Carlson: “Perelman fulfills all the requirements of the Millennium Prize”Cao: “Hamilton and Perelman are the giants and ourheroes!”

INTERVIEW WITH JIM CARLSON

For Perelman to be declared thewinner of one of the MillenniumPrizes he has to fulfill several con-ditions: his work must be publishedand reviewed by a referee; and hemust then wait two years, duringwhich the Clay Institute, whichdrew up the list of problems to besolved, verifies that these condi-tions have been fulfilled. JimCarlson, president of the ClayMathematics Institute (CMI) andone of the members of the CMI’sScientific Advisory Board, speaksabout the million-dollar prize.

Does the Clay Institute considerthat Poincaré’s Conjecture hasbeen proven?

In his talk, John Morgan said thatthe conjecture had been proven by Grisha Perelman. Letme just say that Perelman’s stunning breakthrough is anevent for great celebration. Perelman stood on the shoul-ders of giants – Richard Hamilton, in particular – but hesaw further ahead than did any of us.

Does this mean that Perelman will be officiallydeclared as a winnerof one of the Millennium Prizes,offered by the Clay Institute to those who can solvethe so-called “Millennium Problems”.

The rules of the Institute stipulate a two-year wait afterthe work receives refereed publication and general accep-tance by the mathematical community. The two-yearperiod during which the entire mathematical communitycan scrutinize these articles, now publicly available, iscrucial. Only then does the Scientific Advisory Boardtake up the matter and make a recommendation to theDirectors of the Institute.

Does the fact that Perelman’s work has only beenposted on Internet posed any problem for the Institute?

Detailed expositions of Perelman’s work have now appe-ared in the work of Cao-Zhu, Kleiner-Lott, and Morgan-

Tian, and the work by Gérard Besson will soonappear as well. These works, all refereed or inthe process of being refereed, are enough.

So Perelman gets the prize?He is the one who announced a proof of the

conjecture. The decision to award the prize willbe made in roughly two year’s time. There is aprocedure in place for the decision: appoint-ment of a panel of experts made up mainly ofpersons not connected with the Clay Institute,consideration of their report by the ScientificAdvisory Board, and, finally, decision by thedirectors of the Institute.

Do you think he’ll accept it?I couldn’t say; I have no idea. We will do

what the ICM did when it conferred a FieldsMedal on Perelman. I think John Ball and theFields Medal Committee did exactly the rightthing. They offered the medal based on

Perelmans’s achievements, and did not condition thatdecision on his possible reaction. If the Clay Institutedecides to offer the Millennium Prize to Perelman, it willfollow the same philosophy.

The Clay Institute (Massachusetts, USA) was foundedin 1999 by Boston businessman Landon T. Clay. In 2000,CMI announced the creation of a series of prizes forseven problems, known as the “Millennium Problems,”each one worth one million dollars. The problems wereselected by the Scientific Advisory Board of CMI (AlainConnes, Arthur Jaffe, Andrew Wiles, and Edward Witten)in consultation with other leading mathematicians. Theseven are important classic problems which have longresisted solution. The current Scientific Advisory Boardconsists of James Carlson, Simon Donaldson, GregoryMargulis, Richard Melrose, Yung-Tong Siu, and AndrewWiles. Arthur Jaffe was the first president of CMI. SimonDonaldson, Gregory Margulis, and Edward Witten areFields Medalists.

I C M 2 0 0 6Daily News

Madrid, August 29th 2006

If there was any doubt about who did what when it came to provingthe Poincaré Conjecture, Huai-Dong Cao makes it clear in this shortinterview he gave to the ICM2006 Daily News. Once this point has beenclarified, the next question is: will as many people have predictedGrisha Perelman receive one of the “Millenium Prizes”? The man withthe answer to this question could be Jim Carlson, president of the ClayMathematical Institute. A question nobody can answer, however, iswhether Perelman will agree to accept the prize if he is awarded it.

INTERVIEW WITH HUAI-DONG CAO

In the latest issue of the Asian Journal of Mathematics,published in the U.S.A, the Chinese mathematicians Xi-Ping Zhu (University of Zhongshan,Canton, China), and Huai-Dong Cao(Lehigh University,Pennsylvania,(USA), announced thatthey had arrived at “a complete proof ofthe Poincaré and GeometrizationConjectures”. They appeared in manymedia as the true authors of the solutionto a legendary problem.

You have been at the heart of amedia-storm. It was even announcedyou have finally proved the PoincaréConjecture. How have you livedthr ough all this?

I like mathematics and love workingon geometric flows, in particular Ricciflow. Unfortunately some of themedia’s attention seems to be muchmore interested in something else other than discussingmathematics. I simply focus on the mathematics part.

The controversy has focused on ‘who did what’.Prof. Shing-Tung Yau, for instance, has been quotedsaying that Perelman contributed no more that 25%

to the solution of the problem, while you contributed30%. What is your view?

In my joint paper with Zhu, we have given a detailedaccount. Hamilton and Perelman have done the most

important fundamental works. They arethe giants and our heroes! In my mind,there is no question at all that Perelmandeserves the Fields Medal. We justfollow the footsteps of Hamilton andPerelman and explain the details. I hopeeveryone who read our paper wouldagree that we have given a rather fairaccount.

Most unfortunately, Prof. Yau’s viewpoint has been distorted by the media.As far as I know, Prof. Yau never assig-ned any percentage distribution and hedoesn’t agree with any such distribu-tions. In fact, he has shown me severalemails (...) he wrote to various repor-ters which stated explicitly that hedoesn’t agree with that kind of distribu-

tion. I also like to add that Prof. Yau respectedPerelman’s contribution a great deal. In more than oneoccasion, including when talking to the reporters of theNew Yorker (...) he said that Perelman deserves theFields Medal.

2 ICM 2006 MADRID SPAIN

It’s a great honour for meto present a poster at theInternational Congress ofMathematicians. I’d like totake this opportunity to thankthe Organizing Committee,and in particular the commit-tee responsible for the volun-teers for giving us this oppor-tunity. It all began during adoctorate course when mycolleagues Pablo CentellaBarrio and Elena GálvezMoreno and I solved a problem and decided to submit thepaper for publication. Almost immediately we received ane-mail from Emilio Bujalance informing all the congressvolunteers that we could present a short communicationor a poster. None of us had ever done a poster before (Idon’t think we even knew what one was!), but we askedaround and decided to go for it. A poster is simply ameans of presenting your most important results in anattractive and entertaining way without going into toomuch detail. It’s also good for those who are not comfor-table speaking in public. In our case, the poster’s quitenoticeable because of some computer-generated imagesof surfaces, and also because fortunately the artistic tasteof my colleagues is better than mine. I also have to saythat it wasn’t too hard to make the poster. Basically, we

did the classic cut and paste. As far ascomposition and presentation of theposter is concerned, we spent quite alot of time deciding on colours, back-grounds, framing, etc..

