Pro ceedings of the seventh EWM meeting
EUROPEAN WOMEN IN MATHEMATICS
Universidad Complutense de Madrid
September
Contents
Preface
Europ ean Women in Mathematics
EWM on the Web
I History Since Luminy
EWM since its fth meeting in Luminy
Brief rep ort on the sixth meeting of EWM in Warsaw
II MADRID Organization of EWM
Decisions taken during the general assembly
THE MATHEMATICAL PART
The Mathematical part of EWM Meetings
III Holomorphic Dynamics
A short course organized by Caroline Series
Holomorphic dynamical systems in the complex plane
An introduction
Bo dil Branner
The theory of p olynomial like mappings
The imp ortance of quadratic p olynomials
N uriaFagella
Lo cal prop erties of the Mandelbrot set M
Similarities b etween M and Julia sets
Tan Lei
IV Classi cation in Algebraic Geometry
A Short Course organized by Rosa M Mir o Roigand Raquel Mallavibarrena
On some notions in algebraic geometry
Margarida Mendes Lop es
On classi cation of algebraic space curves
liaison and mo dules
Mireille Martin Deschamps
On classi cation of pro jective varieties of small co dimension
Emilia Mezzetti
V Mathematical Physics
Session organised by Sylvie Paycha
Mirror Symmetry
Claire Voisin written by Sylvie Paycha and Alice Rogers
Mo delling the randomness in physics
Flora Koukiou
Sup er mo duli space
Alice Rogers
Universal measuring coalgebras
The p oints of the matter
Marjorie Batchelor
VI Mo duli Spaces
An Interdisciplinary Workshop organized by Sylvie Paycha
Mo duli spaces and conformal dynamics
Caroline Series
Mo duli spaces and path integrals
in Quantum Field Theory
Sylvie Paycha
Notes on mo duli spaces in
Algebraic Geometry
Rosa M Mir o Roig
An example of mo duli spaces in number theory
Laura Fainsilber
VI I Short Talks
The multidimensional Riemann Hilb ert problem generalized Knizhnik
Zamolo dchikov equations and applications
Valentina A Golub eva
Quadratic and hermitian forms over rings
Laura Fainsilber
On Sampling plans for insp ection by variables
Vera I Pagurova
VI I I Evaluation
Evaluation of the mathematical asp ects of the meeting
OTHER TOPICS
In b etween meetings the everyday life of EWM
Renormalisation in Mathematics and Physics
Fe mail E mail discussion
Mathematical studies at Europ ean universities
IX Family versus Career
Family versus Career in Women Mathematicians
COMMITTEES COORDINATORS AND PARTICIPANTS
Committee Members
Co ordinators
List of Participants in EWM
Preface to the printed edition
This volume contains records of the mathematical activities and some of the other activities
which to ok place during the seventh general meeting of Europ ean Women in Mathematics at
Universidad Complutense de Madrid September
The meeting was attended by participants from countries Denmark England Finland
France Germany Italy Norway Portugal Rumania Russia Spain Sweden and Switzer
land
The main organizers were by far Capi Corrales Ro drig a nez and Raquel Mallavibarrena
from Madrid They were assisted by an organizing committee consisting of Bo dil Branner
Denmark Isab el Lab ouriau Portugal Rosa Maria Mir o Roig Spain Marjatta N a at anen
Finland Sylvie Paycha France Caroline Series England Laura Tedeschini Lalli Italy
In organizing the meeting we build on earlier exp eriences In particular the work done by
Eva Bayer Mich eleAudin and Catherine Goldstein all from France around the fth EWM
meeting in Luminy in December has b een a constant source of inspiration
We are very grateful for the nancial supp ort we received from the spanish Instituto de la
Mujer and the Ministerio de Educaci on y Ciencia b oth for the Madrid meeting and for the
publication of these Pro ceedings
The logo of the EWM meeting and the t shirt pattern shown on the front page was
designed by DODOT
These Pro ceedings were edited by Bo dil Branner and N uriaFagella We thank Christian
Mannes for setting up the TeX style we have used
The photos were taken by Marketa Novak
We wish to express our thanks to all participants for making the Madrid meeting a success
and to all who contributed in writing to this volume Esp ecially we thank Capi and Raquel
for making it all p ossible
February
Bo dil Branner and N uriaFagella
Addendum to the electronic edition
The original version of these pro ceedings was printed at the Universitat de Barcelona
with Dep Legal L Impreso Poblagra c S L Av Estacion s n Pobla de Segur
The main di erence b etween the electronic edition and the printed one is that some mis
prints have b een corrected and that the logo designed by DODOT the photos and the gures
contained in the mathematical pap ers do not app ear here
If you wish to get a copy of the printed version please contact your regional co ordinator
and ask if some are still available
Europ ean Women in Mathematics
EWM is an a liation of women b ound by a common interest in the p osition of women in
mathematics Our purp oses are
To encourage women to take up and continue their studies in mathematics
To supp ort women with or desiring careers in research in mathematics or mathematics
related elds
To provide a meeting place for these women
To foster international scienti c communication among women and men in the mathemat
ical community
To co op erate with groups and organizations in Europ e and elsewhere with similar goals
Our organization was conceived at the International Congress of Mathematicians in Berke
ley August as a result of a panel discussion organized by the Asso ciation for Women in
Mathematics in which several Europ ean women mathematicians to ok part There have since
b een six Europ ean meetings in Paris in Cop enhagen in Warwick England
in Lisb on in Marseilles in Warsaw and in Madrid The
next meeting will b e in The place of the meeting will b e announced later
At the time of writing there are participating members in the following countries
Belgium Czech Republic Denmark Estonia Finland France Germany Greece Italy
Latvia Lithuania the Netherlands Norway Poland Portugal Romania Russia Spain
Sweden Switzerland Turkey Ukraine and the United Kingdom contacts in Albania as
well as with several non Europ ean countries Activities and publicity within each country
are organized by regional co ordinators Each country or region is free to form its own
regional or national organization taking whatever organizational or legal form is appropriate
to the lo cal circumstances Such an organization Femmes et Mathematiques already exists
in France Other members are encouraged to consider the p ossibility of forming such lo cal
regional or national groups themselves There is also an e mail network
For further information contact
The secretary of EWM Riitta Ulmanen
Department of Mathematics
P O Box Yliopistonkatu
FIN University of Helsinki Finland
e mail ulmanen sophie hel sinki
Tel Fax
For details of the e mail network contact
sarah rees newcastle ac uk September
EWM on the Web
EWM now has a page on the Web There are two adresses although b oth contain the
same information One in Helsinki
http www math helsinki EWM
and one in Austria
http www risc uni linz ac at misc info ewm EWM html
The Web account has b een set up by Olga Caprotti Giovanna Ro da Ileana Tomuta and
Daniela Vasaru at
RISC Linz
Research Institute for Symbolic Computation
Johannes Kepler University
A Linz Austria
They can b e reached at the EWM Web account
ewm risc uni linz ac at
or at their p ersonal accounts FirstName LastName risc uni linz ac at
History Since Luminy
EWM since its fth meeting in Luminy
Caroline Series
University of Warwick England
The fth EWM meeting was held in Luminy France December th In the
lengthy rep ort which was published of the Luminy meeting I wrote a brief history of the rst
ve years of EWM Since this is the rst ma jor EWM rep ort to b e published since that time
I have b een asked to write a few words to record what has happ ened in the interim
The Sixth General Meeting Warsaw June th th
The sixth General EWM Meeting to ok place at the Technical University in Warsaw from
June th th The main organiser was Anna Romanowska The meeting was attended
by ab out participants from Europ ean countries The conference featured lectures in
pure and applied mathematics an interesting session on creativity and several business ses
sions There was a rep ort from the Round Table on women in Mathematics at the Europ ean
Congress in We learned that the prop ortion of women among do ctorates in mathe
matics is highest in eastern Europ ean and Mediterranean countries Greece Italy and Spain
were represented at the conference and lowest in Scandinavia the Netherlands the UK and
Germany There has b een much sp eculation ab out this somewhat counter intuitive situation
but it seems more pro ductive to concentrate on how to get more women into mathematics
in the northern countries Mary Gray gave a talk on the history aims and activities of the
AWM and also gave helpful advice ab out the draft statutes of EWM
Legalisation of EWM and establishment of the Helsinki
O ce
One of our early failures as an organisation which I mentioned in the rep ort of the Luminy
meeting was our lo ose structure in which everything dep ended on just one p erson the inter
national co ordinator By the time of the Luminy meeting we had decided that it was time
to set in place some more formal organisation and that we wanted EWM to b e a legal b o dy
During the Luminy meeting we did a lot of work drafting our basic organisational structure
and statutes and a small committee consisting of Marjatta N a at anen Riitta Ulmanen and
myself was formed to carry this work further
In the course of discussing our statutes we were forced to consider the structure and
function of EWM in great detail We wanted to make EWM work by consensus but at
the same time we had learned from exp erience that it is vital to have a core of central
p eople resp onsible for the continuity and smo oth functioning of the organisation This we
achieved by setting up a standing committee led by a convenor to deal with executive matters
and in particular in planning the next meeting regional co ordinators to deal with members
and circulate information in their regions and international co ordinators to watch over and
circulate information among the regions We also established membership pro cedures and a
metho d of collecting dues None of this is easy in an international organisation with members
living in many di erent circumstances At the Warsaw meeting the General Assembly accepted with some changes the statute
prepared by our committee in consultation with Finnish lawyers It was here that the ne
details of the structure of membership and fees were hammered out It was decided to for
malise membership and start collecting dues from to pay for the work in Helsinki and our
international activities There are three rates to allow for the many di erent circumstances
in which we live The regional co ordinators are resp onsible for collecting this money in lo cal
currency and sending it to the general EWM bank account which is held in Helsinki
The establishment of EWM as a legal b o dy was nalised on December nd This
was surely an imp ortant marker in the history of EWM
One reason for the choice of Helsinki as the legal seat of EWM was that Helsinki was
already the seat of the Europ ean Mathematical So ciety EMS We were very fortunate that
Riitta Ulmanen agreed to b e our general secretary Riitta is librarian in the Mathematics
Department in the University of Helsinki We made an application form for EWM mem
b ership which regional co ordinators circulate and collect yearly The o ce in Helsinki is an
information centre and it collects and keeps constantly up dated information ab out members
nances committees and co ordinators Riitta also answers enquiries ab out EWM and mails
information to members usually via the regional co ordinators For a scattered organisation
like ours it is crucial to have a central place where everything is kept together For the last
two years Marjatta has obtained funding from the Finnish Ministry of Education and the
Finish Cultural Fundation to supp ort Riitta s work
The address and telphone numbers of the EWM Helsinki o ce have changed recently and
the new address is
EWM O ce Riitta Ulmanen Secretary
Department of Mathematics PO Box
Yliopistonkatu FIN
University of Helsinki Finland
Tel Fax
e mail ulmanen sophie helsi nki
The Europ ean Mathematical So ciety Committee on Women
and Mathematics
The Europ ean Mathematical So ciety EMS was founded in Octob er Eva Bayer was in
strumental in setting up and chairing the EMS Committee on Women and Mathematics from
January Not surprisingly EWM members particularly Eva have played a prominent
r ole The committee organised a round table at the rst Europ ean Congress of Mathematics
in Paris which had ab out participants short talks and a lively and interesting
discussion
The committee made an analysis of the situation of women mathematicians in Germany
which has one of the lowest prop ortions of women to men among mathematicians in Europ e
The results together with a rep ort based on the discussions of the round table app ear in the
Pro ceedings of the ECM
In spring the EMS committee made an extensive inquiry in Switzerland ab out the
low number of women mathematicians in that country The results were discussed at the
International Congress of Mathematicians in Z urich August EWM and the EMS
committee also organised a more general discussion ab out the countries in Europ ean countries
with an unusually low prop ortion of women mathematicians
At the Zurich ICM there was a very nice No ether lecture by the distinguished Russian
woman mathematician O A Ladizhenskaya There was also a panel discussion jointly organ
ised by AWM the Canadian women mathematician s group and EWM Rep orts of these
events may b e found in our January Newsletter Number
The next EMS meeting is to b e held in Budap est in summer Kari Hag is organising
a round table on the topic Females in Mathematics in the Ib erian and Scandanavian Penin
sulars A pleasing number of women sp eakers have b een invited including Dusa McDu
who is to give a plenary session
Regional Meetings
We now have regional co ordinators in Belgium Czech Republic Denmark Estonia Finland
France Germany Greece Hungary Italy Latvia Lithuania Netherlands Norway Poland
Portugal Roumania Russia Spain Sweden Switzerland Turkey Ukraine United Kingdom
There have b een regional meetings of women mathematicians in among others France
Germany Russia Sweden and the UK Femmes et math ematiques continues to b e an imp or
tant and very active organisation in France During femmes et math ematiques organised
three general assemblies two in Paris and one in Lille On th March International
women s day there were actions in seven universities on the theme Women in mathematics
In several events are planned including a one day forum for young women mathemati
cians in Paris in January and a general assembly in March in Rennes The Russian Women
Mathematicians Asso ciation RMWA was founded at a conference in May in Suzdal
and already has more than members from more than cities of Russia and the FSU A
second International conference to ok place in Voronezh in May and the third is planned
for Volgograd in May see b elow An EWM group AIDIM asso ciazione italiana donne
in matematica has b een formed in Italy which has ab out o cial members with more
interested This group to ok part in a congress in Anacapri in where it presented its
purp oses and prop osed some topics for discussion The British group BWM organised a very
successful one day meeting in London in September attended by women from all over
the UK There is also a functioning inter uk email network
The German group is also active there is an e mail net managed from Magdeburg with
ab out participants where there is also an ftp server with information ab out EWM A
data base of women mathematicians with a Habilitation in Germany since Emmy No ether
has b een set up There have b een some lo cal meetings in there was a Women and
Mathematics meeting in Ob erwolfach organized by Catherine Bantle Basel in which a large
EWM group participated
The e mail network
For some time we have relied heavily for our communications on the email network set up by
Laura Tedeschini Lalli in Italy This is a very easy and e cient metho d of communicating and
saves much time and exp ense In Madrid Sarah Rees o ered to reorganise and administer
the network from her university in Newcastle She has now set up a new network and
hop es eventually to have subnets for each individual country or region To join mail her at sarah rees newcastle ac uk
The Newsletter
We have b een talking for a long time ab out starting a newsletter and three issues roughly
one p er year have now app eared The newsletter is b eing edited by Cathy Hobbs and Marlen
Fritsche It is distributed via the regional co ordinators and in addition is sent by email in
Tex and plain versions to all on our network You are sent a copy automatically on joining
the network otherwise copies hard or electronic can b e obtained from the Helsinki o ce
The Seventh General Meeting Madrid September th
Since the rest of this rep ort is ab out the Madrid meeting I shall not say anything here I
would just like to note that we were very pleased that Capi Corrales managed to get funding
for a planning meeting of the organising committee including ve non Spanish members
who came to Madrid in November Hosted by Capi in her spacious apartment the
committee was able to sp end an entire whole weekend from early till late discussing and
planning all asp ects of the conference This meant that the wider international committee
could take a really active and informed part in organising the meeting and were much b etter
able to supp ort the Spanish organisers throughout the pro cess This kind of supp ort for the
lo cal organisers is invaluable and ideally we should try to have such a pre meeting b efore
every large meeting
The future
One of our main aims should certainly b e to continue to develop more regional activities
These are easier and cheaper to organise than big international events and can b e attended
relatively easily by members who cannot get to the international meetings They can relate
to lo cal needs and are in p eople s own language Already a number of activities of this kind
are lined up for the coming year Besides the Budap est meeting mentioned ab ove EWM is
organising jointly with femmes et math ematiques an interdisciplinary two day workshop
on Renormalisation from June th th in Paris There will also b e a joint Franco
Russian meeting organised by femmes et math ematiques and the Russian Asso ciation for
Women Mathematicians RAWM in Marseille in December The British group BWM
is planning another one day meeting in London next September The third ma jor meeting
of the Russian Asso ciation is planned for May in Volgograd It will part of an
international forum on Problems of survival under the title Mathematics mo delling and
ecology For details contact the organiser Prof G Riznichenko riznich orgmath msk su
The German group is planning a long weekend in June
On the international level to facilitate the ow of information we are hoping to set up
an Internet page
One of our main problems is money and if we could nd some funding on a steady basis it
would take a huge burden o the organisers Funding meetings is a big problem which ideally
should b e taken care of well in advance We feel we should b e able to get more funding from
the EC but it takes hard work and p ersistence to put in applications We badly need p eople
to help in this work We also need more p eople to sign up as members and pay their dues
so that we can b e more active and have more secure ongoing supp ort for our international
work
As EWM b ecomes more established it is very pleasing to see that many p eople already
see us as an established smo othly running organisation They exp ect that EWM will b e able
to answer queries pro duce statistics organise conferences and put in place networks and
supp ort systems and opp ortunities for women to meet Of course we are very glad to b e seen
in this r ole and this is indeed exactly what our original vision was ab out but keeping it all
going is still hard work We badly need more p eople to come forward to take part in running
the organisation There are many jobs small and large to b e done Although hard work
this has many rewards as one gets to know and co op erate with women mathematicians from
all over Europ e and p erhaps it is really the b est way to nd out what our network is all
ab out
In short although there are still many problems and shortcomings in our organisation
we are spreading I b elieve that p eople are b eginning to take us for granted and that means that we have arrived to stay
Brief rep ort on the sixth meeting of EWM in Warsaw
June th th
The meeting was attended by ab out participants from Europ ean countries Den
mark Finland France Germany Greece Italy Lithuania the Netherlands Norway Poland
Portugal Russia Spain Sweden Turkey Ukraine Mary Gray from the USA came as a rep
resentative of the Asso ciation for Women in Mathematics of which she is one of the founding
members The main organiser of the meeting was Anna Romanowska of the Technical Uni
versity Warsaw
Mathematical Programme
There were mathematical talks by
Danuta Preworska Rolewicz Warsaw Di erential calculus as calculus
Janina Kotus Warsaw Fractals arising in holomorphic dynamical systems
Viviane Baladi Lyon Some recent results on random p erturbations of dynamical sys
tems
Krystyna Kup erb erg Auburn Florida The recent revival of continuum theory
