Open are: Daniel Brox,Sentinel Secondary School, West Vancouver BC; Paul Cheng,West Vancouver Secondary School, West Vancouver BC;Lino Demasi, St. Ignatius High School, Thunder Bay ON;Nima Kamoosi, West Vancouver Secondary School, WestVancouver BC; Roger Mong, Don Mills Collegiate Institute,Don Mills ON; Henry Pan, East York Collegiate Institute,Toronto ON; Feng Tian, Vincent Massey Secondary School,Windsor ON.

The top contestant in each region is selected as a Provin-cial Gold Medalist. The 2000 Open Provincial Gold Medal-ists are: Peter Du, Sir Winston Churchill High School, Cal-gary AB; Daniel Brox, Sentinel Secondary School, West Van-couver BC; Michael Hirsh, St. John’s-Ravenscourt School,Winnipeg MN; Ryan Kabir, Saint John High School, SaintJohn NB; Timothy Hopkins, Queen Elizabeth Regional HighSchool, Foxtrap NF; Jeremy Nicholl, Horton High School,Wolfville NS; Brian Choi, Markville Secondary SchoolMarkham ON; Nicolae Petrescu, Lisgar Collegiate Insti-tute, Ottawa ON; Ilinca Popovici, Lisgar Collegiate Insti-tute, Ottawa ON; Wayne Thompson, Earl of March SecondarySchool, Kanata ON; Henry Pan, East York Collegiate Insti-tute, Toronto ON; Lino Demasi, St. Ignatius High School,Thunder Bay ON; Feng Tian, Vincent Massey SecondarySchool, Windsor ON; Rui Dong, Marianopolis College, Mon-tréal PQ; Alexandre Gadbois, Champlain St-Lawrence, Ste-Foy PQ; Jordon Wan, Aden Bowman Collegiate Institute,Saskatoon SK.

Approximately 80 of the top students from the 2000 Cana-dian Open Mathematics Challenge will be invited to write the2001 Canadian Mathematical Olympiad (CMO). The CMO,Canada’s premier mathematics competition, is organized andadministered by the CMS. Results in the CMO help deter-mine the six students who will represent Canada at the 2001International Mathematical Olympiad in Washington DC.

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EDUCATION NOTESEd Barbeau and Harry White, Column Editors

Associations of Mathematics Teachers in Quebec

There are four associations of mathematics teachers andfour “groups of interest” that are also concerned with mathe-matics teaching. Here is the list:

AMQ: Association mathematique du QuebecGRMS: Groupe des responsables de la mathematique au

secondaireAPAME: Association des promoteurs de l’avancement de

la mathematique a l’elementaireQAMT: Quebec Association of Mathematics TeachersGCSM: Groupe des chercheurs en science mathematiquesGDM: Groupe de didactique des mathematiquesGMA: Groupe de mathematiques appliqueesMOIFM: Mouvement international pour les femmes et

l’enseignement des mathematiques

Each has its particular objectives, but is representedin CQEM (Conseil quebecois de l’enseignement desmathematiques), a lobbying group for promoting mathemat-ics teaching at the Ministry of Education and School Boards.

In this issue, I will give some information about the AMQ,the oldest association of teachers of mathematics in Quebec.

AMQ: Historique

Le 25 avril 1923 est cree a Montreal un organisme ap-pele Societe des mathematiques et d’astronomie du Canadadans le but de faire avancer et encourager les etudes demathematiques et d’astronomie. Un mois plus tard, le 23mai, au Cercle universitaire, a lieu une rencontre en vuede creer une Federation des societes savantes canadiennes-francaises qui deviendra l’Association canadienne-francaispour l’avancement des sciences (ACFAS). D’autres groupesverront le jour tels la Societe de physique et de mathematiquesde Montreal et la Societe de mathematiques de Montreal dansles annees 1940, de meme que la Societe de mathematiquesde Quebec anime par Adrien Pouliot de 1929 jusqu’a la findes annees 1950, avant d’en arriver a la fondation de l’AMQle 5 juin 1958.

Les 125 premiers membres provenaient de toutes lesregions du Quebec. La moitie appartenait a des commu-nautes religieuses qui ont beaucoup fait pour l’enseignementdes mathematiques au Quebec. Le premier Bulletin de l’AMQ(journal officiel de l’AMQ) est paru en avril 1959, le premiercongres a eu lieu le 16 avril 1959, et le premier concoursmathematique, le 16 mai 1959. La Societe mathematique duCanada avait offert deux prix de 100$ pour ce concours.

Dans les annees 60, l’AMQ est incorporee comme une

societe a but non lucratif, et propose un plan de recyclagedes enseignants et enseignantes qui sera a l’origine des pro-grammes de perfectionnement des maitres en mathematiques.Au debut des annees 70, le GDM (groupe de didactique desmathematiques) est fonde et devient un groupe d’interet del’AMQ. Il en sera de meme pour le GRMS (groupe des respon-sables de la mathematique au secondaire) et de l’APAME (as-sociation des promoteurs de l’avancement de la mathematiquea l’elementaire) qui se preoccupent de facon plus specifiquede la cause de l’enseignement des mathematiques a l’ordresecondaire et a l’ordre primaire. Le GCSM (groupe deschercheurs en sciences mathematiques), un autre grouped’interet de l’AMQ, voit le jour en 1976. Ce groupe est al’origine des Annales des sciences mathematiques du Quebec,une revue internationale de recherche en mathematiques.