Several months later, here I am, aboutto present a poster for the first time,and nothing more and nothing less thanat the ICM! With of course the addedemotion of taking part in a competition.The poster will be on display fromAugust 25th to 29th, but we all have acertain time (from 5.00pm to 6.00pm;

Saturday 26th in my case) when we have to be present inperson to answer questions from interested parties – ormaybe listening to criticism (I hope not!). That’s okay byme, at least until somebody asks me a question I can’t ans-wer, but that’s all part of the game.

Seriously though, I’m pretty excited about all this, butnot nervous. It’s always gratifying to see something you’-ve spent some time working on displayed like this whereeveryone can view it. I invite you to come and see ourposter “Tetraharmonic Bézier surfaces”, as well as theothers on display with it. I’m sure the standard of thisICM won’t disappoint anybody. Lastly, I’d like to congra-tulate the winners of the competition. That’s all!

Raúl Oset Sinha , Doctorate student at the University of València, ICM Volunteer

“Just seeing ourposterup and people looking at it, I feel satisfied and rewarded”.POSTER COMPETITION

3

Benoit Mandelbrot coined the term ‘fractal’in 1975from the latin word fractus (‘broken’or ‘fractured’), andopened up a new field of mathematics devoted to thestudy of these structures. Today, fractals are used in suchdiverse applications as medicine –i.e. to detect certaintypes of tumors— and cinema –to generate artificiallandscapes—, as well as art. Mandelbrot is currently theSterling Professor Emeritus of Mathematical Sciences atYale University, and IBM Fellow Emeritus (Physics) atthe IBM Research Center. He answered some of ourquestions via e-mail.

Fractals have become one of the most beautifulfaces of modern mathematics, and a useful one topopularise mathematics among the general public.How do you feel about this?

I’m thoroughly delighted to see that fractals help brid-ge the abyss that separates difficult open questions ofmathematics from lay persons’interests -young and old.Some mathematicians have always liked their field to beleft alone but I always thought that isolation is unwise, infact, very harmful.

What did your colleagues first say when you showedthem your results with fractals?

Fractal geometry did not involve any Eureka! moment.It was not a serendipitous discovery but the end of a verylong process which repeated itself for diverse forms ofroughness that occur in pure mathematics and in manyareas of science. Pure mathematicians had come to belie-ve that it was no longer possible for pictures to affecttheir field. To the contrary, my discovery of theMandelbrot set consisted in a number of visual observa-tions phrased into conjectures demanding rigorous mathe-matical treatment. Naturally, the very possibility of deri-ving conjectures from pictures was originally receivedwith scepticism and the extreme difficulty of my conjec-tures came as a very big surprise. Today the situation iscompletely different. For example, my 4/3 conjectureabout Brownian motion has inspired a large amount ofhighly admired purely mathematical work. And the MLCconjecture about the Mandelbrot set -that it is locallyconnected- has so far resisted all attempts to proof or dis-proof. All those conjectures can be rephrased in wayseverybody can understand and provide the layperson withglimpses of active portions of the frontier.

On top of these questions, Mandelbrot gave a pressconference a few days ago in the ICM2006. During theconference he spoke of the relation between art andmathematics “via” fractals. “It’s wonderful to see howboth worlds combine, art and mathematics. In a certainsense, it harks back to the nature of human beings, to theforms that human beings have always perceived in natu-re. Because save for a few exceptions, such as the eye orthe moon, forms in nature are rough, not homogeneous orsimple. Art has always noticed this. Mathematics on theother hand have always remained isolated and have con-centrated on simple figures. I’ve been very lucky to workwith the mathematics of roughness. In a certain sense, thecircle has been completed; in many cultures the rough

Mandelbrot: “Fractals ar e very important tounderstand the evolution of the stock markets”

has always been considered art. Then this concept arrivedto mathematics and afterwards to other sciences, thenback to mathematics again, and finally to art once morethrough fractal art”.

What is also very significant for Mandelbrot is that themultiple applications of fractals will arrive after their“artistic” uses in fractal geometry. In fact, engineers werea rather late in discovering the uses of fractals, but assoon as they realized it the applications soon followed asan almost natural outcome. “Take antennas, for instance;in many modern devices the antennas are fractal. Theyare much more efficient. Or take the noise in houses; thewalls are very bad, because they reflect noise. So why notmake the walls fractal? Now the patents of fractal wallsare being used by a very big company. They have a roughtexture and the noise is not reflected, it is absorbed. Ortake electronics. You can make microcapacitors by fol-ding following a fractal structure. Or take concrete. It is avery important material of course, but most of the concre-te in old buildings is very bad. The water goes into it andrusts metallic bars... Concrete is a porous material thatyou don’t want to be porous. But the thinking for the newconcrete is based entirely on the theory of fractal porousmaterials. It is much stronger and durable. Once the tradi-tion was to think only of smooth shapes, and [by breakingthat tradition] fractals are becoming increasingly useful”.

When Mandelbrot turned 80 his friends organised in hishonour a conference on practical application of fractals.“It was a beautiful conference, because in almost everyfield of engineering, if you accept the rough shapes yousimply have a greater choice”. Mandelbrot has currentlyreturned to the field where fractal geometry started, finan-cial markets. One of his latest projects has been to applyfractals to models for predicting price behaviour. “Theusual theory in price variation, taught in every businessschool, is that prices follow a random motion; they go upa little bit, then down... but never much, and this processis continuous. But prices are not continuous; prices jump.And price jumps are absolutely essential. My theory takesaccount of them, and the theory that used to be taught inevery business school did not. So I think fractals are veryimportant to understand the evolution of stock markets.Fractal geometry has the tools to represent prize varia-tions realistically, and this is a first.

Equations are a way of describingreality, but some equations can beused the other way round; that is, togenerate artificial worlds. Equationsdescribing fractal sets, for example,can give rise to landscapes as fasci-nating as those depicted in famouscanvases. To see it you only have totake a look at the works on display atthe Exhibition of Fractal Art, at theICM2006 International Congress ofMathematicians and at the CentroCultural Conde Duque in Madrid.

It is not necessary to enter into acomplicated mathematical descrip-tion to get an intuitive grasp ofwhat they are: structures which,“when a small portion is observed,preserve a similar, although notnecessarily identical appearance towhat they look like when observed in their entirety”,explains Javier Barrallo, fractal artist. Some examples offractals are: a tree and its branches; a cauliflower, appa-rently made up of endless cauliflowers joined together;the coastline of a country…

The point of departure to make a fractal artwork isindeed a mathematical formula. The first fractal formulaewere described more than a century ago. Today there arehundreds. And, as Barrallo explains, the computer is vital:

“A small image, one of 640 x 480pixels, for example, contains307,200 dots that must be calcula-ted. It may be necessary to calculateeach one of these dots about 1,000times by the formula determiningthe fractal. This means that the for-mula must be calculated more than300 million times. And this is justfor a small-size image!”.So, armedwith both formula and computer, wemust now proceed to iteration. Thisinvolves “calculating a formula overand over again, starting from itsinitial value”, says Barrallo. “Aftercalculating the formula for the firsttime, we take the resulting value andintroduce it into the formula. Thenew result is calculated again, andso on successively”. In the case of

fractals, the initial value has to do with the position of thedot in the frame (the pixel on the screen). Then coloursare assigned according to the value of each dot. The factthat the behaviour of two dots situated very close togethercan be radically different – one diverging toward the infi-nite and the other converging toward a given value - is“what makes fractal exploration so fascinating”, saysBarrallo, and also what leads to the explosion of shapesand colours in the image.