Ina Kersten Bielefeld Linear algebraic groups
Zo a Adamowicz Warsaw On real closed elds
There was also a p oster session in which participants presented asp ects of their work
General Discussions
As decided at the previous EWM meeting the non mathematical theme was Creativity
organised by Coby Geijsel There were two talks
Coby Geijsel Amsterdam Images of creativity
Karin Kwast Amsterdam Creativity a case study
These were followed by discussions in small groups Other activities were a rep ort on the
Round Table on Women in Mathematics which to ok place at the Europ ean Mathematical
Congress in Paris a talk by Mary Gray on the history aims and activities of the
AWM and one by Krystyna Kup erb erg a Polish mathematician who has settled in the USA
comparing the academic environment for women mathematicians in Poland Sweden and the
USA There was also a discussion on the situation of women mathematicians in the former
so cialistic countries Moreover the programme included a talk by Vassiliki Farmaki Athens
Women mathematicians in Ancient Greece and a talk by Magdalena Jaroszewska Poznan Olga Taudsky Todd
Organisation of EWM and the General Assembly
Following the decision in Luminy to go forward with the establishment of EWM as a legal
b o dy much work had b een done mainly by Caroline Series and by Marjatta N a at anenand
Riitta Ulmanen of Helsinki in consultation with Finnish lawyers on preparing a draft of the
statutes This draft was presented to the general assembly and following detailed discussion
the essentials were accepted with some changes There will b e two categories of membership
supp orting members and full members Supp orting members can come to meetings but not
vote and men can only join as supp orting members The suggestion of this formulation was
made by Mary Gray who in addition to her long asso ciation with AWM is a lawyer as well
as a mathematician The legalisation committee was asked to prepare a nal version of the
statutes and the legalisation was completed by December nd of The legal seat of EWM
will b e in Helsinki in Finland where Riitta Ulmanen as the secretary will have an o ce for
EWM at the Department of Mathematics at University of Helsinki There was considerable
discussion on the knotty problem of membership fees and how to collect them It was decided
to start charging membership fees from after legalisation is complete with rates low
ECU standard ECU high ECU ECU equals approximately US dollar
The regional co ordinators should b e resp onsible for collecting the money and sending it or
part of it to a general account The general assembly also app ointed new co ordinators
convenors and standing committee for details see the names and committees list The new
convenor of the standing committee is Anna Romanowska and the international co ordinators
are Capi Corrales west Marketa Novak central Inna Berezowskaya and Marie Demlova
east There will b e some joint activity of AWM and EWM at the International Congress
of Mathematicians in Z urich August This is b eing organised by Cora Sadosky the
president of AWM and Eva Bayer Since the Warsaw meeting it has b een decided that the
next EWM meeting will b e planned to take place in Madrid in July with Mariemi Alonso
Madrid Capi Corrales Madrid and Rosa Maria Miro Barcelona as the main organisers
It is likely that the meeting can b e in Germany
Organising Committees
The Warsaw meeting was organised by the EWM standing committee consisting of Polyna
Agranovich Ukraine Mariemi Alonso Spain Eva Bayer France Bo dil Branner Den
mark Jacqueline Detraz France Sandra Hayes Germany Magdalena Jaroszewska Poland
Anna Romanowska Poland Barbara Roszkowska Poland and Caroline Series England
The lo cal organising committee consisted of Elzbieta Ferenstein Irmina Herburt Felicja
Okulicka Ewa Pawelec Agata Pilitowska Anna Romanowska Barbara Roszkowska and
Krystyna Twardowska The meeting was nancially supp orted by the Technical Univer
sity of Warsaw in particular the Dean of the Department of Technical Physics and Applied Mathematics and by the money left after the third EWM Meeting in Warwick
MADRID
Organization of EWM
Following the statutes of EWM a general assembly was held during this meeting The
decisions taken are valid until the next general assembly which should take place during the
next general meeting of EWM planned for in ICTP Trieste
Decisions taken during the general assembly
Riitta Ulmanen based on notes taken by her and Karin Bauer
who wished everyone The General Assembly was chaired by Marketa Novak from Sweden
present welcome to the General Assembly The General Assembly was announced in
the EWM Newsletter in February and earlier in separate announcements of the
Madrid meeting Thus the requirements for the announcement of the General Assembly
were met and the meeting was valid
Riitta Ulmanen presented an agenda which was approved as the working pro cedure of
the meeting App endix
Approving new members
Because this was the rst meeting since Europ ean Women in Mathematics b ecame an
o cial b o dy everybo dy would b e a new member
It was decided to approve everyone who had sent her application either to her regional
co ordinator or to Riitta Ulmanen or who would leave her application form at the General
Assembly meeting as a member of the EWM
Electing international co ordinators
The following p ersons were elected as the international co ordinators
East Marie Demlova Central Marketa Novak and Inna Berezowskaya
West Capi Corrales Ro driganez
Con rming regional co ordinators
Regional co ordinators were con rmed see XI I I
Electing two auditors and a deputy
Seija Kamari and Kirsi Peltonen were elected as auditors for Marja Kankaan
rinta was elected to b e the deputy auditor They all are from Finland
Con rming the nancial statement and discharging those resp onsible of liabiliti es
Marjatta N a at anenexplained brie y the nancial situation of EWM
Riitta Ulmanen read aloud the nancial statement It was con rmed by the General
Assembly and those resp onsible of liabilitie s were discharged
Cho osing the place and time for the next meeting
The General Assembly decided to have the next meeting of EWM in The month
was not set yet The Nordic countries would b e resp onsible for organizing the meeting
and it was decided that they choose the place and set the time after that It has since
b een decided that the next meeting is to b e held at ICTP Trieste the time will b e
anounced later
Also the p ossibility of Germany organizing the meeting in was discussed
Electing the Standing Committee and convenor for
According to the statutes the Standing Committee consists of members The
term of a member is four years Half of the terms will expire at the general assembly
meeting and half will continue After a lively discussion the Standing Committee was
elected as follows
From the Standing Committee for were elected
Polyna Agranovich Ukraine
Bo dil Branner Denmark
Capi Corrales Ro driganez Spain
Marjatta N a at anen Finland
Rosa Maria Miro Roig Spain
Caroline Series United Kingdom
As new members
Valentina Barucci Italy
Marie Demlova Check Republic
Laura Fainsilber France
Sylvie Paycha France
Ragni Piene Norway
An election of an additional member was left to the Standing Committee to b e made
later Emilia Mezzetti Italy has since b een included
Sylvie Paycha was elected to b e the Convenor and Capi Corrales Ro driganez to b e the
Deputy Convenor Marjatta Naatanen was app ointed Treasurer
Minutes of the previous General Assembly
EWM b ecame an o cial b o dy in December and the previous meeting was in June
in Warsaw It was decided to accept the brief rep ort made from the Warsaw
meeting as the minutes and to approve it
Deciding fees
It was decided to keep the fees as they were ECU low ECU standard and
ECU high
The question of how to collect the fees and how to send it to EWM was raised Marjatta
N a at anenexplained that every regional co ordinator collects the fees whichever way is
most convenient for her She may op en an account for that purp ose After making
deductions necessary for lo cal use she then sends the rest to the EWM account in
Finland either in her own currency or in Finnish currency
Setting up committees for sp eci c issues The decisions made app ear in the list of committee members at the end of this volume
THE MATHEMATICAL PART
The mathematical programme constituted the main part of the EWM meeting and simi
larly the mathematical pap ers form the main part of these Pro ceedings
Included is a description of the philosophy b ehind the organization of the mathematical
part edited versions of ten lectures given on the three chosen topics Holomorphic Dynamics
Algebraic Geometry and Mathematical Physics and of four contributions which formed the
basis of an interdisciplinary discussion on Mo duli Spaces abstracts of three shorter talks
and at the end an evaluation of the mathematical asp ects of the programme
The Mathematical part of EWM Meetings
Capi Corrales and Laura Tedeschini Lalli
The organization of the scienti c part of an EWM meeting is quite di erent from that of
most mathematical meetings Starting at the EWM meeting in Luminy in we decided
to exp eriment with the format trying to reach the following main goals to learn mathematics
which is new to us to learn how to transmit mathematics to learn how to discuss mathematics
with other mathematicians not necessarily sp ecialists in the same eld as we are and nally
to b e able to establish scienti c links which women isolated for a number of reasons can
refer to at any stage in their professional career We have b een using the following structure
as a mo del
Before the meeting
Step A scienti c committee chosen by the standing committee of EWM selects three
topics in mathematics Several considerations are taken into account when choosing the
topics
the topics should b e in the avantgarde of current research
the topics should involve b eautiful mathematics
the topics should try to include also branches of mathematics where historically for what
ever reasons the presence of women seems more di cult to detect
Step Once the topics are chosen the scienti c committee chooses a co ordinator for each
topic Several considerations are taken into account when choosing the co ordinators
their knowledge of the eld
their commitment to the pro ject of making the transmission of mathematics a main goal
of their work
their ability and will to work in team with others
Step The co ordinator selects the sp eakers for her topic Several considerations are taken
into account when choosing the sp eakers
their knowledge of the eld
their ability or their will to improve their ability to transmit knowledge
Step Co ordinators and sp eaker work together as a team in preparing the talks The
di erent talks form a whole and the level of di culty should b e progressive Once a sp eaker
has b een assigned a talk she is invited to give a written draft of her lecture to the co ordinator
To ensure cross eld dissemination and ab ove all understandability the co ordinator then
distributes these drafts among a few women mathematicians NOT sp ecialists in the topic
who will read them and p oint out passages where assumptions are taken for granted or
needing an example or otherwise remaining obscure etc We call the crucial function of these
professionals stupid readers or naive readers The co ordinator sends the comments of
the non sp ecialists back to the sp eakers The sp eakers make the appropriate corrections and
changes and return the text to the co ordinators who send them again to the readers for a
nal check
The lectures
Many are the questions that frame our work within the mathematical talks Here are a few
of them
how do we create an atmosphere in which the audience feels free to ask questions
how do we balance the inevitably di erent levels of knowledge ab out the topic in a general
mathematical audience
how do we balance the ow of questions with the ow of the sp eaker
how do we manage to b e understo o d by non sp ecialists without deca einating our exp osi
tions
mathematics is di cult how can we make something clear and at the same time keep its
richness depth and not hide its di culties
Common sense is a main to ol we count on but we know it is not su cient Common
sense patience and as scientists the will and inclination to exp eriment try and nd by
searching Several strategies have b een tested and as our exp erience develops so do es the
number of strategies that we see work adequately towards answering the ab ove questions
Here are a few
one or two women volunteer to concentrate to their fullest ability in the talk and ask
questions when they do not follow the sp eaker or think this is the case for many in
the audience We lab el this other crucial function planted idiots We think it works
b est if the planted idiot is actually naive in the eld Other questions are welcome as
always
the sp eaker knows ahead of time that when a question is p osed by someone in the audience
if someone else knows a more clear or direct way of answering it this p erson will sp eak
up In this way the ow and rhythm of the talk is easier to mantain and since the
sp eaker knows this might happ en she do es not feel intruded or judged when it do es
if interdisciplinary connections or other interesting discussions start taking place along the
course of a talk the co ordinator of that topic should channel it into organizing a side discussion later making sure there is a time and a space allowed for it and announced
Writing and publishing the lectures
It is our exp erience that this is the step where we should b e more cautious since mathemati
cians have the habit of writing only for sp ecialists Hence a pro cess analogue to that of step
b efore the meeting is followed each sp eaker sends a draft of the text to the co ordinator
The co ordinator should distribute this draft again among naive readers who will make
sure the text is faithful to the version and comments agreed on b efore the actual talk The
co ordinator receives the comments of the non sp ecialists and sends them back to the sp eaker
The sp eaker makes the appropriate corrections and returns the new corrected text to the
co ordinator who in turn sends them again to the readers for a nal check
Conclusion
As one can deduce from the ab ove summary the mathematical part of an EWM meeting is
conceived as a learning exp erience for ALL THE PERSONS taking part in it Ideally
everyone will learn new mathematics even the sp ecialists The advantage of sp eaking
clearly to an interdisciplinary audience of mathematicians is that such situation rarely
fails to give as fruit the bringing out of connections or p oints of view thus far unknown
to us
the sp eakers will improve their ability as lecturers and mathematical writers
everyone will improve her ability to sp eak ab out what she works on
Unfortunately it is still the case in many Europ ean universities that women are singulari
ties within the mathematical departments Frequently this has a well known inhibiting e ect
on us resulting in lack of self consciousness or defensiveness b oth particularly negative when
we start our professional path And if we are inhibited we do not sp eak ab out mathematics
and if we do not sp eak ab out mathematics we do not learn how to sp eak ab out mathematics
and the lo op traps us The vicious circle of communication well known to many creates a
steady isolation which b ecomes sterile and depressing as opp osed to the temp orary isolation
which is necessary to all creative work In fact we think many problems arise for women in
mathematical research from the di erent types of isolation communication life passages
adding to the second necessary one and making it seem unbearable
Other forms The Interdisciplinary workshops
As we went on planning this EWM meeting we came along words which seem to have di er
ent meaning in di erent branches of mathematics But often the use of the same words in
mathematics p oints to a common ro ot a core idea We think it is one of our original con
tributions to organize workshops around a word or an idea to re walk paths and rediscover
if not build common ground on b oth language and conceptual basis The rst such encounter
to ok place in Madrid on Mo duli spaces with sp eakers from algebraic geometry number
theory hyperb olic geometry and quantum eld theory In Madrid the next interdisciplinary
workshop was put forward on the words Renormalization Group It will hop efully take place in June in Paris with contributions from statistical physics quantum eld theory
markov pro cesses holomorphic dynamics and real dynamical systems These workshops are
kept more informal with several p ersons resp onsible for illustrating what they deem neces
sary to the core idea or the strength of the results that follow in their eld Everybo dy else
is welcome to pitch in in workshop style
Poster sessions
Up to now we have only once exp erienced a p oster session We think it is quite a challenge
to our creativity to rethink p oster sessions in a way that makes them a go o d communication
to ol We are working on it
Holomorphic Dynamics
A short course organized by Caroline Series
The aim of the session was to present some of the basic facts and techniques in holomorphic
dynamical systems in particular those de ned through iteration in the complex plane of
complex p olynomials or p olynomial like mappings All three talks discussed di erent asp ects
concerning dynamical b ehaviours in the dynamical plane as well as prop erties of families of
such maps represented in a parameter plane
The rst talk given by Bo dil Branner was an introduction emphasizing the classical notion
of normal families and its imp ortance in the dynamical plane and the parameter plane of
quadratic p olynomials in relation to Julia sets and resp ectively the Mandelbrot set
The second talk given by N uriaFagella fo cused on p olynomial like mappings and families
of such in particular Mandelbrot like families showing the imp ortance of p olynomials as
lo cal mo dels of more general analytic maps
The third talk given by Tan Lei discussed two examples of transfer of results from the
dynamical planes to the parameter plane namely asymptotic self similarities of Julia sets and
the Mandelbrot set and the Hausdor dimension of certain Julia sets and of the b oundary
of the Mandelbrot set
Bo dil Branner
th
Proc of the EWM
meeting Madrid
Holomorphic dynamical systems in the complex
plane
An introduction
Bo dil Branner
Technical University of Denmark Denmark
branner mat dtu dk
A short historic note
The study of complex analytic dynamics b egan at the end of the previous century The work
by Ernst Schr oder Leop old Leau Gabriel Ko enigs Lucyan B ottcher and others was fo cused
on the lo cal b ehavior of a complex analytic function also called a holomorphic function near
a xed p oint With the work of rst Arthur Caley and later Pierre Fatou and Gaston Julia
the fo cus changed from lo cal to global b ehavior
Fatou and Julia studied indep endently of each other iteration of rational functions
A rational function f z p z q z is the quotient of two p olynomials p and q which are
supp osed to b e relatively prime The degree d of f is de ned as the maximum value of the
degrees of the p olynomials p and q A rational function can b e viewed as a map f C C
where C C f g denotes the extended complex plane the Riemann sphere
Figure The Riemann sphere
A rational function is holomorphic and on the other hand any holomorphic map f C
C is a rational function If d then f is surjective in fact each p oint has d preimages
counted with multiplicity Degree d corresp onds to automorphisms of the Riemann
sphere the so called M obius transformations and degree d gives rise to interesting
dynamical systems
Bo dil Branner
Both Fatou and Julia made intensive use of the theory of normal families
which had just b een introduced and developed by Paul Montel at the time
For a detailed description of the early history see A
Only few pap ers were published on complex dynamics b etween and But the
sub ject is again a very active area of research Several new ideas and to ols were introduced
in the b eginning of the eighties Among them are computer graphics quasi conformal map
pings introduced by Dennis Sullivan p olynomial like mappings and transfer of results from
dynamical plane to parameter space introduced by Adrien Douady and John H Hubbard
In this short series of pap ers we will give examples of the ab ove ideas and to ols This
rst pap er contains basic de nitions and results There are very few pro ofs Those which are
sketched are chosen to illustrate the concept of normal families and to stress the imp ortance
of rep elling p erio dic p oints
Polynomials
In the rest of the pap er we restrict our attention to p olynomials of degree d Polynomials
are exactly those rational functions with the prop erty that f f But most
often we just think of a p olynomial as acting in the complex plane
To study a p olynomial P as a dynamical system means to study the long term b ehavior
for di erent seeds z of the sequence
n
z z P z z P z P z
n n
called the orbit of z under iteration The dynamics take place in the z plane the dynamical
plane
The goal is b oth to understand each individual dynamical system for a xed p olynomial
P and to understand how the systems change qualitatively with the p olynomial
In order to understand the dynamics of all p olynomials of degree d it is su cient to
consider monic centered p olynomials of the form
d d
P z z c z c z c
d