D’autres developpements dignes de mention apparaitrontau cours des annees : les Camps mathematiques pourles eleves de l’ordre collegial qui obtiennent les meilleursresultats au concours mathematique, et depuis l’an dernier,il y a un camp mathematique pour les eleves de l’ordre sec-ondaire ; de meme, des prix seront institues pour soulignerdes contributions remarquables : le prix Abel-Gauthier pour lapersonnalite de l’annee dans le domaine des mathematiques,le prix Roland-Brossard pour le meilleur article publie dansle Bulletin, le prix Adrien-Pouliot pour le meilleur livre edite,le prix Frere-Robert pour le meilleur materiel non edite, etle prix Dieter-Lunkenbein pour la meilleure these en didac-tique des mathematiques. Le CQEM (conseil quebecois del’enseignement des mathematiques) verra le jour au debutdes annees 90. Cet organisme a produit plusieurs avis et aservi de table de concertation pour les diverses associationset les differents groupes qui en font partie. L’annee mondi-ale de mathematiques fut l’occasion d’un mega-congres orga-nise conjointement par toutes les associations et les groupesd’interet associes.

References

Bulletin AMQ (octobre 1982) et conference de BernardCourteau lors du 40e congres de l’AMQ.

Buts de l’AMQ

Regrouper en association les personnes interessees auxmathematiques.

Contribuer a l’etude des mathematiques et au progres deson enseignement.

S’engager pedagogiquement et socialement dans ladefense et la promotion de mathematiques au Quebec.

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Organiser et tenir des conferences, reunions, assemblees,expositions pour la promotion, le developpement et la vulgar-isation des mathematiques.

Imprimer, editer, distribuer toute publication pour les finsci-dessus, et etablir en bibliotheque des publications se rap-portant ou connexes aux mathematiques.

Veiller aux interets de ses membres.

Site web: http://www.Mlink.NET /∼amq/AMQ/index.html

Harry White

A message from the Pacific

In December, at the Vancouver meeting, I had the plea-sure to meet Kanwal Neel, currently president of the BritishColumbia Association of Mathematics Teachers (website:www.bctf.bc.ca/bcamt), who was one of the speak-ers at the education session. His organization has been veryactive, particularly in informing the general public about nu-meracy and how it can be fostered in the classroom. Inthe following article, he describes the initiatives taken in hisprovince and how they are in accord with the goals enunciatedby NCTM. (EJB)

Numeracy Initiatives in British Columbia

Almost everyone knows what math is but what is numer-acy? Defined by the British Columbia Association of Mathe-matics Teachers (BCAMT) as “the combination of mathemat-ical knowledge, problem solving and communication skillsrequired by all persons to function successfully within ourtechnological world." Numeracy is a significant part of liter-acy.

Over the past few years, numeracy has been a topical issuewith parents, teachers, Ministry of Education, NCTM, media,and other organizations.

In order to put theory into practice the BCAMT haslaunched a number of numeracy initiatives such as produc-ing a brochure, presenting “Numeracy Workshops" to a widerange of audiences such as students, parents, teachers, admin-istrators, business, media, etc. BCAMT’s initiatives have alsoinfluenced the Ministry of Education’s report on the “Mathe-matics Task Force" and the “BC Numeracy Performance Stan-dards."

Our numeracy initiatives are aligned with the NationalCouncil of Teachers of Mathematics (NCTM) Principlesand Standards for School Mathematics 2000. Overall, theNCTM 2000 Standards documents advocate a broader andmore meaningful mathematics curriculum that is responsiveto changing societal priorities and to changes in instructional

practice that meet the needs of a far greater proportion of thestudent population than has been true in the past. The six Prin-ciples and ten Standards constitute a vision to guide educatorsas they strive for the continual improvement of mathematicseducation in classrooms, schools, and educational systems.

The six Principles for school mathematics address over-arching themes; they are:

The Equity Principle: Excellence in mathematics educa-tion requires equity, high expectations and strong support forall students.

The Curriculum Principle: A curriculum is more than acollection of activities: it must be coherent, focused on im-portant mathematics, and well articulated across the grades.

The Teaching Principle: Effective mathematics teachingrequires understanding what students know and need to learnand then challenging and supporting them to learn it well.

The Learning Principle: Students must learn mathemat-ics with understanding, actively building new knowledge fromexperience and prior knowledge.

The Assessment Principle: Assessment should support thelearning of important mathematics and furnish useful infor-mation to both teachers and students.

The Technology Principle: Technology is essential inteaching and learning mathematics; it influences the math-ematics that is taught and enhances students’ learning.

The ten Standards describe what mathematics instructionshould enable students to know and do. They specify theunderstanding, knowledge, and skills that students should ac-quire from pre kindergarten through grade 12.

The five Content Standards are:

Standard 1: Number and OperationsStandard 2: AlgebraStandard 3: GeometryStandard 4: MeasurementStandard 5: Data Analysis and Probability

The five Process Standards are:Standard 6: Problem SolvingStandard 7: Reasoning and ProofStandard 8: CommunicationStandard 9: ConnectionsStandard 10: Representation

Senses of Numeracy We believe, to be numerate, an indi-vidual should possess a variety of mathematical skills, knowl-edge, attitudes, abilities, understanding, intuition and experi-ence which could be expressed as the following senses:

Spatial Sense

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Statistical SenseSense of RelationshipNumber Sense

Teaching Numeracy. There is a direct association betweenstudents’ attitudes and their levels of success. The mathe-matics classroom should be an environment that engages theinterest and imagination of the learners. Learning activitiesshould help students understand that mathematics is a chang-ing and evolving subject to which many cultural groups havecontributed. Teaching strategies should also place mathemat-ics in context allowing students to see how understanding onemathematical idea can help them understand others.