Fractal Ar t: Painting with numbers4 ICM 2006 MADRID SPAIN

Two years ago, it occurred to the ICM2006Organizing Committee to set up an exhibition based onthe history of the IMU. Right away, however, I realizedthat this would be more or less the same as explainingthe background of the Social Security Office in Soria;i.e. pretty boring (for people who aren’t Spanish). So Idecided to tell the story of the ICM instead, but from agraphic and above all human point of view. That’s whythe exhibition “The ICM through History” is meant tobe an account of the ICM as a human endeavour. In factwe use the activities of mathematicians in ICMs as amirror in which history, culture, technology, fashion andattitudes are reflected.

The background to the exhibition consisted of aquest for the personal stories behind the ICM over theyears: Gaston Julia wearing the mask he wore to con-ceal the wound he sustained in World War I, givingone of the closing addresses of the 1936 ICM in Osloand telling how his life was saved by a Norwegiannurse in a field hospital; or the 1912 congress visitingCayley’s grave in Mill Road cemetery, Cambridge.How could we tell these stories? We needed picturesfor our imagination to feed on, but where could wefind these pictures?

This was no easy task. It took more than a year totrack some of these images down. We had to unearththem one by one from math departments, libraries,archives, and personal records: all together, more thanthirty institutions from between 30 to 40 different coun-tries have made contributions. The result - more than400 images help the visitor to visualize the journey ofICMs through the turbulent waters of the last one hun-dred years.

The outer part of the exhibition reviews the ICMs inchronological order from the first congress held in 1897to the current one in 2006. It consists of a selection ofimages and texts. On the inside we have a cross-sectionof the ICM from the point of view of social life in ICMs(excursions, banquets, parties, concerts), graphic designin ICMs, the buildings that have provided the venue forthe congress, the history of the InternationalMathematical Union, and of course the awards: the FieldsMedals, and the Nevanlinna and Gauss Prizes. At the endof the ICM2006, the exhibition will travel to Spanish uni-versities and different institutions in Europe.

Guillermo Curbera, Curator of the exhibition “The ICM through History”

Personal Stories Behind One Hundred Years of the ICM

5

The Spanish Face of the CongressWho´s Who on the Executive Committee

Alber to Ibor t (Huesca, 1958) is probably the most versatile mem-ber of this group,. Not only does he provide the Daily News cartoon,he also stands in as interpreter in the Congress press conferences.And from the mathematical point of view, he is a man for all seasons.Professor of applied mathematics at the Carlos III University ofMadrid, he has always tried to adopt a broad view of science, combi-ning theoretical physics with pure and applied mathematics. He isalso interested in the relation between computation and pure mathe-matics, as well as the role that geometry and topology could play incomputation. What he likes most about the Congress is the vitality ofthe mathematical community, something that is reflected in all theevents, from sessions "sold out" to the steady stream of exhibitionvisitors. He is also eager to point out "the interaction with the media,which is both constructive and creative. In this way we can learnfrom each other and improve".

The history of the ICMs will not be forgotten; it has its own chronicler,Guillermo Curbera (Madrid, 1961). Formerly a student delegate (Madrid1979 and Sevilla 1985), he is now an associate professor of mathematicalanalysis at the University of Seville. The objects of his study are functionspaces, vectorial measures, operators with kernal and Sobolev spaces. Hisremarks on the Congress leave no room for doubt: “What I’m most proudabout is that it exists”. This pride extends to the exhibition he set up aftertwo years of hard work and which has become one of the star events ofthe Congress: “The ICM Through History”. As far as reactions to theCongress are concerned, he says he’s happy that for at least once a yearmathematicians are in the limelight, “because it’s good for the professionand because it can encourage students to take up this science”.

The separation between science and letters is just a myth. They arenot in the least incompatible. Antonio Durán (Córdoba, 1962) provi-des a proof for this and makes it a theorem. Novelist (among otherbooks, “La luna de nisán”, published by Editorial Debate) and profes-sor of mathematical analysis at the University of Seville, is particu-larly interested in approximation theory and special functions. He hasderived many good things from this Congress, but as the Committeemember responsible for cultural activities he is especially proud of theresults achieved by teamwork. “I feel particularly satisfied by themedia coverage given to the Congress, and the general level of publicinterest about what we’re doing here”.

Mar ta Sanz-Solé(Barcelona, 1952) has the whole ICM2006 scientificprogramme in her head. She is professor of mathematics at the Universityof Barcelona, a specialist in stochastic analysis and chair of the LocalProgramme Committee. From the very beginning, her dealings with theinvited speakers have been one of the most exciting aspects of her work,because as she says, “you appreciate different characters and their ways ofworking, and of course it’s an honour to deal with so many outstandingmathematicians”. She also emphasizes the enthusiasm shown by the parti-cipants: “the excellent attendance at the sessions and the positive com-ments about the programme and the organization are a source of greatsatisfaction”, says Sanz-Solé.

Highlights of Wendelin Werner’s work 6 ICM 2006 MADRID SPAIN

The work of Wendelin Werner and his collaboratorsrepresents one of the most exciting and fruitful interac-tions between mathematics and physics in recent times.Werner’s research has developed a new conceptual frame-work for understanding critical phenomena arising inphysical systems and has brought new geometric insightsthat were missing before. The theoretical ideas arising inthis work, which combines probability theory and ideasfrom classical complex analysis, have had an importantimpact in both mathematics and physics and have poten-tial connections to a wide variety of applications.

A motivation for Wendelin Werner’s work is found instatistical physics, where probability theory is used toanalyze the large-scale behavior of complex, many-parti-cle systems. A standard example of such a system is thatof a gas: Although it would be impossible to know theposition of every molecule of air in the room you are sit-ting in, statistical physics tells you it is extremely unli-kely that all the air molecules will end up in one corner ofthe room. Such systems can exhibit phase transitions thatmark a sudden change in their macroscopic behavior. Forexample, when water is boiled, it undergoes a phase tran-sition from being a liquid to being a gas. Another classi-cal example of a phase transition is the spontaneous mag-netization of iron, which depends on temperature. At suchphase transition points, the systems can exhibit so-calledcritical phenomena. They can appear to be random at anyscale (and in particular at the macroscopic level) andbecome “scale-invariant”, meaning that their generalbehavior appears statistically the same at all scales. Suchcritical phenomena are remarkably complicated and arefar from completely understood.