any p olynomial F of degree d is namely conjugate to a p olynomial of this form through a
global a ne co ordinate change z az b a In other words the following diagram is
commutative
F
C C
z az b z az b
y y
C C
P
d
We identify the set of monic centered p olynomials with the parameter space C
f c c g The p olynomials are called centered since the critical p oints are centered
d
i e
X
P
where P denotes the set of critical p oints i e P fz C j P z g
Bo dil Branner
The monic centered quadratic p olynomials are of the form
Q z z c
c
with one critical p oint All the examples we give will b e from this family of quadratic
p olynomials The family is identi ed with C the c plane also called the parameter plane it
should not b e confused with the dynamical plane
The Julia set the lled Julia set and the Fatou set
For each p olynomial P the dynamical plane is decomp osed into two complementary sets the
set of p oints with b ounded orbit and the set of p oints whose orbit tends to We denote by
K P the l led Julia set that is the set of seeds with b ounded orbit
n
K P fz C j P z as n g
We denote by A the set of seeds with orbit tending to
P
n
A fz C j P z as n g
P
The set A is called the attractive basin of Both sets are completely invariant under
P
iteration i e under b oth forward and backward iteration The common b oundary
K P A J P
P
is called the Julia set
Figure shows in black the Julia sets for di erent quadratic p olynomials Q
c
Example Consider the simplest quadratic p olynomial Q z z The lled Julia set
K Q equals D the closure of the unit disk the attractive basin A equals C n D the
Q
exterior of the closed unit disk and the Julia set J Q equals S the unit circle
The ab ove de nition of the Julia set is simple but only valid for p olynomials Many
results ab out Julia sets can only b e obtained from the classical and general de nition of Julia
sets using the concept of complex analytic normal families We therefore give this de nition
as well
Normal family Let U C b e an arbitrary domain A family F ff U C g of
i i I
analytic functions is said to b e normal if any in nite sequence of functions from F contain a
subsequence that either converges in C or tends to uniformly on each compact subset of
U
n
In complex dynamics we are interested in the families F U fP j U C g of
U n
iterates of the p olynomial in question restricted to arbitrary domains U
A p oint z C is said to b e normal if there exists a neighborho o d U of z such that the
n
family F U fP j g is normal
U n
De nition The Fatou set F P is the set of normality that is F P fz C j z normal g
and the Julia set J P is the complement of F P or the set of non normality A connected
comp onent of the op en set F P is called a Fatou component
Note that the lled Julia set is the Julia set lled with all the b ounded Fatou comp onents
Example revisited Consider again the quadratic p olynomial Q z z The family F D
is normal since each subsequence of the iterates restricted to D converges to the constant
Bo dil Branner
Figure Julia sets of the p olynomials Q for di erent values of c c
Bo dil Branner
function equal to uniformly on any compact set D fz C j jz j r g with r The
r
family F C n D is normal since each subsequence of the iterates restricted to C n D converges
to the constant function equal to uniformly on any set C n D fz C j jz j Rg with
R
R No p oint on the unit circle is normal The Fatou set is therefore C n S and the Julia
set as b efore
Montel proved a useful criterion for normality
Montel s criterion Simplest form Any family of analytic functions de ned on a domain
U taking values in the plane minus two p oints C n fa bg is normal
Generalized form Let h U C j b e two analytic functions satisfying h z
j
h z for all z U Any family of analytic functions de ned on U with values at any z U
which di er from h z and h z is normal
Note that it follows that for any neighborho o d U of a p oint in the Julia set the union of
S
n
orbits which start in U P U is equal to C except at most one p oint Using this
n
prop erty one can prove that the Julia set is a p erfect set that is a closed set where any p oint
is a limit p oint in the set the Julia set has therefore no isolated p oints
Periodic and prep erio dic p oints
A p oint z is called p periodic if
z z and z z for j p
p j
p
a xed point is a p erio dic p oint and a p p erio dic p oint is a xed p oint of P A p erio dic
orbit is called a cycle A p oint z is called preperiodic of preperiod k and period p if
z z is a p p erio dic p oint and z z for j k
k p k j k
see gure
Figure Periodic and prep erio dic orbits
p
The multiplier of a p p erio dic p oint z is de ned as the derivative of P at z Using the
chain rule we obtain
p
P z P z P z
p
p
is therefore the same at all p oints of the cycle For this reason is also the derivative of P
called the multiplier of the cycle We call a cycle
Bo dil Branner
attracting if j j
superattracting if
repelling if j j
indi erent if j j
Note that a cycle is sup erattracting if and only if it contains a critical p oint
i
An indi erent cycle has multiplier of the form e The cycle is called rationally
indi erent or parabolic if is rational and irrationally indi erent otherwise
A p erio dic p oint z of p erio d p is called linearizable if there exists a lo cal holomorphic
p
change of co ordinates so that P in these co ordinates is of the form where is the
multiplier
Theorem Koenigs An attracting but not superattracting periodic point is linearizable
A repelling periodic point is linearizable
Proof A sketch Assume z is an attracting but not sup erattracting xed p oint Since
n n
j j there is a neighborho o d U of such that P U U Set z P z for
n
z U Then
P z z
n n
The holomorphic functions converge in U uniformly on compact subsets to a holomorphic
n
map with derivative z de ning the required lo cal co ordinate change
For a rep elling xed p oint we reduce the situation to the ab ove by considering the branch
of P which xes the xed p oint
p
For p erio dic p oints of p erio d p we consider P instead of P
The linearizing co ordinates are uniquely determined with the extra requirement z
Note that it follows from the implicit function theorem that a p p erio dic p oint with multi
plier j j can b e followed analytically in the parameters in a neighborho o d of the p olyno
mial and in a su ciently small neighborho o d the p oint remains a p p erio dic rep elling p oint
Moreover the linearizing co ordinates vary analytically with the parameters
Example Consider again Q z z The p oint z is a rep elling xed p oint with
multiplier Set z Log z the principal branch of the logarithm de ned in the
domain C n fz C j Re z Imz g Then is the linearizing co ordinate change For
it
z r e r t we have
Q z z and
The xed p oints for the quadratic p olynomials Q are solutions to z c z hence of the
c
p
form z c The analytic continuation of the rep elling xed p oint z is
p
p
given by c c where denotes the principal branch of the ro ot function
A sup erattracting p erio dic p oint z is of course not linearizable In the sup erattracting
p k
case there exists a co ordinate change so that P in these co ordinates is of the form
p
where k is the smallest integer n for which the n th derivative of P at z is di erent from
Such co ordinates play an imp ortant role in the analysis of the lo cal b ehavior around
Bo dil Branner
sup erattracting orbits in particular around which is a sup erattracting xed p oint for
all p olynomials For a monic p olynomial the co ordinates are uniquely determined with the
extra requirement z z as z They are called B ottcher co ordinates and provide a
d
co ordinate change so that P in these co ordinates is of the form where d is the degree
of the p olynomial
To nish the list A rationally indi erent p erio dic p oint z is not linearizable An irra
tionally indi erent p erio dic p oint z is called Siegel if it is linearizable and Cremer if it is
not
A p erio dic p oint is contained in the Fatou set if it is attracting or Siegel All other p erio dic
p oints are contained in the Julia set The rep elling p erio dic p oints are of sp ecial interest
Theorem The Julia set is the closure of the repelling periodic points
The theorem corresp onds to Julia s de nition of the Julia set while Fatou used the concept
of normal families
Proof The pro of has two steps
Step The Julia set is contained in the closure of the p erio dic p oints
Step There are only nitely many non rep elling cycles
We sketch the rst step and comment on the second step at the end of the next section
Assume the statement is false Then there exists a p oint z J P and a neighborho o d U of
z without any p erio dic p oints We may assume that z is not a critical value the image of
a critical p oint since there are only nitely many critical values for P Furthermore we may
assume that P has a lo cal inverse h de ned on U with h U and U disjoint Set h z z
n
It follows that the family F U fP j g satis es Montel s normality criterion with
U n
resp ect to the two analytic functions h j This contradicts that z is in J P and
j
therefore a non normal p oint
Classi cation of p erio dic Fatou comp onents
A Fatou comp onent is mapp ed onto a Fatou comp onent by P We call a Fatou comp onent
p k
V periodic if P V V for some p preperiodic if P V is p erio dic for some k
and wandering otherwise For a p olynomial the attracting basin of is always a p erio dic
Fatou comp onent of p erio d For a p olynomial the di erent p ossibiliti es are listed in the
classi cation theorem b elow
Theorem Let V denote a bounded periodic Fatou component of period p Then V is one
of the fol lowing three types
Attracting basin There is a p periodic attracting point z V and al l points in V
p
converge to z under iteration of P
p
Parabolic basin There is a parabolic point z V which is xed under P and
p p
satis es P z and al l points in V converge to z under iteration of P Note
that the period p of the periodic point z may be a divisor of p and the multiplier of
z a root of unity which raised to the power p p equals
Bo dil Branner
Siegel disk There is a p periodic irrationally indi erent point z V which is lineariz
p
able the linearizing coordinates are de ned on V and the iterate P in these coordinates
i i
expressed as the irrational rotation e where e is the multiplier
In gure we show examples of the three types ab ove A sup erattracting basin is also
called a B ottcher domain and an attracting but not sup erattracting is called a Schr oder
domain A Parabolic basin is also called a Leau domain
Already Fatou made an exhaustive list of p ossible types of Fatou comp onents including
the p ossibility of wandering comp onents Only the rst two p ossibilities in the classi cation
theorem were known by Fatou to exist the existence of the third was proved by Carl Siegel
in
The nal break through came when Sullivan in proved the following theorem using
quasi conformal mappings
Theorem Sul livan There are no wandering Fatou components for a rational function
The non wandering theorem implies that any Fatou comp onent is either itself p erio dic of
one of the ab ove mentioned three types or eventually mapp ed onto such a p erio dic Fatou
comp onent
Relation to critical p oints Each type of p erio dic Fatou comp onents is related to a critical
S
p
j
P V If V is p oint Let V b e a p p erio dic Fatou comp onents as ab ove and set V
j
an attracting basin or a parab olic basin then V contains a critical p oint compare with gure
If V is a Siegel disk then the b oundary of V is contained in the closure of the orbit of a
critical p oint
Note that this implies that a p olynomial of degree d can have at most d cycles which
are attracting or parab olic The statement is in fact also true if we add Cremer and Siegel
cycles to the list The pro of is using the notion of p olynomial like mappings A p olynomial
of degree d can therefore have at most d non rep elling cycles
It also follows from the classi cation theorem that if none of the critical p oints are attracted
to attracting or parab olic cycles and if there are no Siegel disks then K P J P For
quadratic p olynomials this happ ens for instance if the critical p oint is prep erio dic Such a
p olynomial is called a Misiurewicz polynomial The p erio dic orbit which the critical p oint
eventually lands on is always rep elling
Hausdor distance and dep endence of Julia sets and lled
Julia sets on the p olynomial
Both the lled Julia set and the Julia set are non empty compact sets in the complex plane
The Hausdor distance D de nes a metric in the set C omp C of non empty compact
H
subsets of the complex plane Given this metric one can discuss whether the lled Julia set
K P and the Julia set J P dep end continuously on the p olynomial or not
Hausdor distance Let d denote the Euclidean distance in C and de ne for A B
C omp C
A B sup d a B
a A
and
D A B max A B B A H
Bo dil Branner
Figure The three di erent types of Fatou domains attracting parab olic and Siegel
Bo dil Branner
Observe that it follows from the de nition of that for any
A B A B
where B is the neighborho o d of B i e B fz C jd z B g Moreover the triangular
inequality holds i e
A C A B B C for all A B C C omp C
With these two prop erties of it follows that D is a metric in the space C omp C in fact
H
a complete metric
Note that A B A B if B B
The lled Julia set and the Julia set do not in general dep end continuously on the p oly
nomial The general statement is formulated in the following theorem where the p olynomials
are assumed to b e of a xed degree
Theorem The map P K P satis es
K P K P when P P
The map P J P satis es
J P J P when P P
d
Recall that we have identi ed the family of p olynomials of degree d with C That P
tends to P therefore means that the co e cients of P tend to the co e cients of P
Corollary Suppose K P J P for a polynomial P Then both P K P and P
J P are continuous at P
Proof The pro of of is more involved that the pro of of see D The pro of of is
easy and relies essentially on the fact that the rep elling p erio dic p oints are dense in the Julia
set We sketch the pro of of
Fix a p olynomial P and an Cho ose a nite number of rep elling p erio dic p oints in
J P say X fx x g such that
N
N
J P D x
j
j
where D x denotes the op en disk centered at x and of radius We have J P X
j j
Supp ose x is p p erio dic then
j j
p p
j j
P x x and j j j P x j
j j j j
It follows from the implicit function theorem that there exists a neighborho o d U of P in the
parameter space and analytic functions P P for P U so that P x and P is
j j j j
a rep elling p p erio dic p oint We can assume that U is chosen so small that d x P
j j j
for all P U Set X P f P P g then X X P for all P U Since
N
Bo dil Branner
X P J P for all P U it follows that X J P X X P and therefore that
J P J P for all P U
Note that the Julia set and the lled Julia set are functions which are continuous at any
quadratic Misiurewicz p olynomial i e a p olynomial where the critical p oint is prep erio dic
One can prove that the Julia set varies dis continuously at any p olynomial with a Siegel
cycle and that b oth the Julia set and the lled Julia set vary dis continuously at a p olynomial
with a parab olic cycle see again D
Parameter space the Mandelbrot set
The goal is to decomp ose the parameter space into regions corresp onding to qualitatively
di erent dynamical b ehavior This is in general a very hard problem In complex dynamics
we divide the parameter space according to qualitatively di erent b ehaviors of the nitely
many critical p oints This turns out to b e a go o d strategy
We have already seen that the critical p oints play an imp ortant role in connection with
the classi cation theorem of p erio dic Fatou comp onents Another result connected with the
critical p oints is expressed in the following classical theorem known to Fatou and Julia Note
that this is a global result
Theorem Fatou Julia The l led Julia set K P is connected if and only if the critical
points are contained in K P
The parameter space of monic centered p olynomials is decomp osed into two complemen
tary sets the connectedness locus corresp onding to p olynomials with connected lled Julia
set and the rest corresp onding to p olynomials with disconnected lled Julia set
The Mandelbrot set M is de ned as the connectedness lo cus for the family of quadratic
p olynomials Q
c
n
M fc C j K Q is connected g fc C j Q as n g
c
c
see gure
The rst theorem ab out the Mandelbrot set was the following proved by Douady and
Hubbard in
Theorem Douady Hubbard The Mandelbrot set is connected
The de nition of the Mandelbrot set is very rough it is surprising that it turns out to
give the detailed decomp osition of the parameter plane we are interested in The b oundary
of the Mandelbrot set is the bifurcation set i e the set where the qualitative changes o ccur
j It follows from a theorem of Ma n e Sad Sullivan that any two p olynomials Q
c
j
in the same connected comp onent of C n M are J equivalent That means there exists a
homeomorphism H J Q J Q conjugating the dynamics i e the following diagram
c c
is commutative
Q
c
J Q J Q
c c
H H
y y
J Q J Q
c c
Q
c
Bo dil Branner
Figure The b oundary of the Mandelbrot set
Figure shows two Julia sets that are J equivalent The parameters are chosen in M in
the same connected comp onent of C n M
Figure J stability
The Mandelbrot set can also b e de ned within the concept of normal families Set
F c c F c F c c for n
n n
For a xed c the sequence F c F c is just the orbit of the critical p oint
n
under iteration by Q If c M then F c as n If c M then jF c j for all
c n n
n It follows that p oints in M are non normal for the family F fF g while p oints in
n n
the complement are normal The b oundary of M is therefore the set of non normality
This alternative de nition can b e used to prove the following theorem ab out Misiurewicz p olynomials
Bo dil Branner
Prop osition Misiurewicz polynomials are dense in the boundary of the Mandelbrot set
Proof Assume the statement is false Then there exists a p oint c in M and a simply
connected neighborho o d U of c without any Misiurewicz p olynomials We may assume that
c Let g U C j denote the two branches of the square ro ot of c
j
Then h c g c j determine the two xed p oints of Q They di er since
j j c
c It follows that the family F fF j g satis es Montel s normality criterion with
n U n
resp ect to the two analytic functions h j This contradicts that c in M and therefore
j
a non normal p oint
Observe that we have proved more than stated the set of Misiurewicz p oints for which
is eventually mapp ed onto a xed p oint is dense in the b oundary The same kind of pro of
would give that Misiurewicz p oints for which is eventually mapp ed onto a p erio dic orbit of
p erio d p is dense in the b oundary for any xed p
Universality of the Mandelbrot set is discussed in Nuria Fagella s pap er Prop erties of
Misiurewicz p olynomials are discussed further in Tan Lei s pap er
References
A Daniel S Alexander A History of Complex Dynamics From Schr oder to Fatou and
Julia Asp ect of Mathematics Vieweg
B Bo dil Branner The Mandelbrot set in Chaos and Fractals The Mathematics Behind
the Computer Graphics Pro c of Symp osia in Appl Math AMS Vol pp
D Adrien Douady Does the Julia set depend continuously on the Polynomial in Complex
Dynamical Systems Pro c of Symp osia in Appl Math AMS Vol pp
DH Adrien Douady and John H Hubbard Etude dynamique des polyn ome complexes I
II Publications Mathematiques d Orsay
DH Adrien Douady and John H Hubbard On the dynamics of polynomial like mappings
Ann Ecole Norm Sup Vol pp
MSS Ricardo Ma n e Pablo Sad and Dennis Sullivan On the dynamics of rational maps Ann
Ecole Norm Sup Vol pp
S Dennis Sullivan Quasi conformal maps and dynamical systems I Solutions of the
Fatou Julia problem on wandering domains Ann Math pp
St Norb ert Steinmetz Rational Iteration Complex Analytic Dynamical Systems de Gruyter
Studies in Mathematics Vol Walter de Gruyter
th
Proc of the EWM
meeting Madrid
The theory of p olynomial like mappings
The imp ortance of quadratic p olynomials
N uria Fagella
Univ Aut onomade Barcelona Spain
nuria mat uab es
Introduction
In the eld of complex dynamics and in particular iteration of functions of one complex
variable the topic that has by far b een ob ject of the most attention is the iteration of the
family of quadratic p olynomials Q z c In this pap er we aim to answer the question of
c
why this very particular family of p olynomials is imp ortant for the understanding of iteration
of general complex functions
This is the second pap er in the Complex Dynamics series of EWM We assume that
the reader is familiar with the basic de nitions and theorems concerning the dynamics of
quadratic p olynomials which are the topic of the rst article Br For other surveys we refer
also Bl Br and Mi
As a rst observation we may say that often a go o d place to start is the simplest example
in this case the group of M obius transformations which are already very well understo o d
The next simplest class of functions is the class of p olynomials of degree two and even
that early along the way we already bump into complicated dynamics which have o ccupied
mathematicians in this eld for over twenty years and still do
But the real answer to the question has basically one name and that is the theory of
polynomial like mappings of A Douady and J Hubbard This theory explains how the
understanding of p olynomials is not only interesting per s e but helps understand a much
wider class of functions namely those that lo cally b ehave as p olynomials do
Most of the de nitions and results in this pap er may b e found in the work of Douady