Early Numeracy Project. Dr. Heather Kelleher of the De-partment of Curriculum Studies, Faculty of Education, UBChas embarked on a three year Early Numeracy Project. Thisproject proposes to address the best possible start for thosechildren most at risk in mathematics. We believe that a suc-cessful start in mathematics is essential to future success andthat early attention is preferable to later remediation.

Future direction and Challenges. We want our graduatesto enter the work force with a STRONG sense of numeracyand the ability to effectively communicate their understand-

ing. They should be able to use estimation skills, solve prob-lems, and engage in effective reasoning. Our challenge asmathematics educators is to provide all students with thesequalities. In order to achieve this we offer the following sug-gestions:

Specific numeracy standards should be listed that representthe minimum level of numeracy required for high school grad-uates.

Minimum acceptable learning outcomes or stages of numer-acy at grades 4, 7, and 10 also need to be outlined. The intentis not to create standardized tests but instead, performanceindicators at these grade levels.

Multiple forms of assessment with the integration of technol-ogy should be used to assess students’ numeracy level.

School mathematics experiences should provide a sufficientlybroad understanding so that graduates will be prepared to pur-sue a variety of careers in their lifetime.

Establish what constitutes Early Numeracy and develop sup-port materials for the at-risk learners and provide a companionprogram for parents.

Kanwal Neel, President, BCAMT

Millenium Prize Problems

To celebrate mathematics in the new millenium, theClay Mathematics Institute (www.claymath.org) has identi-fied seven old and important questions that have resisted allpast attempts to solve them. The Board of Directors of CMIhave designated a $7 million prize fund for the solution tothese problems, with $1 million allocated to each.

The P versus NP Problem

It is Saturday evening and you arrive at a big party. Feel-ing shy, you wonder whether you already know anyone in theroom. Your host proposes that you must certainly know Rose,the lady in the corner next to the dessert tray. In a fractionof a second you are able to cast a glance and verify that yourhost is correct. However, in the absence of such a suggestion,you are obliged to make a tour of the whole room, checkingout each person one by one, to see if there is anyone you rec-ognize. This is an example of the general phenomenon thatgenerating a solution to a problem often takes far longer thanverifying that a given solution is correct. Similarly, if some-one tells you that the number 13,717,421 can be written as theproduct of two smaller numbers, you might not know whetherto believe him, but if he tells you that it can be factored as3607 times 3803 then you can easily check that it is true using

a hand calculator. One of the outstanding problems in logicand computer science is determining whether questions existwhose answer can be quickly checked (for example by com-puter), but which require a much longer time to solve fromscratch (without knowing the answer). There certainly seemto be many such questions. But so far no one has proved thatany of them really does require a long time to solve; it maybe that we simply have not yet discovered how to solve themquickly. Stephen Cook formulated the P versus NP problemin 1971.

–Ian Stewart ans Stephen Cook

The Hodge Conjecture

In the twentieth century mathematicians discovered powerfulways to investigate the shapes of complicated objects. The ba-sic idea is to ask to what extent we can approximate the shapeof a given object by gluing together simple geometric buildingblocks of increasing dimension. This technique turned out tobe so useful that it got generalized in many different ways,eventually leading to powerful tools that enabled mathemati-cians to make great progress in cataloging the variety of ob-jects they encountered in their investigations. Unfortunately,the geometric origins of the procedure became obscured in

18


this generalization. In some sense it was necessary to addpieces that did not have any geometric interpretation. TheHodge conjecture asserts that for particularly nice types ofspaces called projective algebraic varieties, the pieces calledHodge cycles are actually (rational linear) combinations ofgeometric pieces called algebraic cycles.

–Pierre Deligne

The Poincare ConjectureIf we stretch a rubber band around the surface of an apple, thenwe can shrink it down to a point by moving it slowly, withouttearing it and without allowing it to leave the surface. Onthe other hand, if we imagine that the same rubber band hassomehow been stretched in the appropriate direction around adoughnut, then there is no way of shrinking it to a point with-out breaking either the rubber band or the doughnut. We saythe surface of the apple is “simply connected,” but that the sur-face of the doughnut is not. Poincaré, almost a hundred yearsago, knew that a two dimensional sphere is essentially char-acterized by this property of simple connectivity, and askedthe corresponding question for the three dimensional sphere(the set of points in four dimensional space at unit distancefrom the origin). This question turned out to be extraordinar-ily difficult, and mathematicians have been struggling with itever since.

–John Milnor

The Riemann Hypothesis

Some numbers have the special property that they cannot beexpressed as the product of two smaller numbers, e.g., 2, 3,5, 7, etc. Such numbers are called prime numbers, and theyplay an important role, both in pure mathematics and its ap-plications. The distribution of such prime numbers among allnatural numbers does not follow any regular pattern, howeverthe German mathematician G.F.B. Riemann (1826 – 1866)observed that the frequency of prime numbers is very closelyrelated to the behavior of an elaborate function ζ(s) calledthe Riemann Zeta function. The Riemann hypothesis assertsthat all interesting solutions of the equation ζ(s) = 0 lie on astraight line. This has been checked for the first 1,500,000,000solutions. A proof that it is true for every interesting solutionwould shed light on many of the mysteries surrounding thedistribution of prime numbers.