In 1982 physicist Kenneth G. Wilson received theNobel Prize for his study of critical phenomena, whichhelped explain “universality”: Many different physicalsystems behave in the same way as they get near criticalpoints. This behavior is described by functions in which aquantity (for instance the difference between the actualtemperature and the critical one) is raised to an exponent,called a “critical exponent” of the system. Physicists haveconjectured that these exponents are universal in thesense that they depend only on some qualitative featuresof the system and not on its microscopic details. Althoughthe systems that Wilson was interested in were mainlythree- and four-dimensional, the same phenomena alsoarise in two-dimensional systems.

During the 1980s and 1990s, physicists made big stri-des in developing conformal field theory, which providesan approach to studying two-dimensional critical pheno-mena. However, this approach was difficult to understandin a rigorous mathematical way, and it provided no geo-metric picture of how the systems behaved. One greataccomplishment of Wendelin Werner, together with hiscollaborators Gregory Lawler and Oded Schramm, hasbeen to develop a new approach to critical phenomena intwo dimensions that is mathematically rigorous and thatprovides a direct geometric picture of systems at and neartheir critical points.

Percolation is a model that captures the basic behaviourof, for example, a gas percolating through a randommedium. This medium could be a horizontal network ofpipes where, with a certain probability, each pipe is openor blocked. Another example is the behaviour of pollu-tants in an aquifer. One would like to answer questionssuch as, What does the set of polluted sites look like?Physicists and mathematicians study schematic models ofpercolation such as the following. First, imagine a planetiled with hexagons. A toss of a (possibly biased) coindecides whether a hexagon is colored white or black, sothat for any given hexagon the probability that it getscolored black is p and the probability that it gets coloredwhite is then 1 - p. If we designate one point in the planeas the origin, we can ask, Which parts of the plane areconnected to the origin via monochromatic black paths?This set is called the “cluster” containing the origin. If pis smaller than 1/2, there will be fewer black hexagonsthan white ones, and the cluster containing the origin willbe finite. Conversely, if p is larger than 1/2, there is apositive chance that the cluster containing the origin isinfinite. The system undergoes a phase transition at thecritical value p = 1/2.

This critical value corresponds to the case where onetosses a fair coin to choose the color for each hexagon. Inthis case, one can prove that all clusters are finite and thatwhatever large portion of the lattice one chooses to lookat, one will find (with high probability) clusters of sizecomparable to that portion. The accompanying picturerepresents a sample of a fairly large cluster.

The percolation model has drawn the interest of theore-tical physicists, who used various non-rigorous techni-ques to predict aspects of its critical behavior. In particu-lar, about fifteen years ago, the physicist John Cardy usedconformal field theory to predict some large-scale proper-ties of percolation at its critical point. Werner and hiscollaborators Lawler and Schramm studied the conti-nuous object that appears when one takes the large-scalelimit—-that is, when one allows the hexagon size to getsmaller and smaller. They derived many of the propertiesof this object, such as, for instance, the fractal dimensionof the boundaries of the clusters. Combined withStanislav Smirnov’s 2001 results on the percolationmodel and earlier results by Harry Kesten, this work ledto a complete derivation of the critical exponents for thisparticular model.

Another two-dimensional model is planar Brownianmotion, which can be viewed as the large-scale limit ofthe discrete random walk. The discrete random walk des-cribes the trajectory of a particle that chooses at random anew direction at every unit of time. The geometry of pla-nar Brownian paths is quite complicated. In 1982, BenoitMandelbrot conjectured that the fractal dimension of theouter boundary of the trajectory of a Brownian path (theouter boundary of the blue set in the accompanying pictu-re) is 4/3. Resolving this conjecture seemed out of reachof classical probabilistic techniques. Lawler, Schramm,and Werner proved this conjecture first by showing that

7

the outer frontier of Brownian paths and the outer boun-daries of the continuous percolation clusters are similar,and then by computing their common dimension using adynamical construction of the continuous percolationclusters. Using the same strategy, they also derived thevalues of the closely related “intersection exponents” forBrownian motion and simple random walks that had beenconjectured by physicists B. Duplantier and K.-H. Kwon(one of these intersection exponents describes the proba-bility that the paths of two long walkers remain disjointup to some very large time). Further work of Wernerexhibited additional symmetries of these outer boundariesof Brownian loops.

Another result of Wendelin Werner and his co-workersis the proof of the “conformal invariance” of some two-dimensional models. Conformal invariance is a propertysimilar to, but more subtle and more general than, scaleinvariance and lies at the roots of the definition of thecontinuous objects that Werner has been studying.Roughly speaking, one says that a random two-dimensio-nal object is conformally invariant if its distortion byangle-preserving transformations (these are called confor-mal maps and are basic objects in complex analysis) havethe same law as the object itself. The assumption thatmany critical two-dimensional systems are conformallyinvariant is one of the starting points of conformal fieldtheory. Smirnov’s above-mentioned result proved confor-mal invariance for percolation. Werner and his collabora-

tors proved conformal invariance for two classical two-dimensional models, the loop-erased random walk andthe closely related uniform spanning tree, and describedtheir scaling limits. A big challenge in this area now is toprove conformal invariance results for other two-dimen-sional systems.

Mathematicians and physicists had developed very dif-ferent approaches to understanding two-dimensional criti-cal phenomena. The work of Wendelin Werner has helpedto bridge the chasm between these approaches, enrichingboth fields and opening up fruitful new areas of inquiry.His spectacular work will continue to influence bothmathematics and physics in the decades to come.

BIOGRAPHICAL SKETCHBorn in 1968 in Germany, Wendelin Werner is of

French nationality. He received his PhD at theUniversity of Paris VI in 1993. He has been professor ofmathematics at the University of Paris-Sud in Orsaysince 1997. From 2001 to 2006, he was also a memberof the Institut Universitaire de France, and since 2005he has been seconded part-time to the Ecole NormaleSupérieure in Paris. Among his distinctions are theRollo Davidson Prize (1998), the EuropeanMathematical Society Prize (2000), the Fermat Prize(2001), the Jacques Herbrand Prize (2003), the LoèvePrize (2005) and the Pólya Prize (2006).

by Allyn Jackson

8 ICM 2006 MADRID SPAIN

CARLSON: “PERELMAN CUMPLE LOS REQUISITOS PARA EL PREMIO DELMILENIO”

CAO: “HAMIL TON YPERELMAN SON GRANDES. Y SON NUESTROS HÉROES”

Si quedaba alguna duda sobre quién hizo qué a lahora de demostrar la Conjetura de Poincaré, Huai-DongCao la resuelve en esta breve entrevista concedida alDaily News del ICM2006. La siguiente cuestión es:¿recibirá Grisha Perelman, como muchos han pronosti-cado, uno de los 'premios del Milenio'? La respuestapodría tenerla Jim Carlson, presidente del Instituto Clayde Matemáticas. Algo que nadie podría responder, sinembargo, es si Pereleman aceptará el premio si efecti-vamente le es concedido.