and Hubbard On the Dynamics of Polynomial like Mappings DH Our goal is to state
their most imp ortant results as well as to give several examples that illustrate them These
examples serve also as initial motivation example B concerns families of cubic p olynomials
whose dynamical planes exhibit homeomorphic copies of quadratic lled Julia sets see Figs
and while their parameter spaces contain homeomorphic copies of the Mandelbrot set see
Fig example C deals with the family of entire transcendental functions f z cos z
for which the same phenomena o ccur see Figs and nally example D shows how
we nd copies of the Mandelbrot set in the Mandelbrot set itself see Figs and
Examples of the same phenomena for Newton s metho d may b e found in BC CGS DH T
N uriaFagella
z
and in F for the family z z e
This work is divided in two parts the rst one concerning the dynamical planes and
the second one the parameter spaces Section contains the de nition of a p olynomial like
map and sets up the examples that we follow throughout the pap er In Section we state
the straightening theorem Theorem which explains how p olynomial li ke maps and actual
p olynomials are related Along the way we give a small survey of the di erent types of
conjugacies that may o ccur Section contains the parameter plane version of the straight
ening theorem explaining why we nd homeomorphic copies of the Mandelbrot set in the
parameter planes of other families of functions
Figure was b orrowed from Br by courtesy of Bo dil Branner All other computer
illustrations in this pap er were created with the program It by Christian Mannes whom I
thank for his assistance and patience
Dynamical Plane
The De nition of a Polynomial like Map
De nition A polynomial like map of degree d is a triple f U U where U and U are
U U and f U U is a holomorphic map such op en sets of C isomorphic to discs with
that every p oint in U has exactly d preimages in U when counted with multiplicity
0
Figure The three elements f U U that form a p olynomial like map
For the examples throughout the pap er the following de nition will b e necessary
De nition Let P z b e a p olynomial of degree d and let z b e the B otcher co ordinates
at in nity see Br It is a fact that if all critical p oints of P b elong to the lled Julia set
K P then can b e extended to map the complement of K P to the complement of the
unit disk We de ne an equipotential curve of potential to b e the preimage under of a
circle of radius e It follows then that an equip otential curve of p otential is mapp ed under
P to an equip otential curve of p otential d with degree d
Example A The obvious example is an actual p olynomial of degree d restricted to a large
enough op en set Let P b e a p olynomial of degree d and let b e an equip otential curve
of P of some given p otential such that it is a single simple curve Then P is an
equip otential curve of p otential d If we let U and U b e the op en sets enclosed by and
0
U U is a p olynomial like map see g Note that resp ectively then the triple P j
U
we do not necessarily have to choose the op en sets as regions enclosed by equip otentials In
fact if we let V b e any large enough disk then V P V is an op en set contained in V
0
V V is another p olynomial like map and P j V
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Figure The restriction of two p olynomials of degree two as p olynomial like maps Left Q z
1
2
z with connected Julia set Right Q z where c w i with totally disconnected Julia
c
set
Example B In this example we want to consider some p olynomials of degree three which
restricted to an op en set form a p olynomial like map of degree two Let P b e a cubic
p olynomial with one critical p oint escaping to in nity under iteration and the other
one remaining b ounded Let b e the equip otential curve that has the critical value
v P as one of its p oints and let U b e the op en set b ounded by Then the preimage
of under P is a gure eight curve since all p oints on have three preimages with the
exception of the critical value v that has only two preimages see g This gure eight
b ounds two connected comp onents Let U b e the op en connected comp onent that contains
the critical p oint with a b ounded orbit Then U maps to U with degree two i e every
0
U U is a p olynomial like map p oint in U has exactly two preimages in U The triple P j
U
of degree two Notice that if we choose sets U and U as we did in example A we would
obtain a p olynomial like map of degree three We have chosen a p olynomial of degree three
for the sake of the example but it is clear that similar situations would o ccur with p olynomials
of any degree with critical p oints escaping and not escaping to in nity
Figure The restriction of a cubic p olynomial to create a p olynomial like map of degree two
Example C Let f z cos z and let U b e the op en simply connected domain
U fz C j jIm z j j Re z j g
and set U f U One can check that U U as shown in Fig Since U contains only
one critical p oint it follows that f maps U to U with degree two Hence the triple
0
U U is a p olynomial like of degree two f j U
N uriaFagella
Example D Sometimes a p olynomial like map is created as some iterate of a function
restricted to a domain For example let Q z z c and let c w i
c
Set
U fz C j jIm z j jRe z j g
One can check that the p olynomial Q maps U onto a larger set U with degree as shown
c
0
U U is a p olynomial li ke map of degree two in Fig The triple Q j
U
c
3
z right to create p olynomial like maps Figure The restriction of f z cos z left and Q
c
0
of degree two
This is an example of what is called renormalization We say that a quadratic p olynomial
n
0
U U is renormalizable if there exist op en disks U and U and an integer n such that f j
U
is p olynomial like of degree two Renormalization is a very imp ortant topic in the eld of
complex dynamics See Mc
The Filled Julia Set
The l led Julia set and the Julia set are de ned for p olynomial like maps in the same fashion
as for p olynomials keeping in mind that a p olynomial li ke map is de ned only in an op en
subset of C
De nition Let f U U b e a p olynomial like map The l led Julia set of f is de ned
as the set of p oints in U that never leave U under iteration i e
n
j f z U for all n g K fz U
f
An equivalent de nition is
n
U K f
f
n
and from this expression it is clear that K is a compact set
f
As for p olynomials we de ne the Julia set of f as
J K
f f
Notice that if the map f is the restriction of some p olynomial F to a set U then in
general K K As an example consider example B ab ove where F is a p olynomial of
f F
N uriaFagella
degree three and f its restriction to the set U in Fig Notice that U maps to U with degree
two The other connected comp onent of F U which we denote by V maps to U with
degree one Hence there are p oints in U that map to V and come back to U afterwards
never leaving the set U Such p oints do not b elong to K since they are not in U at all times
f
but they b elong to K since they do not escap e to in nity under iteration Hence K K
F f F
and moreover a connected comp onent C of K is either a connected comp onent of K or it
F f
is disjoint from K since F maps connected comp onents of K to connected comp onents
f F
Therefore K might have more connected comp onents than K but not larger ones
F f
The Relation with Polynomials
The Straightenning Theorem stated in this section shows that the relation b etween p olynomial
like maps and actual p olynomials is actually very strong In order to state it we need to
review the di erent types of equivalences b etween holomorphic maps
Equivalences or conjugacies of maps
Supp ose f U U and g V V are two p olynomials like maps of degree d The
weakest but very imp ortant equivalence b etween f and g is what we call topological equiva
lence or topological conjugacy and denote by
top
De nition We say that f g if there exists a homeomorphism from a neighborho o d
top
N K of K to a neighborho o d N K of K such that the following diagram
f f g g
f
N K N K
f f
y y
g
N K N K
g g
commutes where N K N K and N K N K
f f g g
If two functions are top ologically conjugate their dynamics are qualitatively the same
since the conjugacy must map orbits of f to orbits of g p erio dic p oints of f to p erio dic
p oints of g critical p oints of f to critical p oints of g etc In particular K must b e mapp ed to
f
K but since is only a homeomorphism these sets could lo ok quite di erent For example
g
all quadratic p olynomials that b elong to a given hyperb olic comp onent of the Mandelbrot
set except the center are top ologically equivalent All p olynomials in the complement of
the Mandelbrot set are also top ologically conjugate In fact these conjugacies are global
conjugacies See remark b elow
On the other hand the strongest type of equivalence b etween two holomorphic maps is
conformal equivalence due to the rigidity of holomorphic maps
De nition We say that f g if f g and the homeomorphism is conformal
conf top
Remark If we were dealing with maps de ned in the whole complex plane we could
consider also global conjugacies b etween them In such a case if two maps are conformally
conjugate then they must b e conjugate by an a ne map z az b since holomorphic
isomoprhisms from C to itself are a ne For the quadratic family one can easily check that
there is a unique representative in each a ne class that is if Q and Q are a ne conju
c c
gate then c c
N uriaFagella
The concept of quasi conformal maps app ears when we want to consider conjugacies that
are stronger than top ological but weaker than conformal
Quasi conformal mappings For a homeomorphism we do not have any control whatso
ever in how angles are distorted On the other hand conformal maps have to preserve angles
Intuitively a map is quasi conformal if we have some control on the distortion of angles even
if these are not preserved i e the distortion of angles is b ounded
The precise de nition is very intuitive if we assume that the map is di erentiable This
is not such a crude assumption given the fact that quasi conformal maps are di erentiable
almost everywhere If is a di eomorphism the tangent map at a given p oint z takes a
certain ellipse in the tangent space at z to a circle in the tangent space at z We de ne
the dilatation of at z D z as the quotient of the length of the ma jor axis over the
length of the minor axis of this ellipse
De nition Let U V b e a di eomorphism and D sup D z Then is K quasi
z U
conformal if D K
If we do not assume the map to b e di erentiable we can express its distortion in terms of
mo duli of annuli
De nition Let b e a homeomorphism Then is K quasi conformal if for all annuli A
in the domain
mo d A mo d A K mo d A
K
Note that a map is quasi conformal if and only if it is conformal
For those that prefer analytic de nitions one can de ne quasi conformal maps as follows
De nition Let b e a homeomorphism Then is K quasi conformal if lo cally it has
distributional derivatives in L and the complex dilatation z de ned lo cally as
d z dz
z
z
z
dz dz
z
z
K
k almost everywhere satis es j j
K
For more on quasi conformal mappings see A and LV
Quasi conformal conjugacies and hybrid equivalences We de ne a quasi conformal
conjugacy f g by requiring the homeomorphism in the top ological conjugacy to b e
qc
K quasi conformal for some K We say that f and g are hybrid equivalent f g if
hb
they are quasi conformally conjugate and the conjugacy can b e chosen so that
z
almost everywhere on K If J has measure zero this simply means that is holomorphic
f f
in the interior of K Clearly
f
f g f g f g f g
conf hb qc top
N uriaFagella
The Straightening Theorem
The relation b etween p olynomial like mappings and actual p olynomials is explained in the
following theorem whose pro of can b e found in DH
Theorem Let f U U be a polynomial like map of degree d Then f is hybrid
equivalent to a polynomial P of degree d Moreover if K is connected then P is unique up
f
to global conjugation by an a ne map
This theorem explains why one nds copies of Julia sets of p olynomials in the dynamical
planes of all kinds of functions Notice that if f is p olynomial like of degree two and K
f
is connected then f is hybrid equivalent to a p olynomial of the form Q z z c for a
c
unique value of c by remark This may also b e true for other families of p olynomial like
maps of degree larger than two as long as the resulting class of p olynomials has a unique
d
representative in each a ne class As examples consider the families z z d C n f g
for any d
Example B In the setting of example B in Sect we consider the p olynomial P z
a
z a z a a One can check that for all values of a the critical p oint a is
a xed p oint If we take for example a then the critical p oint a escap es to
in nity By the Straightening Theorem P z restricted to the op en set U as de ned in
example B is hybrid equivalent to a quadratic p olynomial and hence to a p olynomial of the
form Q z z c In this case we know that the parameter c must b e since Q z is
c
the only quadratic p olynomial of this form with the critical p oint b eing xed In Fig we
show the dynamical plane of Q and that of P
2
Figure Left the lled Julia set of Q z z in white Right the lled Julia set for P z
0 0 6
0
in white Note that only the largest comp onent in U corresp onds to the lled Julia set of the
p olynomial like map of degree
Example B Again in the setting of example B in sect we consider the p olynomial
p
R z z a z a a a One can check that for all values of a the
a
critical p oint c a is a p oint of p erio d In this case we take a and then the
critical p oint c a escap es to in nity By the straightening theorem R z restricted
to the op en set U as ab ove is hybrid equivalent to a quadratic p olynomial and hence to a
N uriaFagella
p olynomial of the form Q z z c In this case we know that the parameter c must b e
c
since Q z is the only quadratic p olynomial of this form with the critical p oint b eing
of p erio d two In Fig we show the dynamical plane of R to b e compared with that
of Q in Fig
0
Figure The lled Julia set for R in white Note that only the largest comp onent in U
0 75
corresp onds to the lled Julia set of the p olynomial like map of degree This gure is to b e compared
with Fig left
Example C Even though the function f z cos z is an entire transcendental function
when restricted to the set U as de ned in Sect it is a p olynomial li ke map of degree two
In Fig we see in white the set of p oints that do not escap e to in nity in the imaginary
direction under iteration of f The largest comp onent inside U corresp onds to the lled
Julia set of the p olynomial like map Since the critical p oint is xed under f the lled
Julia set is homeomorphic to that of Q z z
0
Figure The largest white comp onent in U corresp onds to the lled Julia set of f z cos z
0
restricted to the set U
Example D Consider again Q z z c where c w As explained
c
in Sect Q maps the square b ox U centered at and with side length onto a larger
c
set U containing U see Fig By the Straightening Theorem Q is hybrid equivalent to
c
N uriaFagella
Q for some value of c One can check that the critical p oint is p erio dic of p erio d three under
c
iteration of Q hence there are a limited number of p osibilities for c In this case the lled
c
Julia set of the p olynomial like map is homeomorphic to the Douady rabbit see Figs
where c w i Figure The lled Julia set of Q
0 c
0
2
z z c in white where Figure Left the Douady rabbit or the lled Julia set of Q
1 c
1
around the critical p oint The c w i Right magni cation of the lled Julia set of Q
1 c
0
3
copy of the Douady rabbit is the lled Julia set of the p olynomial like map corresp onding to Q
c
0
Parameter Plane
As usual the phenomena in dynamical plane are re ected in parameter space Recall that
the parameter space of the family of quadratic p olynomials Q z z c contains the
c
Mandelbrot set de ned as
n
M fc C j fQ g is b ounded g
n
c
or equivalently the set of c values for which the lled Julia set of Q is connected see
c Fig
N uriaFagella
Figure The Mandelbrot set
If we lo ok at the parameter space for other functions we very often encounter p ortions that
resemble the Mandelbrot set This fact is again explained by the theory of p olynomial like
maps Since the Mandelbrot set app ears when we consider families of quadratic p olynomials
it is reasonable to exp ect that it should also app ear when we consider families of p olynomial
like maps of degree two as long as these families are nice enough
Remark For the sake of exp osition we consider here only one parameter families of
p olynomial like mappings of degree two For other cases see DH
Analytic families of p olynomial like mappings
De nition Let b e a Riemann surface and F ff U U g b e a family of p olynomial
like mappings Set
U f z j z U g
g U f z j z U
f z f z
Then F is an analytic family of p olynomial li ke maps if it satis es the following prop erties
U and U are homeomorphic over to D
The pro jection from the closure of U in U to is prop er
The map f U U is holomorphic and prop er
If these prop erties are satis ed the degree of the maps is constant and it is called the
degree of F We denote K K and J J By the Straightening Theorem for each
f f
the map f is hybrid equivalent to a p olynomial of degree the degree of F By analogy with
p olynomials we de ne
M f j K is connected g
F
In the next section we give some conditions under which the set M is homeomorphic to the
F Mandelbrot set
N uriaFagella
Homeomorphic Copies of the Mandelbrot Set
Let F b e an analytic family of p olynomial like maps of degree two Then for each M
F
f is hybrid equivalent to a unique p olynomial of the form Q z z c Hence the map
c
C M M
F
c C
is well de ned
Theorem Let A be a closed set of parameters homeomorphic to a disc and containing
M Let be the critical point of f and suppose that for each n A the critical value
F
0
Assume also that as goes once around A the vector f turns f U n U
once around see Fig Then the map C is a homeomorphism and it is analytic in the
interior of M
F
Figure Illustration of theorem
Remarks
0
if n A is equivalent to M b eing compact The assumption f U n U
F
If the winding number of f around is then C is a branched covering
of degree
Example A The purp ose of this example is to illustrate that the conditions of the theorem
are satis ed for the Mandelbrot set itself Consider the parameter plane for the quadratic
family and let
fc j G c gA fc j G c g
M M
where G denotes the Green s function of the Mandelbrot set Given the way the Green s
M
function of M is de ned if c A then c lies on an equip otential curve of p otential in
the dynamical plane as well So for each c A let and b e the equip otential curves
c
c
in the dynamical plane of Q of p otentials and resp ectively The op en sets enclosed by
c
0
and are the discs U and U resp ectively and F Q j U U the analytic family
c c c c
U
c c c
c
of p olynomial like maps Note that by construction for each c n A the critical value
Q c lies in U n U Also as c turns once around A the critical value c turns once
c c
c
around the critical p oint In this case M M F
N uriaFagella
Example B Consider the family of cubic p olynomials P z P z z az b For any
a b
given constants and we de ne the parameter space to b e the set of p olynomials
P such that
one critical p oint escap es to in nity with escap e rate
another critical p oint escap es to in nity at a slower rate or stays b ounded
the co critical point of that is the other preimage of P di erent from
b elongs to the external ray R see Br for de nitions of this terms and Br for
more in this example
Note that p olynomials of this type are p olynomial like maps of degree two as shown in
example B in Sect In BH Branner and Hubbard prove
Theorem The parameter space is homeomorphic to a disc
Hence p olynomials in form a one parameter family of p olynomial like maps of degree two
Let B B b e the set of p olynomials in for which the orbit of is b ounded Note
that examples B and B are in B for some values of and Also in BH we nd the
following theorem
Theorem Let B and suppose that the connected component of c in K P is periodic
Then the connected component of in B is a homeomorphic copy of the Mandelbrot set
Figure shows the parameter space with B in black
Figure The set B shown in black with countably many comp onents which are homeomor
0 0
phic copies of the Mandelbrot set
Example C Let f z cos z and let A b e an appropriately chosen disc in the
and U the maps plane around One can check that for appropriate choices of U
0
f j U U form an analytic family of p olynomial like maps As turns once around A
U
the critical p oint stays xed at while the critical value winds once around hence
N uriaFagella
Figure Left Parameter plane of f z cos z Right magni cation of the copy of the
Mandelbrot set centered around
satisfying the conditions of theorem In Fig we see the resulting copy of the Mandelbrot
set with as the center of its main cardioid
Example D
Let A b e a small discs of parameters centered at c w and with c contained
in A where c is as in example D in Sect For Q the critical p oint is p erio dic of p erio d
c
three One can check that for apropiate choices of U U and A the conditions of the
c
c
theorem are satis ed for the family F fQ U U g Figure shows the Mandelbrot
c c
c c
set and a magni cation of the homeomorphic copy that contains c
Figure Copy of the Mandelbrot set in the parameter plane of Q Range
c
N uriaFagella
References
A L Ahlfors Lectures on Quasi conformal Mappings Van Nostrand
Bl P Blanchard Complex Analytic Dynamics on the Riemann Sphere Bul l Amer