–Enrico Bombieri

Yang–Mills Existence and Mass Gaphe laws of quantum physics stand to the world of elementaryparticles in the way that Newton’s laws of classical mechan-ics stand to the macroscopic world. Almost half a century

ago, Yang and Mills discovered that quantum physics revealsa remarkable relationship between the physics of elementaryparticles and the mathematics of geometric objects. Predic-tions based on the Yang-Mills equation have been verified inhigh energy experiments performed at laboratories all over theworld: Brookhaven, Stanford, CERN, and Tskuba. Nonethe-less, there are no known solutions to their equations whichboth describe massive particles and are mathematically rig-orous. In particular, the “mass gap” hypothesis, which mostphysicists take for granted and use in their explanation forthe invisibility of “quarks,” has never received a mathemati-cally satisfactory justification. Progress on this problem willrequire the introduction of fundamental new ideas both inphysics and in mathematics.

–Arthur Jaffe and Edward Witten

Navier–Stokes Existence and Smoothness

Waves follow our boat as we meander across the lake, andturbulent air currents follow our flight in a modern jet. Math-ematicians and physicists believe that an explanation for andthe prediction of both the breeze and the turbulence can befound through an understanding of solutions to the Navier–Stokes equations. Although these equations were writtendown in the 19th Century, our understanding of them remainsminimal. The challenge is to make substantial progress to-ward a mathematical theory which will unlock the secretshidden in the Navier–Stokes equations.

–Charles Fefferman

The Birch and Swinnerton–Dyer Conjecture

Mathematicians have always been fascinated by the problemof describing all solutions in whole numbers x,y,z to algebraicequations like x2 + y2 = z2. Euclid gave the complete solu-tion for that equation, but for more complicated equations thisbecomes extremely difficult. Indeed, in 1970 Yu V. Matiya-sevich showed that Hilbert’s tenth problem is unsolvable, i.e.,there is no general method for determining when such equa-tions have a solution in whole numbers. But in special casesone can hope to say something. When the solutions are thepoints of an abelian variety, the Birch and Swinnerton-Dyerconjecture asserts that the size of the group of rational points isrelated to the behavior of an associated zeta function ζ(s) nearthe point s=1. In particular this amazing conjecture assertsthat if ζ(1) is equal to 0, then there are an infinite number ofrational points (solutions), and conversely, if ζ(1) is not equalto 0, then there is only a finite number of such points.

–Arthur Wiles

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UNIVERSITY OF VICTORIA – VICTORIA, BRITISH COLUMBIADEPARTMENT OF MATHEMATICS AND STATISTICS

Position in Discrete Mathematics

The University of Victoria, Department of Mathematics and Statistics, invites applications for a tenure-track position in DiscreteMathematics. The competition is open to applicants at any level. The successful applicant will be sponsored for an NSERCUniversity Faculty Award in the 2001-2002 competition, consequently only applications from candidates meeting the criteriafor this program will be considered. We quote from the NSERC eligibility requirements: "you must be a Canadian citizen, orpermanent resident of Canada, as of the nomination deadline date; and be a woman or Aboriginal person who holds a doctorate inone of the fields of research that NSERC supports; or expect to have completed all the requirements for such a degree, includingyour thesis defense, by the proposed date of appointment". Applicants should have a Ph.D. in Mathematics and a demonstratedrecord of, or the potential for, excellence in both research and teaching. The successful applicant will be expected to maintain astrong and active research program in Discrete Mathematics which includes the supervision of graduate students.Applicants should send a curriculum vitae, and arrange for three confidential letters of recommendation to be sent to:

Discrete Mathematics SearchDepartment of Mathematics and Statistics

University of VictoriaPO Box 3045 STN CSC

Victoria BC V8W 3P4 CanadaTel: (250) 721-7436Fax: (250) 721-8962

e-mail: [email protected]: http://www.math.uvic.ca/

The closing date for applications is May 15, 2001. The position may be taken up any time between April 1, 2002, and September1, 2002.For further information on NSERC University Faculty Awards, visit the NSERC website www.nserc.ca.In accordance with the University’s Equity Plan and pursuant to Section 42 of the B.C. Human Rights Code, the selection willbe limited to women and aboriginal peoples. Candidates from these groups are encouraged to self-identify.In accordance with Canadian immigration requirements, this advertisement is directed to Canadian citizens and permanentresidents.

(EDITORIAL–continue de page 2)

S. S.: C’est difficile à comparer. Ça dépend de votre apti-tude et de votre attitude. Je constate que vous avez beaucoupde facilité en mathématiques, même si vous réussissez trèsbien dans les deux matières. De plus, comme les étudiantssont nombreux à opter pour l’informatique, la concurrencesera donc plus féroce dans ce domaine.

Étudiant : L’informatique est très utile dans une sociétéaxée sur la technologie comme la nôtre, ce qui n’est pas le casdes mathématiques.