Entrevist a con Jim CarlsonPara que Perelman sea el ganador de uno de los

Problemas del Milenio se deben cumplir varios requisi-tos: que su trabajo se publique y sea revisado por unreferee, que pasen dos años desde ese momento y queel Instituto Clay, la institución que elaboró la lista de losproblemas dé por válidos esos condicionantes. JimCarlson, presidente del Instituto Clay de Matemáticas(CMI, en sus siglas en inglés) y uno de los miembros delComité Científico Asesor del la institución, habla sobreel premio, dotado con un millón de dólares.

¿El Instituto Clay considera que la conjetura dePoincaré está probada?

John Morgan en su conferencia dijo que la conjeturaestá probada por Grisha Perelman. Déjeme decir que elimpresionante avance de Perelman es un evento parauna gran celebración. Perelman basa su trabajo en elgigantes -Richard Hamilton, en particular- pero él ha idomucho más allá que cualquiera de nosotros.

¿Quiere decir eso que Perelman será el próximoPremio del Milenio, el galardón otorgado por elInstituto Clay p ara quien resolviese alguno de losdenominados "Problemas del Milenio"?

Las reglas del Instituto dicen que hay que esperar dosaños desde que se publique el trabajo y haya una acep-tación general por parte de la comunidad matemática.El período durante el cual la totalidad de la comunidadmatemática puede examinar estos artículos, ahora dis-ponibles públicamente, es crucial. Solo entonces elComité Científico tomará una decisión y hará una reco-mendación a los directores del Instituto

Que el trabajo de Perelman se haya publicadosólo en Internet, ¿constituye un problema p ara elInstituto?

La exposición detallada del trabajo de Perelman apa-rece en los trabajos de Cao-Zhu, Kleiner-Lott y Morgan-Tian y el trabajo de Gérard Besson también se publica-rá pronto. Estos trabajos, todos revisados por refereeso en proceso de ser refutado, son suficientes.

¿Entonces el premio es p ara Perelman?Él es el único que ha anunciado la prueba de la con-

jetura. La decisión de a quién otorgar el premio serátomada en dos años. Hay un procedimiento que seguirpara la toma de la decisión: la cita de un grupo deexpertos de personas no relacionadas con el InstitutoClay (en su mayoría), la consideración de su informepor parte del Comité Científico y, finalmente, la decisiónde los directores del instituto.

¿Cree usted que lo acept ará?Eso no lo puedo decir: no tengo ni idea. Nosotros pro-

cederemos exactamente igual que el ICM cuando le hapremiado con la Medalla Fields. Creo que John Ball y elcomité de las medallas Fields lo han hecho perfectamen-te. Ellos le otorgaron la medalla basándose en sus logros,sin pensar en la posible reacción de Perelman. Si elInstituto Clay decide ofrecerle el Premio del Milenio aPerelman, seguirá la misma filosofía.

El Instituto Clay (Massachusetts, EE.UU.) fue fundadoen 1999 por el empresario bostoniano Landon T. Clay.En el año 2000 el CMI anunció la creación de una seriede premios: un millón de dólares para quien resolviesealguno de los siete problemas que el grupo de expertosconvocado al efecto proclamó como los "Problemas delmilenio", considerados los más importantes y elusivosdel inicio del siglo. La lista de los "Problemas del mile-nio" fue elaborada por el Comité Científico Asesor delCMI, tras consultar a algunos de los más prestigiososmatemáticos de todo el mundo.

Entrevist a con Huai-Dong CaoUsted ha est ado en medio de la torment a mediáti -

ca que llegó incluso a anunciar que usted había pro -bado finalmente la conjetura de Poincaré. ¿Cómo havivido este proceso?

A mí me gustan las matemáticas y me encanta traba-jar en fluidos geométricos, en particular el flujo de Ricci.Desafortunadamente la atención de ciertos mediosparece haber estado mucho más interesada en hablarde asuntos ajenos a las matemáticas. A mí sólo me pre-ocupa la parte matemática.

La controversia ha est ado centrada en 'quién hizoqué'. Por ejemplo, se ha cit ado al professor Shing-Tung diciendo que Perelman contribuyó a la resolu -ción en menos de un 25%, mientras que usted lohizo en un 30%. ¿Qué opina al respecto?

En mi trabajo con Zhu detallamos qué ha hecho cadauno. Hamilton y Perelman han realizado los trabajosfundamentales más importantes. ¡Ellos son los gigantesy nuestros heroes! Para mí no hay ninguna duda de quePerelman se merece la Medalla Fields. Nosotros sólohemos seguido el camino abierto por Hamilton yPerelman y explicamos los detalles. Espero que cual-quiera que lea nuestro trabajo esté de acuerdo en quelo hemos planteado de una forma justa.

Desafortunadamente, la prensa ha distorsionado elpunto de vista del professor Yau. Por lo que sé, él nuncase asignó ningún porcentaje y no está de acuerdo conese tipo de distribuciones. Es más; me ha enseñadovarios correos electrónicos que envió a periodistas, en

Daily News (versión en español)

9

los que dejaba claro su desacuerdo con esas distribu-ciones. Quisiera también añadir que el professor Yausiente un enorme respeto por la aportación dePerelman. En más de una ocasión, incluida su conver-sación con los periodistas del New Yorker, dijo quePerelman se merecía la Medalla Fields.

CON VER EL PÓSTER COLGADO Y A GENTE MIRÁNDOLO YA ME SIENTO PREMIADO

Es para mi un gran honor presentar un póster en elInternational Congress of Mathematicians. Quiero apro-vechar esta oportunidad para agradecer al ComitéOrganizador, y en particular al comité encargado delvoluntariado, el habernos brindado esta posibilidad.

Todo empezó cuando, en un curso de doctorado, miscompañeros Pablo Centella Barrio y Elena GálvezMoreno y yo resolvimos un problema que decidimosenviar a publicar. Acto seguido, nos llegó el e-mail deEmilio Bujalance informándonos a todos los voluntariosde que podíamos dar un short communication o presen-tar un póster. Ninguno de nosotros había hecho un pós-ter antes (¡creo que ni siquiera sabíamos lo que era!)pero, tras informarnos un poco, decidimos intentarlo.Un póster es simplemente una forma de dar a conocertus resultados más importantes, sin entrar en detalle, deuna manera divertida y vistosa, evitando el mal trago detener que dar una charla a quienes les suponga un pro-blema hablar en público. En nuestro caso, el resultadoes bastante llamativo por algunos dibujos de superficiesgeneradas por ordenador, y porque, por suerte, el gustoartístico de mis compañeros es mejor que el mío. Debodecir que tampoco supuso un trabajo excesivo darleforma al póster. Básicamente (y todos los que hayanhecho pósters lo sabrán) hicimos el clásico copiar ypegar. Pero, en cuanto a maquetación y presentación,sí que estuvimos bastante tiempo decidiendo colores,fondos, cuadros, etc.