Math Soc
Bl P Blanchard The Dynamics of Newton s Metho d Proc Symp Applied Math
BC P Blanchard A Chiu Complex Dynamics An Informal Discussion Bo ok chapter
in Fractal Geometry and Analysis ed J B elair and S Dubuc Kluwer Academic
Publishers
Br B Branner The Mandelbrot Set Proc Symp Applied Math
Br B Branner Puzzles and para puzzles of quadratic and cubic p olynomials
Proc Symp Applied Math
Br B Branner Holomorphic Dynamical Systems in the Plane An introduction EWM
BH B Branner J H Hubbard The Iteration of Cubic Polynomials Part I The Global
Topology of Parameter Space Acta Mathematica
CGS J Curry L Garnett D Sullivan On the iteration of a rational function Computer
exp eriment with Newton s metho d Comm Math Phys
DH A Douady J Hubbard Iterations des Polynomes Quadratiques Complexes C R
Acad Sci Paris Ser I
DH A Douady J Hubbard Etude dynamique des p olynomes complexes I I I Publ Math
Orsay
DH A Douady J Hubbard On the Dynamics of Polynomial like Mappings Ann Scient
e
Ec Norm Sup series
F N Fagella Limiting Dynamics of the Complex Standard Family Int J of Bif and
Chaos To app ear
LV O Lehto K J Virtanen Quasi conformal Mappings in the Plane Springer Verlag
Mc C T McMullen Complex Dynamics and Renormalization Annals of Mathematics
Studies Princeton University Press
Mi J Milnor Dynamics on One Complex Variable Introductory Lectures IMS at Stony
Bro ok Preprint n
MSS R Ma ne P Sad D Sullivan On the Dynamics of Rational Maps Ann Scie Ecole
Norm Sup
T Tan Lei Cubic Newton s metho d of Thurston s type preprint Lab oratoire de Math
Ecole Norm Sup de Lyon
th
Proc of the EWM
meeting Madrid
Lo cal prop erties of the Mandelbrot set M
Similarities b etween M and Julia sets
Tan Lei
University of Warwick England
tanlei maths warwick ac uk
Introduction
Lo cal prop erties of the Mandelbrot set around a p oint c such as self similarity Hausdor
dimension lo cal connectivity are closely related to prop erties of the lled Julia set K for c
c
in a neighborho o d of c Recall that M fc C j K g For each N let F denote
c N
N
the holomorphic mapping c Q where Q denotes the p olynomial z z c Since
c
c
K is totally invariant for any N N we have
c
M fc C j F c K g
N c
It is often convenient to go to the pro duct space c z C C to study b oth M and K
c
We may regard c K as a map and K f c z j c C z K g as its graph in C C
c c
Then M can b e interpreted as
g r aph F K P r oj
N c
where P r oj denotes the pro jection mapping to the rst co ordinate the c co ordinate
c
to the lo cal A typical way to relate K Let c b e a p oint of M so F c K
c N c
structure of M around c is to study the lo cal structure of K around the p oint c F c
N
which dep ends on the regularity of the mapping c K at c see for example conditions
c
and b elow and the slop e of F at c i e the constant F c In this pap er we
N
N
illustrate this technique by showing two related results
The rst theorem states a similarity result b etween the dynamical plane and the param
eter plane around Misiurewicz p oints This similarity can actually b e observed in computer
exp eriments as in Figures and We know that the set of such p oints form a dense subset
see B Branner s pap er K of M and for each Misiurewicz p oint c we have J
c c
Denote by D the Hausdor distance on the space C omp C of non empty compact
H
subsets of C Let denote the translation z z c For any closed set A C de ne
c
A A r r
r
where z r denotes the op en disc centered at z and with radius r For technical reasons
it is imp ortant to include the circle r when measuring the Hausdor distance of two
closed sets within the disc r
Tan Lei
Figure Enlargement of the Mandelbrot set around the Missurewicz p oint c
0
i to b e compared with Fig
Let b e a complex number with j j A closed set A C is said to b e self similar
ab out x if A A it is said to b e asymptotically self similar ab out x if there is
x x
n
a self similar set L ab out so that the Hausdor distance D A L tends to
H x r r
zero as n tends to in nity for some r hence every r For an example of a self similar
set see the app endix
Theorem T For every Misiurewicz point c there are two constants C j j
and C f g and a closed self similar set L C such that for any r
n
a D K L as n i e K is asymptotically self similar about c
H c c r r c
n
Moreover for c in a neighborhood of c we have D c K L c
H c r r
c
where c c and c c are holomorphic and c L c is a map verifying the
condition below
n
M L i e M is also asymptotically self similar about c b D
r r H c
c lim D t K t M Hence up to multiplication by the
H c c r c r
t C jtj
sets K and M are asymptotically similar about c
c
We wil l give the precise form of L and later in the formulas
Part c is an easy consequence of a and b As an application of this result one can show
that M is lo cally connected at each Misiurewicz p oint see the app endix
Tan Lei
2
Figure The Julia set of the Missurewicz p olynomial z 7! z c where c is as in Fig
0 0
Figure Enlargment of Fig around the p oint c 0
Tan Lei
The second result can b e considered as a quantitative study of the similarity b etween Julia
sets and the Mandelbrot set It gives estimates for the Hausdor dimension of these sets
The exact de nition of the Hausdor dimension is not so imp ortant for the purp ose of this
pap er It can b e found in the app endix We only state two basic prop erties of it For any
compact set K of C we have H dim K and for any compact subset K K we
have H dim K H dim K One should consider that the Hausdor dimension measures
a kind of density or complexity of a set So if K C is a compact set without interior but
with H dim K then K must b e very complicated
Theorem Shishikura For each there is a dense subset of M satisfying that for
every point c in this set there is a closed set X K and a constant r such that
c
a H dim K H dim X Moreover for c c r we have similarly
c
H dim X c where X c is a subset of K and c X c veri es the
c
condition below
b H dim M c r
Corollary We have H dim M
The existence of c and X satisfying a involves deep analysis of parab olic p erturbations
and renormalizations We will give some ideas of it in the app endix The set X is in fact
a hyperb olic set see the app endix It is this hyperb olicity that guarantees the stability
prop erty
We will sketch the pro ofs of a b b in the following sections
Condition Consider a mapping c A c with c c r A c a closed subset of C
such that c A c is continuous at c for every r The mapping admits a dense set
r
of continuous sections at c if there exists a dense subset Z A c and for each z Z a
neighborho o d U c r of c and a mapping
z
s f c z j z Z c U g C
z
such that s c id and s z U C is continuous with s c z A c
z
Condition Consider a mapping c X c with c c r X c a subset of C The
mapping admits a holomorphic motion if there is a mapping i c r X C where
X X c such that i c id i c X C is injective with i c X X c and
C is holomorphic for each z X i z c r
Dynamical planes Pro of of a
First remark that asymptotic similarity is invariant under conformal transformations More
precisely
Prop osition Let U and V be neighborhoods of and u U V be an injective analytic
map satisfying u and u Suppose A is a closed subset of U containing and
suppose u A is asymptotically self similar for some C with j j i e for any r
n
D u A L as n
H r r
Tan Lei
where L is a closed self similar set Then A is asymptotically self similar to u L
i e for any r
n
L D A
H r
u
r
Now assume that c is a Misiurewicz p oint If there is no ambiguity we simplify the
notation by setting Q Q and K K By de nition there is a smallest number k such
c c
k p
that Q c is a p erio dic p oint Let Q denote the multiplier
It follows from classical results of Fatou and Julia that the p oint is rep elling i e j j
Let U r denote linearizing co ordinates in a neighborho o d U of hence satisfying
p
and Q z z for all z r j j for some r
Since K is totally invariant Q K Q K K and is a linearizing co ordinate we
have
K U K U
r r
Applying prop osition to u A K U we get
n
0 0
U K D K
H
r r
for any r r
k k
Since Q c there exists a neighborho o d V of c and r such that Q
k
V we V r is a homeomorphism Applying prop osition to u A Q K
obtain
n
00
D K K U
H c
r
k
Q c
00
r
This proves the rst assertion of a Note that is the multiplier of the p erio dic p oint and
that the self similar limit set L is determined lo cally by
k
V Q K
k
Q c
where V is any neighborho o d of c which is mapp ed homeomorphically onto its image under
k
Q
Example Let us take the example of c i For Q z z i the orbit of i is
i
i i i i In our notation k p i and Q i
i
p
i
i e
Remark Note that a could have b een stated in greater generality The statement is true for
any rep elling p erio dic p oint with multiplier and linearizing co ordinates and similarly
for any pre p erio dic rep elling p oint We have only chosen the sp ecial pre p erio dic p oint c
in order to b e able to compare with the parameter plane These are the only prop erties we
have used together with the invariance of K
The second assertion of a is a consequence of a stability result As b efore let c b e a
Misiurewicz p oint and let k p b e as ab ove for the map Q
c
As an application of the implicit function theorem pre rep elling p erio dic p oints are
stable with resp ect to the parameter That is for any c in a neighborho o d W of c the
p olynomial Q z z c has a p p erio dic p oint c dep ending analytically on c with
c
multiplier c and with a k th pre image c of c b oth dep ending analytically on c
satisfying
k
c c c c and Q c
c
Tan Lei
Let denote the linearization co ordinate around c The same pro of as ab ove shows
c
k
that there is a neighborho o d V of c which is mapp ed by Q comformally onto its
c
c
c
image and that K is asymptotically c self similar ab out c to the limit set lo cally
c
k
Q K V L c
c c
c
c
k
Q z j
z c
c
As for the condition the mapping c L c is continuous at c b ecause c K is
c
continuous at c Douady Hubbard see B Branner s pap er each rep elling p erio dic p oint
has a continuous section and the set of rep elling p oints is dense in J K
c c
Parameter plane Pro ofs of b b
The pro of of b is done in two steps one consists of a general result one is the adaptation
Prop osition Suppose is a neighborhood of in C Assume we have a mapping
A satisfying the condition at and that A is self similar about where
is holomorphic with j j Assume u C is a holomorphic mapping with
u and u transversality Set
M f ju A g
u
Then M is asymptotically self similar about to the self similar set A u
u
Proof sketch Assume For z C a p oint and K C a compact set we use d z K
0
to denote the euclidean distance from z to K i e d z K min jz z j
z K
To x our ideas we treat rst two simple cases Assume that A is a constant map
i e A A and u z u z is linear Then obviously M is self similar ab out
u
to the set A u Assume now that A is still constant but u z is no longer
linear Then M coincides with u A and the conclusion holds by Prop osition
u
Now let us come back to the setting of our prop osition Set u u and
n
A A Cho ose r su ciently small We must prove that D u M A
H u r r
as n Recall that by de nition D A B max A B B A see B Branner s
H
pap er
n
First we prove u M A Let
u r r
r
M
u
n
u
n n
r and j j j j so that u
n n n n n n
d u A j u u j j u u j d u A I I I
r r
n n n
We have I j j j j j j j j and
n
X
n i i n n n
j j j j j j ju j nj j j j I j j ju j
i
n n n
I b ecause M so u A and j u j j u j r moreover
u
A A as
r r
Tan Lei
n n
To prove A u M we only need to show d z u M for each z Z
r u r u r
where Z is the dense set in A where we have continuous sections s z see condition
r
Let z Z By Rouch e stheorem for n large there is a U as a solution of the equation
n z
n n
s z u Hence M and j j C ju j j j Moreover
n u n n
n
j u j js z j js z j jz j r
n n n
n n n n
So d z u M jz u j jz s z j j u u j I I
u r n n n n n
We have I as ab ove and I by the continuity of s z and the fact s z z
Remark This prop osition is also true in a higher dimensional setting and under a weaker
hypothesis of di erentiability for and u See T for details
Let c b e a Misiurewicz p oint We now will adapt the situation just considered Recall
from the pro of of the second part of a that there exist a neighborho o d W of c and a
holomorphic mapping c such that K is asymptotically c self similar ab out the p oint
c
c to the c self similar limit set L c see Shrinking W if necessary we have
c V whenever c W Hence the Mandelbrot set in the region W can b e interpreted as
c
k
M W fc W j c K g c W Q c L c
c c
c
k
Q z j
z c
c
To apply the ab ove prop osition set W c A c L c and
k
u c Q c
c
c
k
Q z j
z c
c
We have proved in part a that c L c satis es the condition at c and L c is c
self similar ab out Moreover c c is holomorphic with j c j It is clear that
u c is also holomorphic As a consequence of the connectedness of M Douady and Hubbard
showed that u c Set L c L Now since M W M Prop osition shows that
u
for any r
n
L D M
r H r c
u c
Moreover an elementary calculation see T for details shows that the required is
k
z j Q
z c
c
d d
u c
k
Q c j c j
c c c c
c
dc dc
This ends the pro of of b
Example Let us take again the example of c i To nd the value of we rst obtain
k
Q z j Q z j i
z c i z i
c
d d
k
Q c j c c j c j i
c i c i c i
c
dc dc
The function c is the solution near i of the implicit equation Q z z i e z
c
c c z for c close to i Hence
d i i i i
c j and i
c i
i
dc i i i i
i
i
Tan Lei
Computer exp eriments con rm the similarity very impressively
The pro of of b is also done in two steps
C where denotes Prop osition Assume that we have a holomorphic motion i X
C with v z X v i z weak the unit disc and an analytic mapping v
transversality Set M f j v X g Then
v
H dim M lim H dim X z r
v
r
Proof sketch For simplicity we assume v In the simple case that X X
for we have M v X Since v is bi Lipschitz near and hence preserves
v
the Hausdor dimension we have H dim M H dim X z r for some r As a
v
consequence H dim M lim H dim X z r
v r
Now let us come back to the setting of our prop osition We will apply Rouch e stheorem
r
to prove that for r r where r is a small constant and Y v X z r there
r r
is R with R as r and a holomorphic motion j fj j R g Y C
r r r
r r
such that j Y M Therefore
v
r r r
H dim M H dim j Y C R H dim Y C R H dim X z r
v r r
where the existence of C R in the second inequality is due to a non trivial prop erty of
r
holomorphic motions see b elow Furthermore C R as r
r
r
More precisely by Slo dkowski s theorem see for example D the mapping j
extends to a K R quasi conformal mapping and K R as R On the
r r r
other hand by Mori s inequality K quasi conformal maps are K bi H older continuous
Furthermore a simple calculation shows that for any K bi H older continuous map j and
any set Y we have
K H dim Y H dim j Y K H dim Y
Setting C R K R we get the desired inequality
r r
r
To de ne the constant R and the holomorphic motion j we pro ceed as follows As
r
sume z and i Since the family of holomorphic maps fi z g is nor
z X
mal and i z for any z and any by injectivity any limit function of
the family corresp onding to a sequence z must b e the constant function Fix
n
s and a such that in s v is injective and aj j jv j Then for
b sup fji z j j z X r j j s g we have b as r
r r
Fix r such that as b for r r Take r r set R as b Take R
r r r r
and z X r The equation v i z has a unique solution r z in the
disc minfs s j jg we just apply Rouch e stheorem here
r r r r
Now set Y v X r and de ne j fj j R g Y C by j y
r
r r r
r v y It is then easy to check that j is a holomorphic motion and j Y M
v
Now we should adapt the ab ove result to the situation of the b oundary M of the Man
delbrot set It is much less trivial than the similarity case b ecause we want M to b e a subset
v
of M
Tan Lei
Lemma For the holomorphic motion in part a there are z X c with
N
0
a neighborhood of c and v c Q z v c i z for some N with v c i
c
c
c
0 0
z r i such that fc j v c X c g M and lim H dim X
c r c
Proof sketch By compactness of X there exists a p oint z X such that H dim X
lim H dim X r Let i c r X C b e the holomorphic motion
r
given in part a Since for c close to c the mapping i c do es not change to o much the
Hausdor dimension see the pro of of the ab ove prop osition there is a small neighborho o d
c r of c such that lim H dim X c i c z r for c
r
n
Recall that F denotes the map c Q The b oundary of M coincides with the set
n
c
f c C j the family F fF n N g is not normal at c g
n
see B Branner s pap er By part a c M We claim that there is c an integer
N such that F c i c z For otherwise the family F would satisfy Montel s
N
normality criterion at c with resp ect to the two analytic functions c i c z and one
branch of c Q i c z which contradicts the fact that F is not normal at c
c
Set v c F c In order to apply the ab ove prop osition we need to know that v c
N
i c z and the set M fc j v c X c g is a subset of M
v
One thing that was not explicitly stated in part a is that b esides the other prop erties
X X moreover the mapping i c conjugates the dynamics i e in a we have also Q
c
z Q i c z i c Q
c c
As a consequence Q X c X c and X c J
c c
There are at least two ways to see that v c i c z I Since c M there is
N
00 0 0
I I We may also use But i c z X c J J c M So F c Q
00
c c N
c
the normality argument Assume F c i c z in Then for any integer k
N
k k k
z F c Q F c Q i c z i c Q
N k N
c c c
where the last equality is due to the formula So the family F is uniformly b ounded in
hence normal This contradicts the fact that F is not normal at c
To prove M M we need Ma n e Sad Sullivan s characterization of M We say that
v
Q is J stable at c if there is a continuous map h c r J C such that h h c
c c c
I d Then to J Q and h Q is a conjugacy from J
c c c c c
M fc C j Q is NOT J stable at c g
c
This formula can b e used to get another way to prove v c i c z for otherwise one
S
n
can pull back the formula to get a holomorphic motion of Q v c Since this is
c
n
a dense subset of J we can apply the lemma of Ma n e Sad Sullivan to show that Q is
c c
J stable at c thus a contradiction
Now assume that c is a p oint in M M So Q is J stable at c and admits a conjugacy
v c
map h c r J C Decreasing r if necessary we may assume c r Set
c
X i c X We claim that hj must coincide with the holomorphic motion i
c r X
The reason is that b oth maps are continuous and h c z i c z for z any rep elling p erio dic
p oint and rep elling p erio dic p oints are dense in X this is b ecause X is a hyperb olic subset
Tan Lei
On the other hand h must preserve the critical p oint i e h c So h c v c v c
This gives rise to a contradiction since v c i c z but v c i c z
Hence all the conditions required by the ab ove prop osition are satis ed So
H dim M H dim M lim H dim X c i c z r
v
r
This completes the pro of of b
App endix
An example of a self similar set Let A denote the standard middle third Cantor
set Take all logarithmic spirals through p oints in A which in logarithmic co ordinates are
t it
straight lines parallel to the vector log i This set is self similar for e for
any t R
Lo cal connectivity For c a Misiurewicz p oint T constructed a sequence of Jordan
S
curves in the dynamical plane such that A fc g is asymptotically self similar
n n
n
and the set of b ounded comp onents U of C forms a basis of nested neighborho o ds of
n n
connected Moreover there is a holomorphic motion i c r A U K C c with
n c
with A c asymptotically c self similar As n the sequence of subsets P fc
n
c r j c c g b ounds a nested sequence of neighborho o ds W of c in the parameter
n n
W M connected Applying Prop osition to P we see that it is also plane with
n n
asymptotically self similar In particular the diameters of W shrink to zero exp onentially
n
fast This is a stronger statement than saying that M is lo cally connected at c
Hausdor dimension Let E d b e a metric space For A E denote by jAj its diameter
For X E t we de ne
X
t
jA j X A jA j m X inf
j j j
t
fA g
j
j N j N
Fixing t m X increases as decreases We can then de ne m X lim m X
t
t t
X Note that m X can b e An easy calculation shows that if for some t sup m
t
t
m X then m X for any T t As a consequence there is a unique number
t T
such that m X for t and m X for t This is called the
t t
Hausdor dimension of X
It is not easy to calculate the Hausdor dimension in general However in the following
situation there is an easy lower b ound Let G V U b e an analytic covering with U an
op en disc V a nite union of op en discs with disjoint closures and V U Then for the
m
non escaping set X fz V j G z V for all m g one has
log number of comp onents of V
H dim X
log max jG z j
The set X is a sp ecial example of hyperbolic sets and is automatically stable under p ertur
bation
Pro of of theorem a As an example we give the main steps to show that there are
c c c M and hyperb olic subsets X K such that H dim X as
n n n c n
n
n The technique is called geometric limits of a parab olic map and parab olic implosion
Tan Lei
A similar study can b e done for each c in a dense subset of M namely the set of ro ots of
primitive hyperb olic comp onents
Denote by f the map z z There will b e two holomorphic functions g and h the
rst and a second geometric limit of f generating the family of maps
k l m
L ff g h j k l m Z m g
with a certain convention on f and g satisfying the following two prop erties
There exists a sequence c with c M but one can also choose c in the
n n n
main cardioid or c C M such that for each G L there are integers j n i
n i
j n i
converges to G uniformly on compact sets in the domain of de nition of such that Q
c
i
n
G
i
There exists a small neighborho o d U of and constants a C C such that for