S. S.: L’utilité des mathématiques pour la société n’estgénéralement pas bien comprise. En fait, les mathématiquesjouent un rôle considérable dans la société à plus d’un égard.Je vous donne quelques exemples. En février, presque exacte-ment deux cents ans après la découverte du premier astéroïde,la sonde NEAR (Near Earth Asteroid Rendezvous), lancée deCape Canaveral (Floride) en 1996, a fait un atterrissage forcémais en douceur sur l’astéroïde Eros, en orbite à une vitessevertigineuse, à 315 millions de kilomètres de la Terre. La

sonde, qui s’est posée sur la surface accidentée de l’astéroïde,était commandée de la Terre par des scientifiques qui ont baséleurs calculs sur des processus mathématiques complexes.Par ailleurs, le public reconnaît davantage l’importance desmathématiques depuis qu’Andrew Wiles a réussi à prouverle dernier théorème de Fermat. De plus, la pièce de théâtreProof, dont trois des quatre personnages étaient des mathé-maticiens, a été bien reçue à New York à l’automne 2000. Lapièce a été suivie d’un symposium à l’Université de New Yorkau cours duquel trois groupes de mathématiciens et de non-mathématiciens ont discuté des thèmes et des idées abordésdans la pièce. Et je pourrais en dire encore long sur l’utilité desmathématiques, mais j’imagine que ça suffit pour l’instant.

Étudiant : On me dit que les informaticiens font beaucoupd’argent. Est-ce qu’un mathématicien peut devenir riche?

S. S.: C’est vrai que les informaticiens sont mieux payéstant dans le milieu universitaire qu’en entreprise. Un math-ématicien peut devenir millionnaire s’il parvient à résoudrel’un des problèmes de Clay. Problèmes à la page 18.

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(EXEC–continued from page 1)

Office staff provides a lot of administrative support, the suc-cess of the meetings in Hamilton (June 2000) and Vancouver(December 2000) was also due to the extensive help receivedfrom the meeting directors, the local organizers and the nu-merous sessions organizers. The CMS is always indebtedto the host university, to other participating universities and,through the National Program Committee, to the three re-search institutes for their indispensable support.

A highly successful meeting program together with on-going expansion of our publications and educational programscontributed to a very productive year. The Society also orga-nized and supported, via a grant from the CMS EndowmentFund, a number of special activities to help celebrate WorldMath Year 2000. These activities ranged from posters in theMontreal Transit System to museum exhibits in Sherbrookeand Regina, and from special lectures at provincial math as-sociation meetings to regional Math Camps. The CMS isgrateful to all of those who helped make these activities areality and who contributed largely to their success.

Our publication activities continue to be of a high standardand, in 2000, all periodicals were shipped on-time with issuesavailable to subscribers on Camel approximately one weekbefore the shipping date. With all CMS periodicals availableonline for both individual and institutional subscribers, morework is required of the Executive Office to ensure the neces-sary access information is provided to Camel in a timely andaccurate manner. The Executive Office, in conjunction withthose responsible for our web site, are continually trying tofind ways to streamline the process, to provide subscriberswith improved ways to update their information, and to im-prove the accounting system that provides electronic accessto each subscriber.

The new CMS Book Series with Springer-Verlag is pro-gressing well and the first four books in the Series appeared in2000. The new agreement with the American MathematicalSociety to publish the CMS Tracts in Mathematics has been

signed and work commenced to attract and publish books tothis new series. “A Taste of Mathematics” (ATOM) – a seriesof work booklets for high school students – continues to bewell received and is a series that has a large market potential.All of the Society’s book and periodicals must be promotedmore extensively so that each receives the necessary exposureand market potential.

Through the Canadian International MathematicalOlympiad team, the 2000 Canadian Mathematical Olympiad,the 2000 Canadian Open Mathematics Challenge, the CMSproblem solving journal (CRUX with MAYHEM), special ed-ucation sessions at our semi-annual meetings, public lectures,regional and national and Math Camps, and other activities,the CMS provides a wide array of educational enrichment ac-tivities. These activities are only possible because of the sig-nificant support received from provincial governments, cor-porations, foundations and CMS members. The 2000 EssoMath Camps program was extremely successful. Many of thestudents who participated have written to express their enthu-siasm for the program and indicated how much they valuedthe opportunity to participate in this unique program. In 2001,it is hoped the program will include Math Camps at SimonFraser University and Memorial University of Newfoundland.It is extremely encouraging that, in only three years, the pro-gram will have grown to provide a Math Camp in almost everyprovince.

In June 2000, Richard Kane (Western) ended his termas CMS President and Jonathan Borwein (SFU) assumed theposition as President. Richard’s leadership and direction con-tributed significantly to the development of the Society dur-ing his term. The continual growth of the Society impactsconsiderably on the work required by committee members,editors, organizers, directors, and particularly the President.The Society’s enviable national and international reputationwould not be possible without the efforts and guidance of allof these individuals. Many thanks to all of those who helpedmake 2000 a most successful year.

NEWS FROM DEPARTMENTS

Universite Laval, Quebec, QuebecAppointment: Daniel Le Roux (assistant professor, appliedmathematics, January 2001).

Retirement: Norbert Lacroix (July 2001).

GEORGE F. D. DUFF 1926-2001The CMS Notes has been informed that Professor GeorgeDuff, a former president of the Society and a member of theUniversity of Toronto faculty since 1952, passed away on Fri-day, March 2nd. Please see next month’s issue for an obituaryand further information about his life and career.

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DU BUREAU DU DIRECTEUR ADMINISTRATIFRapport annuel 2000

En mars 1998, sous la gouverne de la présidente de l’époque,Katherine Heinrich, la Société a entrepris un examen appro-fondi de ses activités et de son fonctionnement. Richard Kane,président de 1998 à 2000, et Jonathan Borwein, le présidentactuel, ont à tour de rôle supervisé cet important exercice deplanification stratégique. Les rapports finals de sept groupesde travail (Budget et politique, Représentation au Conseil,Fonds de dotation de la SMC, Publications, Finances et col-lecte de fonds, Soutien à la communauté mathématique etStratégies administratives) et d’un comité spécial (Servicesélectroniques et Camel) sont désormais publiés sur Camel(www.cms.math.ca/Projects/).