Varios meses después, aquí me encuentro, a punto depresentar por primera vez un póster. ¡Y nada más y nadamenos que en el ICM! Por supuesto, con la emoción aña-dida de entrar en concurso. El póster estará colgadodesde el día 25 hasta el 29, pero todos tenemos una horaconcreta (de 17:00 a 18:00 del sábado 26 en mi caso) enla que tenemos que estar ahí, atendiendo a los interesa-dos y respondiendo preguntas. O recibiendo críticas(espero que no). En principio, no me supone ningún pro-blema. Al menos no hasta que alguien me pregunte algoque no sepa contestar; pero bueno, eso es parte deljuego.Sinceramente estoy bastante ilusionado, que nonervioso, con todo esto. Siempre es gratificante ver algoen lo que has invertido bastante trabajo colgado de esamanera y al alcance de todos.Os invito a todos a pasarosa ver nuestro póster “Tetraharmonic Bézier surfaces” y,por supuesto, los de todos los demás participantes. Estoyconvencido de que el nivel de los pósters en ICM no nosdefraudará a ninguno. Por último, quiero dar mi más sin-cera enhorabuena a los ganadores del concurso. Y nadamás, a seguir disfrutando del congreso.

Raúl Oset Sinha, Alumno de doctorado en laUniversitat de València, Voluntario del ICM

ARTE FRACTALLas ecuaciones son una manera de describir la reali-

dad, pero algunas ecuaciones pueden usarse comogeneradoras de mundos artificiales. De las ecuacionesque describen conjuntos fractales, por ejemplo, puedenemanar paisajes tan sugerentes como los creados porel mejor de los pinceles. La mejor prueba: la Exposiciónde Arte Fractal del Congreso Internacional delMatemáticos ICM2006, que se puede ver la sede delICM2006 y en el Centro Cultural Conde Duque (Madrid).Muchas de las obras expuestas proceden del ConcursoInternacional de Arte Fractal ICM2006 BenoitMandelbrot, considerado el padre del arte factral.

No hace falta recurrir a una descripción matemática -bastante compleja- de lo que son los fractales. Paratener una idea intuitiva bastaría decir que son estructu-ras que "al ser observadas en una pequeña porciónmantienen un aspecto similar, aunque no necesaria-mente idéntico, al que presentan al ser observadas deforma completa", explica Javier Barrallo, artista fractal.Algunos ejemplos: un árbol, con sus ramas; una coliflor,aparentemente formada por un sinfín de minicolifloresunidas; el litoral de un país... El ejemplo de la línea decosta sirve para explicar otra de las propiedades de losfractales, el hecho de que por mucho que disminuya laescala a la que son observados -por mucho que sehaga 'zoom' en ellos-- siempre mantendrán el mismoaspecto, hasta el infinito. Aunque el litoral de un país noes, obviamente, infinito -los fractales 'auténticos' sonuna idealización matemática--, el efecto del fenómenofractal se puede apreciar en la 'paradoja de la costa',que sí es del todo real. Al medir una costa, o cualquiersuperficie rugosa, el resultado variará en función de laprecisión a que se aspire: si se tiene en cuenta el con-torno de las bahías, de las rocas, de los granos dearena... la costa será teóricamente más y más larga, yen un fractal ideal llegaría a hacerse infinita.

Cada fractal parte de una fórmula matemática -las pri-meras fórmulas fractales fueron descritas hace más deun siglo; hoy hay centenares-. Y el ordenador es indis-pensable, como explica Barrallo: "Una pequeña imagen,por ejemplo de 640x480 píxeles, contiene 307.200 pun-tos que deben ser calculados. Cada uno de estos pun-tos puede requerir ser calculado por la fórmula quedetermina el fractal unas 1.000 veces. Es decir, cadafórmula ha de ser calculada más de 300 millones deveces. Una vez armados de fórmula y computadora,hay que recurrir a la iteración. Se trata de "calcular unafórmula repetidas veces a partir de un valor inicial. Unavez calculada la fórmula por primera vez, tomamos elvalor resultante y volvemos a introducirlo en la fórmula.El nuevo resultado se vuelve a calcular y así sucesiva-mente", dice Barrallo. En el caso de los fractales el valorinicial tiene que ver con la posición del punto en el plano(el píxel en la pantalla). Luego se asignan colores enfunción del valor de cada punto. El hecho de que elcomportamiento de dos puntos muy próximos pueda serradicalmente opuesto -uno divergiendo hacia el infinitoy otro convergiendo hacia un valor dado- es "lo quehace fascinante la exploración fractal", dice Barrallo. Ytambién lo que permite la espléndida explosión de for-mas y colores en la imagen.

10 ICM 2006 MADRID SPAIN

QUIÉN ES QUIÉN EN ELCOMITÉ EJECUTIVO ESPAÑOL

- La separación de ciencias y letras es puro mito. Noson incompatibles. Antonio Durán (Córdoba, 1962) lodemuestra y lo hace teorema. Escritor (La luna denisán, Editorial Debate, entre otras novelas) y catedrá-tico de análisis matemático de la Universidad deSevilla, tiene como intereses de estudio la teoría deaproximación y funciones especiales. De esteCongreso extrae muchas cosas buenas, pero se enor-gullece en particular, desde su cargo como vocal deactividades culturales del Comité, del resultado obteni-do, fruto del trabajo del equipo. "Me siento especial-mente satisfecho de la acogida que los medios handado al Congreso y el interés del público en general porlo que hacíamos en él".

- Tiene en la cabeza el programa científico delICM2006. Marta Sanz-Solé (Barcelona, 1952) es cate-drática de matemáticas en la Universidad de Barcelona.Especialista en análisis estocástico y presidenta delcomité local de Programa. Desde que comenzó, lo másemocionante ha sido el trato con los conferenciantes einvitados porque, dice, "se aprecian las diferentesmaneras de trabajar, los caracteres y, además, es unhonor tratar con tantos matemáticos de prestigio",comenta. También subraya el entusiasmo por parte delos participantes, "la alta asistencia a las sesiones y loscomentarios positivos sobre el programa y la organiza-ción me llenan de satisfacción".

- La historia de los ICM no caerá en el olvido: tiene uncronista, Guillermo Curbera (Madrid, 1961). Fue dele-gado estudiantil (Madrid 1979 y Sevilla 1985) y actual-mente es profesor titular de análisis matemático en laUniversidad de Sevilla. Su objeto de estudio son losespacios de funciones, medidas vectoriales, operadorescon núcleo y espacios de Sobolev. Cuando habla delCongreso es rotundo: "Lo que más me enorgullece esque exista". El orgullo se extiende a la exposición queha montado tras dos años de intenso trabajo y que esuna de las estrellas del Congreso, El ICM a través de laHistoria. Respecto a la repercusión del evento, señalaque está contento de que, por una vez al año, los mate-máticos sean los protagonistas, "porque es bueno parala profesión y porque puede animar a los estudiantes aacercarse a esta ciencia".