large there are op en sets U U with N a U U U U and G L
N i j i i
j C log U U is bijective and C log jG j such that G j
i i
i
U U
i i
As a consequnce for n large i N there is j n i large and U n i close to U
i
j n i
maps U n i bijectively onto U with derivative close that of G For X n such that Q
c
i
n
the non escaping set the formula in App endix gives us
log log a
H dim X n
log log log constant
We will skip the pro of of which can b e found in the pap ers of Shishikura and in D
and give a sketch of the construction of g h U U G and the estimate of jG j
i i
i
a Denote by B the basin of the parab olic p oint for f By classical results there
are holomorphic surjective maps C C w z and B C z w such that
f T and f T where T denotes the translation w w the
mappings are called Fatou coordinate changes of f Set g g B C
iw
g B C and g g B C where w e Then
there exist choices unique up to addition of integers of such that g w w o
as I m w and For this g g g g we have g and
g j j
b Because B is simply connected and contains only one critical value the immediate
g has the same prop erty Similarly one can nd but with basin B of for the map
g with the same asymptotic b ehavior at and C C instead and de ne g g
There are lifts of g by and successively We call them h and h
c Here is the diagram of our construction
U W
i i
m
i l
i
g G s s T
i
U W
l l m
k i m i i
i
where s T g and s h g g The other terms are going to b e de ned b elow
Fix U a small disc neighborho o d of There is a disc W and a choice of such that
W onto U bijectively with b ounded derivative maps
Since is rep elling for g the mapg b ehaves like a translation w w with I m
as I m w So for m N there are W disjoint discs such that g W W
m m
m
Tan Lei
m
W W is a bijection with b ounded derivative I m W as m and g
m
m
indep endent of m
Fix large In the rectangle R fw j w j I m w g there are N
l
i
disjoint discs W W with N a and l m Z m such that T W W
N i i i i m
i
m
l
i
i
W W is a bijection with b ounded derivative Therefore for each i the map T g
i
indep endent of i and
Let C R fw j w g b e the sp ecial branch of Set U
i
W
i
One can easily check that for large and a go o d choice of h we have U U and there
i
k l m k l m
i i i i
is k Z indep endent of i and such that f g h U U Set G f g h j
i i U
i
W R the rest has d The derivative jG j is controlled by j w j w
i
i
b ounded e ects On the other hand when is large b oth and can b e approximated
by I w w A simple calculation shows that jG j log
i
Some key words dynamical spaces parameter space pro duct space implicit function theorem
Rouch e stheorem holomorphic motion
References
D A Douady Prolongement de mouvements holomorphes d apr es Slo dkowski et autres
S eminaireBourbaki number November
D A Douady Do es a Julia set dep end continuously on the p olynomial Pro ceedings of
Symp osia in Applied Mathematics Vol
S M Shishikura The parab olic bifurcation of rational maps Col oquio Brasileiro de
Matem atica IMPA
S M Shishikura The Hausdor dimension of the b oundary of the Mandelbrot set and Julia
sets preprint SUNY Stony Bro ok
S M Shishikura The b oundary of the Mandelbrot set has Hausdor dimension two S M F
Ast erisque
T Tan Lei Similarity b etween Mandelbrot set and Julia sets Commun Math Phys
T Tan Lei Voisinages connexes des p oints de Misiurewicz Ann Inst Fourier Grenoble
UMR CNRS ENS Lyon all eed Italie Lyon France Department of Mathematics University of Warwick Coventry CV AL U K
Classi cation in Algebraic Geometry
A Short Course organized by Rosa M Mir o Roigand Raquel Mallavibarrena
The session consisted of three talks organized with the aim of present to a non sp ecializ ed
audience some basic facts and main problems that b elong to this relatively recent but vast
eld of Algebraic Geometry
The rst talk was given by Margarida Mendes Lop es and it was essentially an introduction
with basic concepts and de nitions all this with the goal of making the other two talks
understandable
Mireille Martin Deschamps sp oke in the second place She presented some asp ects of the
problem of classi cation of space curves results techniques
The third talk was given by Emilia Mezzetti She also fo cused on a classi cation problem
the one for pro jective varieties of small co dimension Here as well as in the previous talk
the use of sheaves schemes and cohomology was proven to b e a p owerful to ol In fact these
techniques have made p ossible to talk of the so called mo dern algebraic geometry
There was also a kind of example session where several computations were made concerning
some well known problems that have an elementary treatment Rosa Maria Mir o Roig
th
Proc of the EWM
meeting Madrid
On some notions in algebraic geometry
Margarida Mendes Lop es
Universidad de Lisb oa Portugal
mmlopes ptmat lmc fc ul pt
th
Foreword This pap er is an expanded version of the talk given to the meeting of EWM
It was meant as a very basic introductory talk and do es not prop ose to b e a survey on
surface theory
Some notions
n
We will b e dealing with the pro jective space P over the complex numbers C The pro jective
n n n
space P is the set of dimensional vector spaces of C and for each p oint p P its
homogeneous co ordinates x x are de ned up to scalar multiple
n
Given an homogeneous p olynomial f x x its lo cus of zero es is a well de ned subset
n
n
of P called an algebraic hypersurface
n
More generally a pro jective closed algebraic set V P is the common lo cus of zero es of
a nite set of homogeneous p olynomials i e V fx x x f x f x g
n k
A closed algebraic set is a pro jective algebraic variety if V is irreducible i e V cannot b e
written as the union of two prop er subsets which are algebraic sets
Given an algebraic closed set we consider the ideal I V formed by all homogeneous
p olynomials which vanish identically on the p oints of V This ideal is a nitely generated
homogeneous ideal and it will b e a prime ideal if and only if V is irreducible
n
Asso ciated to a variety V P we have its dimension and its degree
There are various equivalent ways of de ning the dimension but p ossibly since we are
talking ab out algebraic varieties over the complex numbers the easiest way is de ning the
dimension of V as its dimension as a complex top ological space Other p ossible ways which
are equivalent are de ning the dimension as d n r where r is the maximum rank of the
g
i
matrix evaluated over every p oint p V where g are a set of generators for I V or
i
x
j
equivalently we can de ne the dimension as the maximal m such that V pro jects surjectively
m
to P
Now the degree can b e again de ned alternatively as the number of p oints of intersection
of V with a general linear subspace of complementar dimension or then as the number of
p oints on a general bre of a pro jection which realizes the dimension For an hypersurface
the degree is simply the degree of a p olynomial generating I V and the dimension is n
A variety of dimension is a curve whilst one of dimension is a surface As simple
examples one has for instance the cuspidal cubic curve in P de ned by y z x the
Margarida Mendes Lop es
twisted cubic curve in P which is de ned by the equations
x x x x x x x and x x x
or a complete intersection of two quadrics in P
Let us p oint out that similarly to what happ ens with manifolds we can also give a notion
of an abstract algebraic pro jective variety which do es not dep end on the ambient space we are
considering and the dimension do es not dep end on the pro jective space we are considering
Of course the degree will not b e invariant in this context Anyway for our purp oses it will
b e enough to keep this de nition as the lo cus of solutions of a nite system of p olynomial
equations in mind
Given an algebraic variety we can de ne a rational function on it as b eing a function
de ned lo cally by a quotient of homogeneous p olynomials of same degree A rational map
n
b etween algebraic varieties V W P is de ned by v f v f v where f
n i
is a rational function on V Now the main thing to remark is that a rational map is not
necessarily de ned everywhere b ecause there are p oints which are p oles of every rational
function app earing A morphism is a rational map everywhere de ned and we will say that a
rational map is birational if it has an inverse which is a rational map In this case basically
what happ ens is that the two varieties are the same in the complement of a closed algebraic
set
We will b e mainly concerned with smo oth also called non singular pro jective varieties
which are those such that the rank of the matrix ab ove is constant at every p oint Although
apparently we are missing out a lot that is not the case at least up to birational equivalence
due to Hironaka s desingularization theorem which says it is p ossible given a variety to nd
a non singular variety which is birational to it by blowing ups
What is classi cation
There are various problems of classi cation that arise One is exactly classifying algebraic
pro jective varieties up to birational equivalence and nding a way of describing all equivalence
classes and this puts us into theory of mo duli
Another is trying to nd out which smo oth varieties and of a given dimension live in some
n
P and what degrees can turn up Yet another is trying to classify according to the the
n
minimal number of generators for its homogeneous ideal in P
It turns out that every smo oth algebraic variety of dimension d can b e isomorphically
d
pro jected in P So for instance one p ossible problem of classi cation is trying to nd for
given degrees which smo oth varieties of dimension d are embedded and with which degrees
k
in P where k d
Supp ose then we want to classify smo oth varieties of a given dimension up to birational
equivalence We would like to have a way of individuating in each birational equivalence class
a representative For curves it can b e shown that each birational equivalence class contains
a unique smo oth curve In the case of surfaces this is no longer true but nevertheless we can
nd minimal smo oth surfaces A surface S is said to b e minimal if any birational morphism
0 0
of S S with S another smo oth surface is an isomorphism Given any smo oth surface
X it is always p ossible by blowing down the exceptional curves to nd a smo oth minimal
surface S together with a birational morphism X S S is said to b e a minimal mo del
of X Now what happ ens for surfaces is that except for the surfaces in a certain class i e
Margarida Mendes Lop es
those with k od X see section every birational equivalence class contains an unique
minimal non singular mo del and this helps a lot with the classi cation
In this talk I ll fo cus precisely on the classi cation of surfaces up to birational equivalence
more precisely on the usually called Enriques Ko daira classi cation For this I will need
Some more notions
P
A divisor D on a smo oth variety X is a formal nite sum D a C where a is an integer
i i i
and C is any subvariety of X of co dimension A divisor is e ective if a for every i The
i i
set of all divisors with the obvious op eration is a group To every non zero rational function
f on X corresp onds the divisor of f f which is de ned as b eing the di erence f f
of the two e ective divisors f f given resp ectively by the zero es and the p oles of f
Two divisors are linearly equivalent if their diference is the divisor of some rational function
If X is a smo oth surface we can de ne the intersection number of two divisors This
intersection number for two curves meeting tranversally at smo oth p oints is exactly the
number of p oints in which the curves meet
In the complex case it can b e de ned as an intersection number in homology In fact each
divisor D corresp onds to a homology class in H X Z Given two divisors D D we can
de ne the intersection number D D as b eing the intersection number of their homology
classes This intersection number can also b e de ned in purely algebraic terms
Given a divisor D on an algebraic variety one can asso ciate to it a vector space L D
consisting of and the rational functions f such that div f D is an e ective divisor If
n
this vector space is non zero and n dimensional we can consider the map X P
D
given by the functions of a basis of L D
In particular one can sp eak ab out the pluricanonical maps of the variety X which are
asso ciated to the multiples of the canonical divisors of X What are those
The canonical divisors can b e de ned in purely algebraic terms but in this context since
we are considering varieties over the complex numbers we are going to use the language of
complex manifolds
In fact any smo oth algebraic variety over C is a compact complex manifold with the
n
holomorphic structure inherited from P
Let us remark that the class of compact complex manifolds is bigger than the class of
smo oth algebraic varieties A compact complex manifold which is also an algebraic variety
is said to b e pro jective algebraic Nevertheless one has
Theorem Every compact complex manifold of dimension Riemann surface is algebraic
A compact complex manifold X of dimension is algebraic if and only if a tr deg M X
C
where M X is the eld of meromorphic functions on X
If the compact complex manifold X is pro jective algebraic it turns out that any meromor
phic function is actually a rational function
Asso ciated with each compact complex manifold of dimension n one has the canonical line
n
is the holomorphic cotangent bundle So K is the where T bundle which is K T
X X
X X
line bundle which holomorphic sections are the n forms on X for instance in the surface case
lo cally if z z are lo cal co ordinates lo cal holomorphic sections are given by f z z dz dz
where f z z is an holomorphic function
Margarida Mendes Lop es
m
Having the canonical bundle we de ne the pluricanonical bundles K
X
The canonical resp pluricanonical divisors on the smo oth pro jective variety X are
then de ned as the divisors obtained as the divisor of zero es minus the divisor of p oles of a
m
The canonical divisors are denoted by K and the meromorphic section of K resp K
X X
X
pluricanonical by mK omiting the subscript when there is no danger of confusion
X
This takes us to yet another de nition Ko daira dimension of X k od X as the maximum
dimension of the image of for m N One has always k od X dim X and if
mK
k od X dim X X is said to b e of general type If L mK by convention dim
X
mK
The Ko daira dimension is a birational invariant and this gives us a very rough classi cation
of varieties with resp ect to birational equivalence which is done by dividing them in various
classes according to the Ko daira dimension
The Ko daira dimension in fact can b e de ned for any compact complex manifold In the
case of surfaces it turns out that if the compact complex surface X has k od X then it
is an algebraic surface
The rough classi cation of minimal surfaces
For surfaces one can describ e the surfaces in some of these classes in more detail To explain
this still rough classi cation of we will need some more notions
A surface S is called elliptic if it admits an elliptic bration i e a morphism f onto a
smo oth curve B such that almost all bres f p are smo oth elliptic curves i e Riemann
surfaces of genus
Asso ciated with a surface we have various invariants Some are linked to the canonical
divisors
p dim L K number of linearly indep endent forms geometric genus
g C
p dim L mK plurigenera
i C
q number of linearly indep endent forms called the irregularity
O p q
S g
the self intersection of a canonical divisor K also denoted by c
c top ological Euler Poincar echaracteristic
These invariants are linked by various relations of which maybe the most relevant is the
No ether s formula
c O K
S
S
Let us notice that all these invariants can b e recovered from the top ology of S Also p p and
g i
q and O are birational invariants whilst K and c are not For birational equivalence
S
is maximum for the surfaces in C classes C with a unique minimal mo del S one has that K
S
Let us notice also that in spite of not lo oking so via Ho dge theory it can b e shown that
all these invariants are in fact top ological and computable from triangulations
Now we can give
The rough classi cation of minimal algebraic surfaces Enriques
Margarida Mendes Lop es
I Kodaira dim These are the surfaces for which p for all i N and are either
i
P or ruled surfaces i e a P bundle over a smo oth curve
I I Kodaira dimension These are the surfaces for which p for all i N and p
i i
for some i N and are divided in the following di erent classes
a bielliptic sometimes called hyperelliptic surfaces admit a lo cally trivial bration
over an elliptic curve with elliptic bre p q
g
b K surfaces These are the simply connected surfaces with K O p
X X g
q
c Enriques surfaces These are such that K O and K O p q
X X X X g
and are quotients of K surfaces by a xed p oint free involution
d Ab elian surfaces These are quotients of C by some lattice p q
g
I I I Kodaira dimension These surfaces have p for some i N but the corresp onding
i
pluricanonical image is a curve and are all elliptic surfaces
IV Kodaira dimension These are the surfaces for which some pluricanonical image is a
surface and are called surfaces of general type
Examples of surfaces in I are the pro jective plane any smo oth surface of degree n
n
in P any surface isomorphic to the pro duct P C with C a smo oth curve
Let us notice that the surfaces with Ko daira dimension form a much bigger class
For instance for the smo oth surfaces in P which are hypersurfaces hence de ned by some
p olynomial of degree d one has that if d S is birationally equivalent to P if d S
is a K surface whilst if d S is of general type
Let us also remark that this classi cation can b e extended to compact complex surfaces
not necessarily algebraic For details see BPV or s Pe
Surfaces of general type geographical questions and pluri
canonical mo dels
Most surfaces will b e of general type and one do es not have a neat description like the ones
for Ko d One has
Theorem Bombieri B Let S be a minimal smooth surface of general type Then for
m is a birational morphism and the images S are al l isomorphic Fur
mK mK
thermore S is a surface having at most double points as singularities and such that its
mK
minimal desingularization is isomorphic to S
Corollary Minimal surfaces of general type with given with given K c or equivalently
N
given K O are surfaces of given degree in a given P and therefore are parametrized
S
by a nite union of irreducible algebraic varieties i e belong to a nite number of families
By the Riemann Ro ch theorem K O completely determine p i for minimal
S i
surfaces of general type and hence we have this corollary which Gieseker used to prove the
existence of a coarse mo duli space for surfaces of general type
Margarida Mendes Lop es
Theorem Gieseker For each pair of integers such that x y x there exists a
coarse moduli space M for al l the isomorphism classes of minimal surfaces of general type
x y
with invariants K y and O x Furthermore M is the union of a nite number
S x y
of quasi projective varieties in particular it has a nite number of connected components
Where do these relations app earing in Gieseker s theorem come from For minimal surfaces
of general type one has the following relations b etween the invariants
i K c and O
S
ii K c O No ether s formula again
S
iii M No ether Bombieri K O and if q K O
S S
iv Bogomolov Miyaoka Yau K c equivalently K O
S
Within these restrictions one type of question known usually as a geographical question
is for what pairs of natural numbers n m satisfying the ab ove relations there are minimal
surfaces of general type with K n O m
S
For what concerns geographical questions roughly the answer is for almost all pairs and
this is proved by constructing very sp ecial surfaces
The type of geographical results one has
Theorem chronologically Persson P Sommese S Chen C C Ashikaga A For
each pair of natural numbers x y such that x y x or x y x
with the possible exceptions of y x d d there exists a minimal surface S
of general type such that K x O y
S
S
Another type of questions are the so called b otanical questions given a pair of numbers
K O or equivalently K c in the allowed region classify all minimal surfaces of
S
general type with these invariants describ e families and if p ossible describ e the mo duli space
This last is almost imp ossible as follows from the following
Theorem Catanese Ca Ca Ca Manetti M For al l n N there is a pair
K O such that the moduli space has more than n irreducible components pairwise of
S
di erent dimensions
Nevertheless in some cases one has descriptions of the mo duli spaces like the ones given
by Horikawa for surfaces with K O see Ho
S
There are a lot of other problems on surfaces I did not refer to as the problems of
equivalence of di erentiable stuctures on the top ological manifold underlying an algebraic
surface or which of the ab ove prop erties and classi cation hold for surfaces de ned over a
eld of characteristic p
Margarida Mendes Lop es
References
This set of references is very incomplete For an extended bibliography on surfaces see sC
Basic b o oks
GH P Gri ths J Harris Principles of algebraic geometry J Wiley and Sons
Ha R Hartshorne Algebraic geometry Springer
H J Harris Algebraic geometry a rst course GTM Springer
R M Reid Undergraduate algebraic geometry LMS Student Texts C U P
S I R Shafarevich Basic algebraic geometry Springer
Basic theory of surfaces
Be A Beauville Surfaces alg ebriquescomplexes Ast erisque Paris
BPV W Barth C Peters A Van de Ven Compact Complex Surfaces Ergebnisse der Math