On a ensuite demandé au Comité exécutif d’étudier tousles rapports et les recommandations qu’ils contenaient, puisd’élaborer une structure et une stratégie d’ensemble pour lesannées à venir. Au cours de l’année 2000, chacun des comitéspermanents, ainsi que l’ensemble des membres de la Société,ont eu l’occasion de commenter les rapports et les recom-mandations. Le Comité exécutif a déjà enclenché l’examendes commentaires et présentera ses conclusions au Conseild’administration en juin 2001. Nous avons consacré beau-coup d’énergie à cet exercice de planification stratégique, etmême si le tout n’est pas encore terminé, plusieurs change-ments ont déjà été apportés. À la fin du processus, la SMCse sera donné les outils nécessaires pour mener ses activitéset offrir des services de la façon la plus efficace, efficiente etrentable possible.

Comme on le constate à la lecture des rapports annuels2000 des comités permanents, la Société soutient une vastegamme d’activités éducatives et scientifiques, et tout un pro-gramme de publication. Toutes ces activités sont renduespossibles grâce à l’apport remarquable des membres de laSociété et d’autres personnes de l’extérieur.

Au cours des dernières années, le nombre de séancestenues à nos Réunions semestrielles a considérablement aug-menté, tout comme le nombre de participants à ces rencon-tres. Nous devons sans contredit le succès des Réunions deHamilton (juin 2000) et de Vancouver (décembre 2000) aupersonnel du bureau administratif, qui joue un important rôlede soutien, mais également au travail monstre accompli parles directeurs des Réunions, les organisateurs locaux et lesnombreux responsables de séances. La SMC doit aussi unefière chandelle aux universités hôtes, aux autres universitésparticipantes et, par l’entremise du Comité du programmenational, aux trois instituts de recherche, pour leur soutieninestimable.

En somme, comme en font foi le vif succès remportépar nos Réunions et l’intensification continuelle de nos pro-grammes éducatifs et de nos activités de publication, la SMC

aura connu une année très productive. La Société a parailleurs organisé et financé, par l’entremise d’une subven-tion de son Fonds de dotation, un certain nombre d’activitésspéciales dans le cadre de l’année internationale des mathéma-tiques. Au nombre de ces activités, mentionnons la réalisationd’affiches destinées au réseau de transport de la communautéurbaine de Montréal; des expositions dans des musées deSherbrooke et de Regina, ainsi que des conférences spécialesprésentées à l’occasion de congrès d’associations provincialesde mathématiques et des camps de mathématiques régionaux.La SMC tient à remercier toutes les personnes qui ont con-tribué à la concrétisation et au succès de ces entreprises.

Côté publications, nos productions sont toujoursd’excellente qualité. De surcroît, tous nos périodiques ontété envoyés aux dates prévues cette année, et les abonnés auxversions en ligne ont eu accès à leurs numéros environ unesemaine avant la date d’expédition des versions papier. Toute-fois, depuis que tous nos périodiques sont accessibles en ligne,tant pour les abonnés à titre individuel que les établissements,la tâche de travail du personnel du bureau administratif a sen-siblement augmenté. Il faut en effet voir à ce que les abon-nés aient tous les renseignements nécessaires pour accéder àCamel, et cela de la façon la plus efficace possible. Le bureauadministratif, en collaboration avec les responsables du siteWeb, tente toujours de trouver des moyens d’alléger le pro-cessus, de faciliter aux abonnés la mise à jour de leur compteet d’améliorer le système qui donne accès aux publicationsen ligne à chaque abonné.

La nouvelle collection d’ouvrages de la SMC chezSpringer-Verlag progresse rapidement. Les quatre premiersouvrages sont d’ailleurs parus en 2000. Une nouvelle ententeavec la American Mathematical Society concernant la publi-cation de la collection Traités de mathématiques de la SMCa été signée, et nous avons déjà entrepris les démarches pourattirer les auteurs et publier des ouvrages dans cette nouvellecollection. Quant à notre collection ATOM - Aime-t-on lesmathématiques, livrets destinés aux élèves du secondaire, ellecontinue de recevoir des éloges. Cette collection a d’ailleursun fort potentiel commercial. Ajoutons que tous les ouvrageset périodiques de la Société devront faire l’objet d’une pro-motion accrue si l’on souhaite les faire connaître davantage àtous les clients potentiels.

La SMC offre par ailleurs une vaste gamme d’activitésd’enrichissement. L’équipe canadienne qui participe àl’Olympiade internationale de mathématiques, l’Olympiadede mathématiques du Canada 2000, le Défi ouvert canadien demathématiques 2000, le journal de résolution de problèmes dela SMC (CRUX with MAYHEM), des séances spéciales por-tant sur l’éducation dans le cadre des Réunions semestrielles,

22


des conférences publiques ainsi que des camps mathéma-tiques régionaux et nationaux sont au nombre des activitésqu’elle appuie ou qu’elle organise. Encore une fois, ces ac-tivités doivent leur réalisation à l’appui considérable des gou-vernements provinciaux, de sociétés privées, de fondations etdes membres de la SMC. Mentionnons aussi que les campsde mathématiques Esso 2000 ont connu un vif succès. Ungrand nombre de participants nous ont écrit pour nous fairepart de leur enthousiasme envers ce programme unique et ontsouligné à quel point ils était heureux d’avoir eu la chanced’y prendre part. En 2001, on espère élargir le programmeet offrir des camps aux universités Simon Fraser et Memorial(Terre-Neuve).Il est très stimulant de constater qu’après trois

ans seulement, un camp de mathématiques sera offert danspresque chacune des provinces.