- Es, probablemente, el más polifacético del grupo. Nosólo aporta su toque de humor gráfico al Daily News;también, si es preciso, hace traductor en las ruedas deprensa del Congreso. Y, desde el punto de vista mate-mático, Alberto Ibort (Huesca, 1958) es, simplemente,un todoterreno. Catedrático de matemática aplicada dela Universidad Carlos III de Madrid, siempre ha intenta-do tener una visión amplia de la ciencia y por ello hacombinado la física teórica con la matemática pura yaplicada. Entre sus intereses se cuentan la relaciónentre la computación y la matemática pura, además delpapel que juega o puede jugar la geometría y la topolo-gía en la computación. Lo que más le gusta delCongreso es la vitalidad de la comunidad matemática,

que se refleja en todos los eventos, sesiones repletas,visitantes continuos a las exposiciones… Resalta tam-bién "la interacción que se está produciendo con losmedios, que es constructiva y creativa. Así, aprende-mos unos de otros y mejoramos".

ASPECTOS DESTACADOS DEL TRABAJO DE WERNER

El trabajo de Wendelin Werner y sus colaboradoresrepresenta una de las interacciones más emocionantesy fructíferas entre las matemáticas y la física de los últi-mos tiempos. La investigación de Werner ha desarrolla-do un nuevo marco conceptual para entender fenóme-nos críticos de sistemas físicos, y ha puesto en eviden-cia nuevos aspectos geométricos antes desconocidos.Las ideas teóricas que emergen en este trabajo, quecombina teoría de la probabilidad e ideas de análisiscomplejo clásico, han tenido un gran impacto tanto enmatemáticas como en física, y tienen conexiones poten-ciales con una amplia variedad de aplicaciones.

Una de las motivaciones del trabajo de WendelinWerner es la física estadística, área en que la teoría deprobabilidad es usada para analizar el comportamientoa gran escala de sistemas complejos, integrados pormuchas partículas. Un ejemplo tipo de un sistema com-plejo es el de un gas: aunque sería imposible conocer laposición de cada molécula de aire en una habitación, lafísica estadística dice que es muy improbable que todaslas moléculas acaben en un rincón de la habitación.Estos sistemas pueden mostrar transiciones de fase quemarcan un cambio repentino en su comportamientomicroscópico. Por ejemplo, cuando el agua hierve seproduce una transición de fase de líquido a gas. Otroejemplo clásico de transición de fase es la magnetiza-ción espontánea del hierro, que depende de la tempera-tura. En estos puntos de transición de fase el sistemapuede exhibir los llamados 'fenómenos críticos'. Puedenparecer aleatorios a cualquier escala (y en particular anivel macroscópico), y se convierten en "invariantes deescala", lo que significa que su comportamiento generalaparenta ser estadísticamente el mismo a cualquierescala. Estos fenómenos críticos son muy complejos, yaún se está lejos de entenderlos completamente.

En 1982 el físico Kenneth G. Wilson recibió el premioNobel por sus estudios sobre los fenómenos críticos, quecontribuyeron a comprender la 'universalidad'. Muchossistemas físicos diferentes se comportan de la mismamanera a medida que se acercan a los puntos críticos.Este comportamiento está descrito por funciones en lasque una cantidad (por ejemplo la diferencia entre la tem-peratura real y la crítica) es elevada a un exponente, lla-mado un "exponente crítico" del sistema. Los físicos hanconjeturado que estos exponentes son universales en elsentido de que dependen sólo de determinadas caracte-rísticas cualitativas del sistema, y no de sus detallesmicroscópicos. Aunque los sistemas en que Wilson esta-ba interesado eran sobre todo de tres y cuatro dimensio-nes, en dos dimensiones se da el mismo fenómeno.Durante los años ochenta y noventa los físicos hicierongrandes esfuerzos por desarrollar la teoría conforme decampos, que proporciona una aproximación al estudio delos fenómenos críticos de dos dimensiones. Pero esta

11

aproximación era difícil de entender de una maneramatemática rigurosa, y no proporcionaba una imagengeométrica de cómo se comportaban los sistemas. Ungran logro de Werner y sus colaboradores GregoryLawler y Oded Schramm ha sido el desarrollo de unanueva aproximación a los fenómenos críticos de dosdimensiones matemáticamente riguroso y que proporcio-na una imagen geométrica directa de sistemas cerca de,y en, sus puntos críticos.

La percolación es un modelo que capta el comporta-miento básico de, por ejemplo, un gas filtrándose a tra-vés de un medio aleatorio. Este medio podría ser unared horizontal de tuberías en las que, con una cierta pro-babilidad, cada tubería está abierta o bloqueada. Otroejemplo es el comportamiento de contaminantes en unacuífero. Uno quisiera responder a cuestiones como¿qué aspecto tiene el conjunto de sitios contaminados?Los físicos y los matemáticos estudian modelos esque-máticos de percolación como el siguiente. Primero, ima-gine un plano cubierto de losetas hexagonales. Con unamoneda (posiblemente trucada) lanzada al aire se deci-de si un hexágono es blanco o negro, de forma que paracualquier hexágono dado, la probabilidad de ser decolor negro es 'p' y la probabilidad de ser de color blan-co es '1 -p'. Si designamos un punto en el plano comoel origen, podemos preguntar, ¿Qué partes del planoestán conectadas al origen por una franja de hexágonosnegra? Este conjunto se denomina 'cluster' que contie-ne el origen. Si p es menor que ½, habrá menos hexá-gonos negros que blancos, y el cluster que contiene elorigen será finito. Por el contrario, si p es mayor que ½hay una probabilidad positiva de que el cluster que con-tiene el origen sea infinito.

El sistema presenta una transición de fase en el valorcrítico p= 1/2. Este valor crítico corresponde a la situa-ción en la que uno lanza una moneda no trucada paraescoger el color de cada hexágono. En este caso sepuede probar que todos los clusters son finitos y que encualquier porción de la superficie que uno escoja mirarencontrará, con gran probabilidad, clusters de tamañocomparable a dicha porción. La imagen adjunta repre-senta una muestra de un cluster bastante grande.

El modelo de percolación ha atraído el interés de losfísicos teóricos, que han usado varias técnicas no rigu-rosas para predecir aspectos de su comportamientocrítico. En particular, hace unos quince años, el físicoJohn Cardy usó la teoría conforme de campos parapredecir algunas propiedades a gran escala de perco-lación en su punto crítico. Werner y sus colaboradoresLawler y Schramm estudiaron los objetos continuosque aparecen cuando se toma el límite de gran escala-esto es, cuando se permite que el tamaño de los hexá-gonos sea más y más pequeño--. Obtuvieron muchasde las propiedades de estos objetos, como por ejemplola dimensión fractal de los bordes de los clusters. Estetrabajo, combinado con los resultados de StanislavSmirnov en 2001 sobre el modelo de percolación y contrabajos anteriores de Harry Kesten, condujo a laobtención completa de los exponentes críticos de estemodelo particular.