Springer Verlag
Surveys
sC F Catanese Some old and new results on algebraic surfaces Pro ceedings of the First
Europ ean Congress of Mathematicians
sP U Persson An introduction to the geography of surfaces of general type Pro c Symp in
Pure Math
sPe C Peters Introduction to the theory of compact complex surfaces Rep ort of the Univ
of Leiden Octob er
Papers refered to
A T Ashikaga A remark on the geography of surfaces with birational canonical morphisms
Math Ann
B E Bombieri Canonical models of surfaces of general type Publ Math IHES
Ca F Catanese On the mo duli space of surfaces of general type J Di Geom
Ca F Catanese Connected comp onents of mo duli spaces J Di Geom
Ca F Catanese Automorphisms of rational double p oints and mo duli spaces of surfaces of
general type Comp Math
C Z Chen On the geography of surfaces simply connected minimal surfaces with positive
index Math Ann
Margarida Mendes Lop es
C Z Chen On the existence of algebraic surfaces with preassigned Chern numbers
Math Z
Ho E Horikawa Algebraic surfaces with small c I Ann Math I I
Invent Math I I I Invent Math IV Invent
Math
M M Manetti On some comp onents of mo duli spaces of surfaces of general type Publ
Scuola Normale Pisa
P U Persson On Chern invariants of surfaces of general type Comp Math
S A J Sommese On the density of ratios of Chern numbers of algebraic surfaces Math Ann
th
Proc of the EWM
meeting Madrid
On classi cation of algebraic space curves
liaison and mo dules
Mireille Martin Deschamps
Ecole Normale Sup erieure Paris France
deschamps dmi ens fr
Smo oth space curves
In the rst pap er on algebraic geometry Margarida Mendes Lop es explained the structure
of the pro jective space over the eld C A space curve is a closed algebraic subset C of the
pro jective space P which has dimension over C note that it has dimension over R
It is de ned by homogenous p olynomials F F b elonging to the ring of p olynomials in
r
variables S C X X X X Since a single p olynomial de nes a surface of dimension
over C we have r
The pro jective space P is obtained by gluing together a ne spaces so a space curve
C can b e covered by a ne pieces If we want to study local prop erties of the curves that is
prop erties in a neighbouro o d of a p oint we can always restrict ourself to a p oint contained
in an a ne piece of the curve For example we say that a p oint P a a a a of C
is smooth or that C is smo oth at P if the tangent space to C at P has dimension To
express it algebraically we can supp ose that a and work in the op en subset X
Let f x x x F x x x for i r Then P is a smo oth p oint of C if and
i i
f
i
only if the rank of the matrix P is
x
j
A curve is smooth if and only if it is smo oth at every p oint
Examples the curve de ned by the p olynomials X X is smo oth it is a line and the
X is not smo oth it has a cusp at the p oint X curve de ned by the p olynomials X X
It is p ossible to show that a general plane intersects transversally the curve in d distinct
p oints where the number d which do esn t dep end on the plane is called the degree of the
curve a plane which is not general is a plane which is tangent
Now a question why are we interested in space curves We can of course de ne pro jective
n n
curves in P for n but one proves that if C is a smo oth curve contained in P for n
n n
there exists a pro jection from P to an hyperplane P such that the restriction of to C
is an algebraic isomorphism from C onto its image In other words every smo oth pro jective
curve can b e embedded in P On the other hand plane curves are in some sense sp ecial
Mireille Martin Deschamps
Example let J b e the ideal generated by the minors of the matrix
X X X
X X X
The quadric surfaces de ned resp ectively by X X X and X X X have a line de ned
by X X in common Their intersection is a curve of degree by Bezout theorem
therefore it is the union of the line and a curve C of degree called a twisted cubic whose
ideal of zero es is J More generally any curve pro jectively equivalent to C is called a twisted
cubic If one pro jects C to a plane one gets a plane cubic curve with a singularity
Complete intersections Liaison
We have seen that a space curve is de ned by at least two homogenous p olynomials From
this p oint of view the simplest case is therefore the case when the curve is de ned by exactly
two homogenous p olynomials F and G without common factor since the dimension is
Such a curve is called a complete intersection of two surfaces c i for short Its degree is
simply the pro duct of the degrees of F and G For example plane curves of degree d are c i
of a plane and a surface of degree d
Like plane curves c i are in some sense sp ecial The smo oth curves of least degree which
are not c i are the twisted cubics In fact if the degree of a c i is a prime number one of
the two surfaces has degree so is a plane And we have seen that the twisted cubics are
not plane
Let again C b e a twisted cubic We have seen that it is contained in the intersection of
two quadrics which is a c i and that the residual intersection is a line We say that the
cubic and the line are linked by the c i
De nition Two curves C and C without common comp onent are linked by a complete
intersection X if X is the union of C and C
In fact it is p ossible to enlarge this concept of liaison to curves having some common
comp onents but I will not explain it here
Now this relation is re exive and symmetric but not transitive it generates an equivalence
relation called liaison or linkage
De nition Two curves C and C are in the same liaison resp biliaison class if and only
if there exist an integer n and a sequence C C C C C such that C and C
n i i
are linked by some complete intersection resp and n is even
This concept has b een introduced rst by Ap ery A A and Gaeta G and devel
opp ed by Peskine and Szpiro PS It turns out that it is a very useful to ol in the classi cation
of space curves There are precise relations b etween the numerical invariants of the curves
degree genus dimension of the cohomology spaces and we will see further that it is p ossible
to characterize the biliaison classes But even if a curve is smo oth a linked curve can b e non
smo oth So even if one is interested in smo oth curves it is necessary to study more general
locally Cohen Macaulay l C M for short curves It is not necessary to give here a precise
de nition but these curves have two imp ortant prop erties
i the complete intersections are l C M
ii if two curves C and C are linked C is l C M if and only if C is l C M
Therefore it is the natural context for liaison problems
Mireille Martin Deschamps
Functions on smo oth space curves Hartshorne Rao mo dule
Let f b e a p olynomial in variables x x x It de nes in a natural way a function from
the op en subset of C where X to C Morover if f is one of the f or more generally
i
if f b elongs to the ideal f f generated by the f the corresp onding function is the
r i
zero function So we obtain a map which is a ring homomorphism from the quotient ring
A C x x x f f to the ring of functions on C One can prove that if C is
r
smo oth this map is injective So A can b e identi ed with a subring of the ring of functions
on C Elements of A are called rational algebraic functions on C de ned where X
Supp ose for simplicity that the plane at in nity X intersects transversally the
curves in d distinct p oints P P If f is a non zero element of A we want to describ e
d
the corresp onding function in a neighbouro o d of the P For that purp ose we have the notion
i
of order which is de ned in the following way
X X X
n
F X X X X X where F is an homogenous p olynomial of Let f
X X X
degree n It is the equation of a surface S and we can de ne the multiplicity of intersection
m C S P of C and S at P it is rather complicate but it is if S do esn t go through P
i i i
if C and S intersect transversally at P and if S is tangent to C at P Then the order
i i
f m C S P n If this order is p ositive resp negative of f at P is the number or d
i i P
i
it means that f can b e de ned resp cannot b e de ned we say that P is a p ole of f at P
i i
f n With this de nition we see that or d
P
i
X X X It X Example let C b e the curve de ned by the p olynomials X
has p oints at in nity P P Let f b e the function de ned by the
f m C S P p olynomial x x Then one has F X X X X and or d
i P
i
Since S do esn t go through P we have or d f If resp S is
P
f In any f resp or d transversal resp tangent to C at P so we have or d
P P
case P is a p ole of f but f is de ned at P
Linear systems on smo oth space curves
Thanks to this notion of order we can introduce a ltration on the ring A by setting for
n N
A ff A jf or or d f n i dg
n P
i
This is a nite dimensional vector space over C which is called by the algebraic geometers
for reasons that I can t develop here the space of sections of the line bundle O n It is
C
p ossible to de ne it in an intrinsic way
For example for n we obtain the set of global functions on the curve functions
de ned everywhere
For every p ositive n recall that S is the set of homogenous p olynomials of degree n there
n
exists a natural map of vector spaces S A de ned in the following way if F S is
n n n n
non zero F is the image of f F x x x in the quotient ring A we have already
n
seen that if F S the order of f at P is n
n i
Mireille Martin Deschamps
Moreover we can put in a natural way a structure of graded S algebra on the direct sum
A A so that the direct sum of all the gives an homomorphism of graded
C n N n n C
S algebra from S to A Of course this map is not injective every F de ning the curve go es
C i
to but we will see now that it can also b e not surjective However
if n is or if n is then is surjective
n
if C is a complete intersection de ned by two p olynomials F and G then is
C
surjective and A S F G
C
De nition The Hartshorne Rao mo dule or Rao mo dule M of a curve C is the
C
cokernel of the map It is a graded S mo dule of nite length b ecause it has only a nite
C
number of non zero homogenous comp onents
This mo dule which was introduced by Hartshorne and studied by Rao has very nice
prop erties and in some sense it re ects algebraic prop erties of the curve Its non zero
elements corresp ond to functions on the curve which not come from functions de ned on the
pro jective space
Example let C b e the curve de ned by the p olynomials Y X Y X X X X X where
Y is a linear form indep endant of X X X and not a multiple of X It is the union
of the two lines L de ned by Y X and L de ned by X X which don t intersect
There are two p oints at in nity P on L and P on L
Since the two lines are disjoint one proves easily that the ring of rational algebraic func
tions on C de ned where X is the pro duct of the two corresp onding rings of functions
on L and L Therefore A is in a natural way the pro duct of A S Y X and
C L
0
S X X The map S S Y X S X X is the pro duct of the two A
C
L
natural pro jections Hence the Rao mo dule M is S Y X X X It has only one non
C
zero comp onent of dimension and in degree zero which corresp ond to the function taking
the value on one of the lines and on the other one there are two such functions but their
sum can b e lifted on the pro jective space
More generally if C is the disjoint union of two c i F F and F F one proves that
M S F F F F
C
Liaison and Rao mo dule
There are nice connexions b etween these two notions and they are summarized in the fol
lowing results
Two curves C and C are in the same biliaison class if and only if there exists h Z
0
h R if M M is a graded S mo dule M h is the graded such that M M
n C
C
S mo dule de ned by M h M we say that the degrees are shifted by h to the
n n h
left
Example the curves C with M are in the same biliaison class which contains
C
all the c i but not only c i for example the twisted cubics
Let M b e a graded S mo dule of nite length There exist a smo oth curve C and an
integer h Zsuch that M M h R
C
Hence the Rao mo dules up to a shift characterize the biliaison classes
Mireille Martin Deschamps
It is easy to prove by an elementary geometric construction that starting from an
existing Rao mo dule every right shift can b e obtained But there exists a minimal left
shift Mi A corresp onding curve is called a minimal curve in the biliaison class
One knows how to construct explicitely the minimal curves from the mo dule asso ciated
with the biliaison class MD P
One knows how to obtain the curves in the biliaison class from a minimal one BBM MD
P
So the classi cation of these mo dules which is an algebraic problem will help us for the
classi cation of space curves
References
A R Ap ery Sur certains caract eresnum riques d un id eal sans comp osant impropre
C R Acad Sci Paris t p
A R Ap ery Sur les courb es de premi ere esp ece de l espace a trois dimensions C R
Acad Sci Paris t p
B G Bolondi Irreducible families of curves with xed cohomology Arch der Math
BB E Ballico et G Bolondi The variety of mo dule structures Arch der Math
BBM E Ballico G Bolondi et J Migliore The Lazarsfeld Rao problem for liaison classes
n
of two codimensional subschemes of P apara trein Amer J of Maths
BM G Bolondi et J Migliore Classi cation of maximal rank curves in the liaison class
L Math Ann
n
BM G Bolondi et J Migliore Buchsbaum Liaison classes J of Algebra
e
Ein L Ein Hilb ert scheme of smo oth space curves Ann Scient E N S s erie
e
E G Ellingsrud Sur le sch emade Hilb ert des vari et esde co dimension dans P ac one
e
de Cohen Macaulay Ann Scient E N S s erie
G F Gaeta Quelques progr es r ecents dans la classi cation des vari et es alg ebriques
d un espace pro jectif Li ege C B R M
GP L Gruson et C Peskine Genre des courb es de l espace pro jectif Lecture Notes
Springer Verlag
GP L Gruson et C Peskine Genre des courb es de l espace pro jectif I I Ann Scient
e
E N S s erie
H G Halphen M emoiresur la classi cation des courb es gauches alg ebriques Oeuvres compl etes t
Mireille Martin Deschamps
jH J Harris Curves in pro jective space Presses de l Universit ede Montr eal
rH R Hartshorne On the classi cation of algebraic space curves in Vector bundles
and di erential equations Pro ceedings Nice France Progress in Math
Birkh auser
rH R Hartshorne Algebraic Geometry Springer Verlag
LR R Lazarsfeld et A P Rao Linkage of general curves of large degree Lecture
Notes Springer Verlag
MD P M Martin Deschamps et D Perrin Sur la classi cation des courb es gauches I C R
Acad Sci Paris t S erieI p
MD P M Martin Deschamps et D Perrin Sur la classi cation des courb es gauches Ast erisque
Mi J Migliore Geometric Invariants of Liaison J of Algebra
PS C Peskine et L Szpiro Liaison des vari et esalg ebriques Invent Math
R A P Rao Liaison among curves in P Invent Math
th
Proc of the EWM
meeting Madrid
On classi cation of pro jective varieties of small
co dimension
Emilia Mezzetti
Universit adi Trieste Italy
mezzette univ trieste it
Introduction
The sub ject of my pap er is the classi cation of embedded varieties esp ecially in low co di
mension To clarify the meaning to b e given to this term let me start with the following
theorem which is classical
Theorem Let X be a smooth projective variety of dimension n over C Then X can be
n
isomorphically projected in P
Idea of proof the pro jection centered at a p oint P is an isomorphism from X to its image
if and only if P do es not b elong to any secant or tangent line to X The secant variety of X
S ecX which is the union of all secant and tangent lines to X has dimension at most n
So if X is a curve natural setting is P for surfaces P and so on In other words in P
we nd al l curves up to isomorphism So to study the isomorphism classes of smo oth curves
the moduli space it is enough to study those of curves lying in P This was the sub ject of
the pap er by Mireille Martin Descamps
If X may b e embedded in a pro jective space of dimension n then X is in some sense
sp ecial For example plane curves surfaces in P and P We will try to understand in
what sense X is sp ecial Moreover there is the natural problem of the classi cation of such
sp ecial varieties
Recently there has b een new interest in this sub ject due particularly to a conjecture
Hartshorne s conjecture and a theorem by Ellingsrud Peskine Moreover new tech
niques as for example adjunction theorems and computational metho ds have led to consid
erable progress
Preliminary facts
n
The rst imp ortant observation is that the varieties of dimension n lying in P i e of co di
mension are precisely the hypersurfaces they are sets of zero es of a unique homogeneous
p olynomial in n variables Moreover for any degree d a general p olynomial of degree d
de nes a smo oth hypersurface
We may say that in co dimension one sp ecial means only one equation is needed
Emilia Mezzetti
When we take the intersection of two hypersurfaces then we get a variety which is not
necessarily irreducible
A simple example in P take the irreducible quadrics of equations
x x x x x x x
The intersection contains the line x x the other comp onent is the skew cubic X
given parametrically by
x s x s t x st x t
So the two comp onents are curves intersecting at the p oint If we intersect
further with the quadric
x x x
then the dimension do es not decrease we nd precisely X
In general we have the following
m
Theorem Let X Y P have dimension r s respectively Let Z be an irreducible com
ponent of X Y Then dim Z r s m
m
In particular if X X X are hypersurfaces in P and Z is an irreducible comp onent
r
of their intersection then dim Z m r Note that in the previous example the inequality
was strict
m
Now we can de ne the varieties complete intersection X of dimension n in P is a
complete intersection c i for short if X is the transversal intersection of m n hypersurfaces
or equivalently the homogeneous ideal of X I X is generated by m n homogeneous
p olynomials
It is p ossible to prove that a general c i variety is smo oth The canonical sheaf of a c i
m
X of type r r in P is O r r m this implies that except for a few
s X s
particular cases the c i are varieties of general type
By Bezout s theorem if X is a c i such that
I X F F r m n
r
then the degree of X satis es
deg X deg F deg F deg F
r
Hartshorne s conjecture
It says the following H
m
m
then X is a complete Let X P be a smooth variety of dimension n If n
intersection
The assumption is that the co dimension of X is small enough with resp ect to the
n
So in this range to b e sp ecial should mean to b e dimension i e codim X m n
a complete intersection
Emilia Mezzetti
The rst signi cant case is in P with co dimension It is easy to construct examples of
non c i smo oth varieties of small dimension For example any skew cubic curve in P is not
c i by Bezout b ecause its degree is prime any curve in P with Hartshorne Rao mo dule
not is not c i Moreover cones over non c i curves provide examples of singular non c i
varieties in the range of the conjecture
Supp ort to the Hartshorne s conjecture
lack of examples
a theorem of Barth B which shows that from a top ological p oint of view
smo oth varieties of small co dimension are similar to complete intersections
m
Precisely by the theorem of Lefschetz if X is c i in P then the restriction maps
i m i
H P C H X C
are isomorphisms for i n The same over Z
Barth s theorem if X is smo oth then the analogous restriction maps are isomorphisms
for i n m n m n The same over Zwas successively proven by Larsen L
Hence the more the co dimension is small the more the cohomology is similar to that of
complete intersections
Two classical examples show that the b ound in the conjecture is sharp We have the
following non c i varieties with n m
a G P of dimension degree non degenerate Bezout
b the spinor variety S P of dimension parametrizing the planes of a family
in a dimensional quadric
No examples in P P etc Zak Z these are the unique manifolds with n m non
c i and corresp onding to orbits of linear algebraic groups
For co dimension varieties the range of the Hartshorne s conjecture is from P on But
there are no examples already in P
Another interpretation of the conjecture is as follows From Barth Larsen it follows that
n
the smo oth varieties X of co dimension in P are sub canonical for n i e the canonical
bundle is O k for some integer k By the Serre corresp ondence OSS each
X X
sub canonical X is the zero lo cus of a section of an algebraic vector bundle E of rank Such
a X is c i if and only if E is decomp osable E O a O b So the Hartshorne s conjecture
for co dimension varieties b ecomes if n then each algebraic vector bundle of rank is
of the form O a O b In fact there are no examples of indecomp osable rank two bundles
already for n
Some partial results ab out Hartshorne s conjecture
for small degree the non necessarily smo oth varieties of small degree are all classi ed
Weil X for d Swinnerton Dyer SD for d Ionescu Io Io and many
others for successive degrees
concerning k normality by de nition X is k normal if the hypersurfaces of degree k
cut on X a complete linear series c i are k normal for any k
there exists a function N d of the degree such that each non c i smo oth X of degree
N d
d is contained in P H The b est known estimates are
Emilia Mezzetti
d for any co dimension Barth Van de Ven BV N d
p
d for co dimension Holme Schneider HS N d
N d r d d for co dimension r Bertram Ein Lazarsfeld BEL
Varieties of co dimension in P and P
We consider now varieties of small co dimension but out of the range of Hartshorne s conjec
ture
Theorem Ellingsrud Peskine EP There is a nite number of families of smooth
surfaces of P not of general type
Here family means irreducible comp onent of the Hilb ert scheme of surfaces of P
m