En juin 2000, Richard Kane (Western) a terminé son man-dant à la présidence de la SMC, et Jonathan Borwein (SimonFraser) a pris la relève. Sous la direction de Richard, la So-ciété a fait de grands pas. Évidemment, la croissance continuede la SMC se répercute sur la charge de travail des comités,des équipes de rédaction, des organisateurs, des dirigeants et,en particulier, du président. La Société n’aurait pas la répu-tation enviable dont elle jouit maintenant tant au pays qu’àl’étranger sans la contribution et l’appui de toutes ces person-nes. Un grand merci à tous ceux et celles qui ont fait de l’an2000 une année si réussie!

CALL FOR NOMINATIONS / APPEL DE CANDIDATURESCoxeter-James / Jeffery-Williams / Krieger-Nelson Prize Lectureships

Prix de conference Coxeter-James / Jeffery-Williams / Krieger-Nelson

The CMS Research Committee is inviting nominations forthree prize lectureships.

The Coxeter-James Prize Lectureship recognizes out-standing young research mathematicians in Canada. The se-lected candidate will deliver the prize lecture at the Winter2001 Meeting in Toronto, Ontario. Nomination letters shouldinclude at least three names of suggested referees.

The Jeffery-Williams Prize Lectureship recognizes out-standing leaders in mathematics in a Canadian context. Theprize lecture will be delivered at the Summer 2002 Meetingin Quebec, Quebec. Nomination letters should include threenames of suggested referees.

The Krieger-Nelson Prize Lectureship recognizes out-standing female mathematicians. The prize lecture will bedelivered at the Summer 2002 Meeting in Quebec, Quebec.Nomination letters should include three names of suggestedreferees.

The deadline for nominations is September 1, 2001. Let-ters of nomination should be sent to the address below:

*****

Le Comite de recherche de la SMC invite les mises en can-didatures pour les trois prix de conference de la Societe, la

Conference Coxeter-James, la Conference Jeffery-Williamset la Conference Krieger-Nelson.

Le prix Coxeter-James rend hommage a l’apport excep-tionnel des jeunes mathematiciens au Canada. Le candidatchoisi presentera sa conference lors de la reunion d’hiver2001 a Toronto (Ontario). Les lettres de mises en candida-tures devraient inclure les noms d’au moins trois repondantspossibles.

Le prix Jeffery-Williams rend hommage a l’apport ex-ceptionnel des mathematiciens d’experience au Canada. LaConference sera presentee lors de la reunion d’ete 2002 auQuebec, (Quebec). Les lettres de mises en candidature de-vraient inclure les noms d’au moins trois repondants possi-bles.

Le prix Krieger-Nelson rend hommage a l’apport excep-tionnel des mathematiciennes au Canada. La Conference serapresentee lors de la reunion d’ete 2002 au Quebec, (Quebec).Les lettres de mises en candidatures devraient inclure les nomsd’au moins trois repondants possibles.

La date limite pour les mises en candidatures est le 1septembre 2001. Les lettres de mises en candidatures de-vraient etre envoyees a :

Douglas Stinson, CMS Research Committee / Comité de recherche de la SMCDepartment of Pure Mathematics,

University of Waterloo200 University Ave West, Waterloo,ON

Canada N2L 3G1

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CMS Summer Meeting 2001University of SaskatchewanSaskatoon, Saskatchewan

June 2-4, 2001Programme Update

The most up-to-date information concerning the programmes,including scheduling, and electronic registration is availableat the following world wide web address:

http://www.cms.math.ca/Events/summer01

Meeting registration forms and hotel accommodation formscan be found in the February 2001 issue of the CMS Notes andare also available on the website, along with on-line forms forregistration and submission of abstracts.

Updates on Symposia SpeakersThere have been a number of additions to the list of invitedspeakers. Please refer to the web site for the most up-to-dateinformation.

Abstracts will also appear on the web site as they becomeavailable.

Reunion d’ete 2001 de la SMCUniversite de la Saskatchewan

Saskatoon (Saskatchewan)2-4 juin 2001

Mise a jour du programme

Vous trouverez l’information la plus recente sur les pro-grammes, y compris les horaires et le formulaire d’inscriptionelectronique, a l’adresse Web suivante :

http://www.cms.math.ca/Events/summer01

Les formulaires d’inscription et de reservation d’hotel serontaussi publies dans le numero de fevrier 2001 des Notes de laSMC. Vous les trouverez egalement sur notre site web, ainsique les formulaires de resumes de conferences.

Liste de conferenciersIl y a eu quelques additions a la liste de conferenciers. Veuillezconsulter le site Web pour l’information la plus recente.

Les resumes de conferences paraıtront sur le site des quenous les recevrons.