Otro modelo de dos dimensiones es el movimientobrowniano en el plano, que puede ser visto como el lími-te a gran escala del paseo aleatorio discreto. El paseo

aleatorio discreto describe la trayectoria de una partícu-la que escoge aleatoriamente una nueva dirección encada unidad de tiempo. La geometría de las trayecto-rias del movimiento browniano en el plano es bastantecomplicada. En 1982 Benoit Mandelbrot conjeturó quela dimensión fractal de la frontera exterior de la trayec-toria de un movimiento browniano en el plano (la fron-tera exterior del conjunto azul en la imagen adjunta) es4/3. Resolver esta conjetura parecía fuera del alcancede las técnicas probabilísticas clásicas. Lawler,Schramm, y Werner la demostraron probando en pri-mer lugar que la frontera exterior de las trayectoriasbrownianas y las fronteras exteriores de los clusters depercolación continuos son similares; después calcula-ron su dimensión común usando una construccióndinámica de los clusters de percolación continuos. Conla misma estrategia también obtuvieron los valores delos muy relacionados 'exponentes de intersección' parael movimiento browniano y el paseo aleatorio discreto,que habían sido conjeturados por los físicos B.Duplantier y K.-H. Kwon (uno de estos exponentes deintersección describe la probabilidad de que las trayec-torias de dos caminantes no se crucen durante un largoperiodo de tiempo). Otro trabajo posterior de Wernermostró simetrías adicionales de estas fronteras exterio-res de los bucles brownianos.

Otro resultado de Wendelin Werner y de sus colabo-radores es la demostración de la "invariancia confor-me" de algunos modelos de dos dimensiones. La inva-riancia conforme es una propiedad parecida a la inva-riancia de escala, aunque más sutil y general. Está enla raíz de la definición de los objetos continuos queWerner ha estado estudiando. Sin precisar demasiado,se dice que un objeto aleatorio bi-dimensional es uninvariante conforme si su distorsión por transformacio-nes que preservan el ángulo, es decir, por las transfor-maciones que en el análisis complejo se denominanaplicaciones conformes, tiene la misma ley de proba-bilidad que el objeto inicial.

Suponer que muchos sistemas críticos de dosdimensiones son invariantes conformes es uno de lospuntos de partida de la teoría conforme de campos. Elresultado de Smirnov mencionado anteriormentedemostró la invariancia conforme para la percolación.Werner y sus colaboradores demostraron la invarian-cia conforme para dos modelos clásicos de dos dimen-siones, el paseo aleatorio con bucles suprimidos (eninglés, loop-erased random walk) y el modelo relacio-nado con el mismo denominado árbol de expansiónuniforme (en inglés uniform spanning tree), y descri-bieron su comportamiento límite normalizado. Uno delos grandes desafíos actuales en esta área es demos-trar la invariancia conforme para otros sistemas de dosdimensiones.

Los matemáticos y los físicos han desarrollado abor-dajes muy distintos para entender los fenómenos críti-cos de dos dimensiones. El trabajo de Wendelin Wernerha contribuido a reducir la brecha entre estas estrate-gias, enriqueciendo ambos campos y abriendo nuevasy fructíferas áreas de investigación. Su espectacular tra-bajo seguirá influenciando tanto las matemáticas comola física en las décadas venideras.

Allyn Jackson

PROGRAMME CHANGES AND NOTES OF INTEREST

NOTES OF INTERESTThe lectures by A. Okounkov and W. Werner (Section

13) and J. Kleinberg (Section 15) have been rescheduledas Prize Winner lectures and therefore cancelled as invi-ted section.

TUE 29, 14:00-14:45 ILA, Lecture by a Fields MedalistAndrei Okounkov, Princeton Univ., Princeton, USAEnumerative geometry of curves in threefolds

Chair: Percy Deift

TUE 29, 15:00-15:45 Andrei Okounkov,(Cancelled)TUE 29, 15:00-15:45 Jon M. Kleinberg, (Cancelled)TUE 29, 16:00-16:45 Wendelin Werner, (Cancelled)

SHORT COMMUNICA TIONS (new)TUE 29, 15:35-15:55 SC 08 L401Mikhail Korobkov, Sobolev Institute of Mathematics,

Novosibirsk, Russia FederationOn a Sard type theorem for C1-smooth functions of twovariablesChair: Hélène Airault

TUE 29, 16:55-17:15 SC 08 L401Mingzhe Gao, Normal College of Jishou University,

Jishou, ChinaA new Hardy-Hilbert’s type inequality for double seriesand its application

TUE 29, 17:35-17:55 SC 16 R403Emil Catinas, ``T. Popoviciu’’ Institute of Numerical

Analysis, Cluj-Napoca, RomaniaNew results in local convergence of iterative methods fornonlinear systems and fixed point problemsChair: Sergio Amat

SHORT COMMUNICA TIONS (cancelled)TUE 29, 16:05-16:25 SC 08 L401Andrei Pokrovskii, Institute of Mathematics of the

National Academy of Sciences of Ukraine, Kiev, UkraineStrong unicity in uniform approximations with a sign-sen-sitive weightChair: Dragan Vukotic

TUE 29, 16:05-16:25 SC 16 R403Mauricio Andrés Barrientos Barría, University of

Antofagasta, Antofagasta, ChileConvergence of adaptive FEM-BEM for linear exteriorproblems via DtN mappingsChair: Juan M. Viaño

SHORT COMMUNICA TIONS (changes respect tothe printed program)

TUE 29 15:35-15:55 SC 06 R204, Agostino Prastaro àrescheduled WED 30, 17:35-17:55 R204

TUE 29 16:05-16:25 SC 04 R103, Brend Kreussler(already done)

TUE 29 16:05-16:25 SC 06 R203, Takashi Tsuboi (alre-ady done)

TUE 29 16:55-17:15 SC 08 l402, Manav Das (alreadydone)

ANNOUNCEMENTSTuesday, August 29, 13:00-13:45, Auditorium BPrize Winners of the PosterCompetition at the ICM

2006In an informal ceremony, the jury of the competition

will make public its decision on the names of the prizewinners in each scientific section. Awardees will receivetheir diplomas and monetary prize.

INFORMAL SEMINARS 29TH AUGUST16:00 – 16:40 Computation of connection coefficients,

Room L205Mijiddorj. R., Departement of Programming,Mongolian

State University of Education, r_mjddrj@yahoo.comSUMMARY: based on some properties of connection

coefficients we propose an algorithm for calculation one,two and three-factor connection coefficients.Asymptotical formulas for one and two factor connectioncoefficients were also presented. These are illustratedwith numerical examples.

18:00 – 18:40 Solving Logic Problems using someSymbolic Dynamics, Room L205

Paseman Gerhard, Sunwave Communications (USA),grpadmin@sunwave.com

SUMMARY: Certain second-order statements (hyperi-dentities) can be represented as first order theories T.Sometimes T is equivalent to a finite theory U. We showhow Symbolic Dynamics answers when there is a logicalequivalence, and ask if Symbolic Dynamics can help set-tle some conjectures related to computability of theseequivalences.

OPENING OF KEIZO USHIO´S SCULPTURETuesday 29th at 16:00 at the main entrance of the

Palacio Municipal de Congresos.

I C M 2 0 0 6Daily News

Madrid, August 29th 2006