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CALENDAR OF EVENTS / CALENDRIER DES EVENEMENTS

APRIL 2001 AVRIL 2001

25–26 Workshop on Mathematical Formalisms for RNAStructure, (CRM, Montreal)http://www.CRM.UMontreal.CA/biomath/

MAY 2001 MAI 2001

18–20 ISM Gradute Student Conference / Colloque ISM desEtudiants Avances (McGIll University, Montreal)www.math.uqam.ca/ISM/francais/colloque2001.html

25–26 2001 Seaway Number Theory Conference (CarletonUniversity, Ottawa)http://www.math.carleton.ca (see Upcoming Events)

25–29 Annual Meeting of the Canadian Mathematics Educa-tion Study Group (University of Alberta, Edmonton)http://cmesg.math.ca

25–29 Workshop on Groups and 3-Manifolds, (CRM, Univ.de Montreal, Quebec)Organizer: S. Boyer (UQAM) [email protected]

25–27 Annual meeting and special session on Frenchmathematics, Canadian Society for History and Philoso-phy of Mathematics / Societe canadienne d’histoire et dephilosophie des mathematiques (Universite Laval, Quebec)http://www.cshpm.org

JUNE 2001 JUIN 2001

2–4 CMS Summer Meeting / Reunion d’ete de la SMC(University of Saskatchewan, Saskatoon, Saskatchewan)http://www.cms.math.ca/CMS/Events/summer01

2–5 One Hundred Years of Russell’s Paradox (Munich)http://www.lrz-muenchen.de/ godeherd.link/russell1101.html,Ulrich.Albert @lrz.uni.muenchen.de

4–8 International Conference on Computational HarmonicAnalysis (City University of Hong Kong)[email protected]

4–13 Hamiltonian Group Actions and Quantization, in theSymplectic Topology, Geometry, and Gauge Theory Program(Fields Institute, Toronto and CRM, Montreal)http://www.fields.utoronto.ca/symplectic.html

12–17 8th Annual Canadian Undergraduate MathematicsConference 8e conference canadienne des etudiants enmathematiques (Laval University, Quebechttp://cumc.math.ca or http://ccem.math.ca

JULY 2001 JUILLET 2001

1–14 42nd International Mathematical Olympiad (Washing-ton D.C., USA)imo2001.usa.unl.edu

9–13 Workshop on Geometric Group Theory, (CRM, Univ.de Montreal, Quebec)Organizer: D. Wise (Brandeis & McGill Univ.) [email protected]

9–20 Seminaire de mathematiques superieures NATO Ad-vanced Study Group (Universite de Montreal)http://www.dms.umontreal.ca/sms

16–21 COCOA VII - The Seventh International Conferenceon Computational Commutative Algebra (Queen’s Univer-sity, Kingston)A. Geramita ([email protected])http://cocoa.dima.unige.it/

17–21 First Joint International Meeting between AMS andSociete Math. de France, History of Math. special session,Tom Archibald (Acadia Univ.)http:///www.ams.org/meetings/

22–25 International Symposium on Symbolic and AlgebraicComputation, (University of Western Ontario, London, On-tario)http://www.orcca.on.ca/issac2001/

23–Aug.3 Combinatorics and Matrix Theory, (Laramie,Wyoming)[email protected], http://math.uwyo.edu/

AUGUST 2001 AOUT 2001

7–9 Nordic Conference on Topology and its applications,NORDTOP 2001 (Sophus Lie Centre at Nordfjordeid, Nor-way)[email protected]

7–10 The 4th Conference on Information Fusion, (CRM,Univ. de Montreal, Quebec)[email protected]

12–18 Thirty-ninth International Symposium on Func-tional Equations (Sandjberg, Denmark, organized byAarhus University) Henrik Stetkaer: [email protected]://www.imf.au.dk/isfe39

13–15 13th Canadian Conference on Computational Geome-try, (University of Waterloo)http://compgeo.math.uwaterloo.ca/ cccg01

13–15 Second Gilles Fournier Memorial Conference / Sec-onde Conférence à la mémoire de Gilles Fournier (Universitéde Sherbrooke, Sherbrooke, Québec)http://www.dmi.usherb.ca/evenements

15–18 Second Workshop on the Conley Index and relatedtopics / Deuxième atelier sur l’indice de Conley et sujets con-nexes (Universite de Sherbrooke, Sherbrooke, Quebec)http://www.dmi.usherb.ca/evenements

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20–23 Second Canada-China Mathematics Congress (Van-couver)http://www.pims.math.ca/science/2001/canada-china/

SEPTEMBER 2001 SEPTEMBRE 2001

22–26 Applications of Discrete Mathematics, AustralianMathematical Society (Australian National University, Can-berra) Ian Roberts: [email protected] Lynn Batten: [email protected]

DECEMBER 2001 DECEMBRE 2001

8–10 CMS Winter Meeting / Reunion d’hiver de la SMC(Toronto Colony Hotel, Toronto, Ontario)http://www.cms.math.ca/CMS/Events/winter01

MAY 2002 MAI 2002

3–5 AMS Eastern Section Meeting (CRM, Universite deMontreal)http://www.ams.math.org/meetings/

JUNE 2002 JUIN 2002

6–8 CAIMS 2002 (University of Calgary)Samuel Shen: [email protected]

15– 17 CMS Summer Meeting / Reunion d’ete de la SMC(Universite Laval, Quebec, Quebec)Monique Bouchard: [email protected]

24–28 Special Activity in Analytic Number Theory (MaxPlanck Institute, Bonn) [email protected]

JULY 2002 JUILLET 2002

22–30 44rd International Mathematical Olympiad (Universityof Strathclyde, Glasgow, UK0

AUGUST 2002 AOUT 2002

20–28 International Congress of Mathematicians (Beijing,China) http://icm2002.org.cn/

DECEMBER 2002 DECEMBRE 2002

8–10 CMS Winter Meeting / Reunion d’hiver de la SMC(University of Ottawa / Universite d’Ottawa,Ottawa, Ontario)Monique Bouchard: [email protected]

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