Some word ab out the Hilb ert scheme if X is a pro jective variety in P and I X is its
homogeneous ideal then the quotient ring
C x x
m
S X
I X
is naturally graded and its homogeneous elements of degree d can b e interpreted as hyper
surfaces of degree d not containing X The Hilb ert function of X is de ned by h t
X
dimS X as a C vector space There is a unique numerical p olynomial P t the Hilb ert
d X
p olynomial of X such that P t h t for t t integer The co e cients of P have
X X X
imp ortant geometrical meaning they give the dimension of X the degree the Euler Poincar e
characteristic the sectional genus of X i e the genus of a curve intersection of X with a
general linear space of the right dimension etc
An imp ortant theorem of Grothendieck G says that xed any numerical p olynomial
m
P the set of subschemes of P having P as Hilb ert p olynomial has a natural structure of
pro jective scheme enjoying a nice universal prop erty
For example for curves xing P means xing degree and genus for surfaces degree
and sectional genus etc
So in particular the theorem says that there is an upp er b ound d on the degree of non
general type surfaces lying in P
Braun Fl ystad BF d Then M Co ok C gave a b etter b ound But all
known examples have d
Analogous result for dimensional manifolds in P was proven by Braun Ottaviani
Schneider Schreyer BOSS In this case there are examples up to degree
After the theorem of Ellingsrud Peskine the problem arises of giving a list of al l families of
surfaces of P not of general type in particular rational and birationally ruled surfaces Sim
ilar problem for folds in P This has b een made for small degrees Okonek O O O
Ranestad R Aure Ranestad AR Popescu P for surfaces Beltrametti Schneider Sommese
BSS for threefolds I would like to mention two imp ortant metho ds
a by the adjunction map This is a classical metho d for studying algebraic surfaces
which go es back to Castelnuovo and Enriques Given X smo oth connected variety of dimen
sion n and a very ample divisor H on X corresp onding to a line bundle L one considers the
adjoint linear system i e j K n H j where K is the canonical divisor
X X
Emilia Mezzetti
Sommese Van de Ven S and V have characterized the varieties X such that the
n
adjoint linear system is not base p oint free or equivalently the line bundle L is
X
not generated by global sections
n
They are P with L O P with L O i e the Veronese surface smo oth
quadrics scrolls over a smo oth curve i e varieties X which are ruled by linear spaces of
dimension n over a smo oth curve For any other X there exists the regular adjunction
mapping The next step is the classi cation of the varieties X such that
jK n H j
X
X has dimension n Fano varieties quadric bundles scrolls over a surface or is not
birational there are four examples For the remaining varieties contracts exceptional
divisors contained in X
For surfaces by iteration of this pro cedure one gets either a minimal mo del or a surface
for which the adjunction map is not de ned or not birational For example for rational
surfaces of low degree in P one can explicitly nd a linear system of plane curves de ning
it For threefolds it is necessary to study also the linear system j K H j and p ossibly its
X
multiples
b the computational metho d by Decker Ein Schreyer DES The ideal sheaf of a
m
co dimension variety X in P has a lo cally free resolution of the form
O F G I k
X
m
for some integer k and lo cally free sheaves F G on P of ranks f f The idea is try to
construct X starting from F and G Taking the cohomology table of I assuming that X
X
exists one lo oks for sheaves F and G with the right cohomology Then one takes a map
F G and checks with a computer if the degeneracy lo cus is smo oth By this metho d
it has b een p ossible to re nd all known examples of surfaces in P and threefolds in P
Related problems
Study the geometry of varieties which are not general For example the Veronese surface in
P It is a general pro jection of the Veronese surface V of P i e the plane embedded via
the complete linear system of the conics This pro jection is isomorphic b ecause S ec V is a
hypersurface
i The theorem of Severi fS says that all smo oth surfaces of P except the
Veronese surface are linearly normal i e they are not isomorphic pro jections of surfaces
contained in spaces of higher dimension In fact if X P X not the Veronese surface
then S ecX P The smo oth folds of P are all linearly normal by a theorem of Zak
Z There is a unique example of a fold X which is not normal this means that the
quadrics of P don t cut a complete linear series on X This example is called the Palatini
scroll it is ruled by lines over a cubic surface of P Its ideal has a resolution as follows
I O
X
P P
so h I h I k for k
X X
There is a strict analogy of X with the Veronese surface of P S b ecause I has a very
S
similar resolution
O I
S
P P
so h I S
Emilia Mezzetti
Conjecture Peskine Van de Ven X is the unique non normal fold of P
ii A classical theorem of C Segre cS states that if a smo oth surface S of P
contains a dimensional family of plane irreducible curves then these curves are conics and
S is a pro jection of the Veronese surface of P hence a Veronese surface of P or a cubic
scroll It is p ossible to prove M that if a smo oth fold X of P contains a family of
dimension of plane curves then either X is contained in a quadric or the degree of these
curves is at most Let us assume that X is not contained in a quadric then MP there
are only two folds satisfying this assumption the Palatini scroll again and the Bordiga
scroll this last one may b e obtained by an exact sequence like by a particular bundle
map
iii A classical way for studying pro jective manifolds of dimension n is to study their
n
pro jections into P A theorem of Franchetta F says that if S P is not the
Veronese surface then the double lo cus of its general pro jection in P is an irreducible curve
For folds in P the general pro jection in P has always an irreducible surface as double
lo cus The triple lo cus is a curve and if X is a Palatini scroll then this curve is reducible
this is the unique known example
Work in progress of Mezzetti and Portelli is related to the p oints i ii iii
References
Basic b o oks here one can nd the classical theorems quoted in this pap er
P Gri ths J Harris Principles of Algebraic Geometry J Wiley Sons
R Hartshorne Algebraic Geometry Springer
I Shafarevich Basic Algebraic Geometry Springer
Articles
AR A B Aure K Ranestad The smooth surfaces of degree in P London Math So c
LNS
B W Barth Transplanting cohomology classes in complex projective space Amer J
Mat
BV W Barth A Van de Ven A decomposability criterion for algebraic bun d les
Invent Math
BSS M Beltrametti M Schneider A Sommese Threefolds of degree and in P
Math Ann
BEL A Bertram L Ein R Lazarsfeld Vanishing theorems a theorem of Severi and the
equations de ning projective varieties Journal Amer Math So c
BF R Braun G Floystad A bound for the degree of smooth surfaces in P not of general
type Comp ositio Math
BOSS R Braun G Ottaviani M Schneider F Schreyer Boundedness for non general
type folds in P to app ear
Emilia Mezzetti
C M Co ok An improved bound for the degree of smooth surfaces in P not of general
type preprint
DES W Decker L Ein F Schreyer Construction of surfaces in P J Alg Geometry
EP G Ellingsrud Ch Peskine Sur les surfaces lisses de P Invent Math
F A Franchetta Sul la curva doppia del la proiezione del la super cie generale del l S
da un punto generico su un S Lincei Rend Sc s mat e nat Vol II
G A Grothendieck Techniques de construction et th eor emesd existence en g eo m etrie
alg ebrique IV Les sch emasde Hilbert S eminaireBourbaki exp
H R Hartshorne Varieties of smal l codimension in projective space Bull Amer Math
So c
HS A Holme M Schneider A computer aided approach to codimension two subvarieties
n
of P n Crelle J
Io P Ionescu Embedded projective varieties of smal l invariants Lect Notes in Math
Io P Ionescu Embedded projective varieties of smal l invariants II Rev Roum Math
Pures Appl
L M E Larsen On the topology of complex projective manifolds Invent Math
M E Mezzetti Projective varieties with many degenerate subvarieties BUMI B
MP E Mezzetti D Portelli Threefolds in P with a dimensional family of plane curves
preprint
O C Okonek Moduli re exiver Garben und Fl achenvom kleinem Grad in P Math Z
O C Okonek Uber codimensionale Untermannigfaltigkeiten vom Grad in P und
P Math Z
O C Okonek Fl achenvom Grad in P Math Z
OSS C Okonek M Schneider H Spindler Vector Bundles on Complex Pro jective
Spaces Birkhauser
P S Popescu On smooth surfaces of degree in P Dissertation Univ Saarbrucken
R K Ranestad On smooth surfaces of degree ten in the projective four space Thesis
Univ Oslo
Emilia Mezzetti
cS C Segre Le super cie degli iperspazi con una doppia in nit adi curve piane o spaziali
Atti R Acc Scienze di Torino Vol LVI also in Opere Vol I I XXXI I I
fS F Severi Intorno ai punti doppi improprii di una super cie generale del lo spazio a
quattro dimensioni e ai suoi punti tripli apparenti Rend Circolo Matematico di
Palermo also in Opere Matematiche Vol I VI
aS A Sommese Hyperplane sections of projective surfaces I The adjunction mapping
Duke Math J
SD H P F Swinnerton Dyer An enumeration of al l varieties of degree Amer J Math
V A Van de Ven On the connectedness of very ample divisors on a surface Duke
Math J
Z F L Zak Tangents and Secants of Algebraic varieties Translations of Mathematical
Monographs AMS
X XXX Correspondence Amer J Math
Mathematical Physics
Session organised by Sylvie Paycha
The aim of the session organised on mathematical techniques in statistical physics
and quantum eld theory was to give an idea of some of the mathematical problems
that have recently arisen in relation to b oth these elds Because of the wide sp ectrum of
the topics presented within this session the sp eakers were asked not so much to go into
technical details but rather to give a brief overview of the mathematical techniques involved
in their work and to try to show the motivations b ehind them arising from statistical physics
of quantum eld theory
The topics of the talks were chosen in such a way that the sp ectrum of mathematical tech
niques involved would b e as broad as p ossible probability theory Flora Koukiou algebraic
techniques Marjorie Batchelor analysis on sup er manifolds Alice Rogers and algebraic
geometry Claire Voisin In this sense the emphasis during this session was not put on
a systematic presentation of the mathematical to ols but rather on how they can b e used
to understand problems of physical origin such as phase transition Flora Koukiou the
structure of spaces of maps involved in the path space description of certain quantum eld
theories Marjorie Batchelor quantisation of sup ersymmetric theories Alice Rogers and
some asp ects of sup er conformal theories Claire Voisin
We hop e these talks will give the reader some insight in these elds of mathematical
physics and convince her him that theoretical physicists and mathematicians still have a lot
to learn from each other
Sylvie Paycha
th
Proc of the EWM
meeting Madrid
Mirror Symmetry
Claire Voisin written by Sylvie Paycha and Alice Rogers
Universit ede Paris XI France
voisin diam ens cachan fr
Warning This is a summary of two introductory talks given by Claire Voisin on mirror
symmetry These notes revised by C Voisin were written by S Paycha and A Rogers
fol lowing the lectures given by Claire Voisin Because of the length of the talks we could only
o er here a summary of these lectures However we have tried to keep the spirit of Claire
Voisin s talk in which her main concern was not so much to go into technical details as to
give the audience an idea of the main topics of research related to mirror symmetry a eld
which is evolving rapidly We have also tried to make these notes accessible to non specialists
at the cost of remaining vague The reader interested in further details is referred to Claire
Voisin s book on Mirror Symmetry and references therein
Abstract Calabi Yau manifolds are de ned and the notion of a mirror pair is introduced
The physical origins of mirror symmetry are describ ed together with steps towards a more
mathematical understanding
Introduction
This pap er is devoted to the description of the mirror symmetry phenomena rst discovered
p q
by physicists it should asso ciate to a Calabi Yau variety X a mirror X satisfying H X
n p q
H X n dim X dim X Furthermore one should have an identi cation for
C C
the mo duli space M parametrising marked complex structures plus complexi ed K ahler
X
0
on X in such a way that the parameters on X with the corresp onding mo duli space M
X
two factors are interchanged
Physicists discovered mirror symmetry via the study of sup erconformal eld theories de
rived from sup ersymmetric mo dels on Calabi Yau manifolds Further information with
imp ortant mathematical applications is obtained from the Yukawa couplings of these theo
ries
Yau s Theorem
In this paragraph we recall some basic notions in complex analysis and algebraic geometry
which o ccur in the description of mirror symmetry that follows see e g GH
K ahlerian manifolds
k
Let X b e a di erentiable manifold of dimension N and X R denote the space of real
di erential forms of degree k on X the elements of which can lo cally b e seen as linear
Claire Voisin written by Sylvie Paycha and Alice Rogers
combinations of expressions of the type dx dx with co e cients given by di erentiable
i i
k
functions on X where fi i g f N g when using lo cal co ordinates x x
k N
k k
Using the exterior di erentiation d X R X R one can de ne De Rham
k k
cohomology spaces H X R resp H X C as quotients Ker d Imd of the space of real
whenever d on X by the space resp complex closed k forms recall that is closed
form whenever there is a form of real resp complex exact k forms recall that is an exact
such that d
p q
Let now X b e a complex manifold of dimension n and let X denote the space of
complex forms of type p q on X the elements of which can b e seen as linear combinations of
expressions of the type dz dz dz dz with co e cients given by di erentiable
i i j j
p q
functions on X where fi i g fj j g are subsets of f ng when using lo cal
p q
co ordinates z z
n
More generally In particular a complex two form on X is an element of
X X X
we have
X
p q
k
X
X
p q k
P
p q p q
n
i i
Using the op erator de ned by dz z one can de ne in
i
X X
p q
a similar way as ab ove the Dolb eault cohomology spaces H X as quotients K er I m
p q p
p q
Since for xed p is a resolution of the holomorphic sheaf we have H X
X X
p
q
H
X
h on a complex manifold X is a hermitian structure h on each tangent A hermitian metric
x
space T X which varies in a di erentiable way with x
x
is a complex manifold X which can b e equipp ed with a hermitian A K ahler manifold
of type structure the imaginary part of which is a closed form called the K ahlerform
that is J invariant in other words such that h g i with d g resp
b eing a symmetric resp antisymmetric real form It can also b e seen as a complex
manifold X equipp ed with a hermitian metric h such that the Levi Civita connexion r for
the metric g h is compatible with the hermitian structure i e r r r and
dh u v h r u v h u r v
Let us now assume X is compact and K ahlerian There is a Ho dge decomp osition for
k
the cohomology spaces H X C into a direct sum of Dolb eaut cohomology spaces The
k p q
cohomology spaces H X C and H X are related by
k p q
H X C H X
p q k
Since a K ahlerform is a real closed form one can consider its K ahlerclass in H X R
and we have H X H X R b ecause is everywhere of type
Lemma The set of K ahler classes in H X R is an open cone the K ahlercone in
H X H X R
Remark In the case when H X H X C the K ahler cone is op en in H X R
and hence contains integer classes X is then algebraic by Ko daira s theorem
Claire Voisin written by Sylvie Paycha and Alice Rogers
Calabi Yau manifolds and Yau s theorem
Let X b e a complex manifold and let T denote its tangent bundle the bre of which is
X
the tangent space T X at p oint x T the cotangent bundle the bre of which is the dual
x
X
space T X to the tangent space at p oint x Both these bundles split into a direct sum
x
corresp onding to the holomorphic resp T T T T T T T T
X
X
X X X X X X
to its antiholomorphic resp T part of the tangent resp cotangent bundle and T
X X
part
n
of rank on X K is the holomorphic vector bundle T The canonical line bundle
X
X
the bre of which is generated by the n form dz dz using holomorphic lo cal co ordinates
n
z z It is trivial whenever there is a nowhere vanishing holomorphic n form on X
n
A Calabi Yau manifold is a compact K ahler manifold with trivial canonical line bun
dle Elliptic curves ab elian surfaces are examples of Calabi Yau manifolds They are often
obtained as hypersurfaces inside larger manifolfds like quintics which are hypersurfaces of
degree see GH for the notion of degree of a variety in the pro jective space P
n
Let K denote the vector bundle T X of rank on X the bre of which is generated
X
by using lo cal co ordinates z z A hermitian metric h on X induces a
n
z z
n
hermitian structure on the tangent bundle T which means a hermitian structure on each
X
bre T X the hermitian structures dep ending in a di erentiable way on the base p oint x and
x
hence a metric h on K The K ahlermetric h will b e called K ahler Einstein whenever
K X
the curvature
h
K K
i
of h vanishes This two form is related to the Ricci curvature of the Riemannian metric g
K
underlying the hermitian structure h so these metrics are in fact Ricci at
Whereas given in the K ahlercone of a given K ahlermanifold there are various hermitian
metrics h g i such that the following theorem by Yau shows that on a Calabi
Yau manifold Ricci at K ahler metrics are in one to one corresp ondance with cohomology
classes of K ahlerforms
Theorem Yau On a Calabi Yau manifold X h for any in the K ahler cone there
is a unique Ricci at K ahlermetric h g i on X with K ahlerform in the cohomology
class H X R of
Mirror symmetry
Deformations of complex structures
A complex structure on X is describ ed in terms of a eld of linear op erators J x X
x
each acting on the tangent space T X with the prop erty that J I this yields a
x
x
pseudo complex structure and satisfying an integrability condition which makes the pseudo
The eigenspaces of J corresp onding to the complex structure into a complex structure
x
of the complex eigenvalues i and i yield a splitting T X T X T X A deformation
x x
x
structure is a deformation J of this eld of op erators J which yields another splitting
t
T X T X T X
x
x t x t
Claire Voisin written by Sylvie Paycha and Alice Rogers
Let us introduce the Dolb eault cohomology group de ned as the quotient space
H T K er T T I m T T
X
X X X X X X X
It is a general fact that H T parametrizes rst order deformations of the complex struc
X
tures on X An imp ortant fact is a result due to Bogomolov Tian and Todorov and proved
recently in a more algebraic way by Ran concerning deformations of complex structures on a
Calabi Yau manifold
Theorem The smal l deformations of the complex structure J on a Calabi Yau manifold
form a smooth family the Kuranishi family whose tangent space at X is H T
X
Remark Deformations of the complex structure do not mo dify the triviality of the canon
ical bundle K
X
Mo duli space
Although the cohomology group H X in which lies the K ahlercone is naturally real one
is led to complexifying the K ahler cone thus introducing a complexi ed K ahlerparameter
i where is in the K ahlercone and H X R is de ned mo dulo H X Z
For each complex structure J on X making X into a complex manifold X we can de ne
t t
such a complexi ed K ahler parameter H X C mo dulo i H X Z Assuming that
t
the manifold X satis es the condition H X C H X the K ahler cone is op en in
H X R and hence varies lo cally in an op en set of H X C H X C b eing lo cally a
t t
constant vector space indep endent of t this means that deformations of the complex structure
X X followed by deformations of the K ahlerstructure on X can b e lo cally interpreted
t t
as pro ducts of complex deformations and K ahlerdeformations
M f X g is the set of couples Let X b e a Calabi Yau manifold The mo duli space
X t
with rst term given by an isomorphism class of complex structures X on X obtained
t
by deformations J of the initial complex structure J on X and second term given by a
t
complexi ed K ahler parameter on X From the ab ove discussion it follows that if X
t
satis es the condition H X C H X then M is lo cally a direct pro duct However
X
this pro duct structure do es not hold in general globally in particular since the K ahler cone
can dep end on t
This lo cal pro duct structure reads on the level of tangent spaces as
T M H T H
X X X
since H T describ es in nitesimal deformations of complex structures and H de
X X
scrib es in nitesimal deformations of K ahlerstrutures
Mirror symmetry
Mirror symmetry predicts the existence for a given Calabi Yau manifold X of a Calabi
Yau mirror manifold X of same dimension as X and such that there is an isomorphism
0
M M by which complex deformations and K ahler deformations are swapped or M
X X
in other words with the prop erty that the lo cal pro duct structure of mo duli space is preserved but the factors exchanged
Claire Voisin written by Sylvie Paycha and Alice Rogers
This conjecture predicted by physicits cannot b e completely true b ecause there are rigid
Calabi Yau manifolds which have no complex structure mo duli whose mirror could thus not
even b e K ahlerian Although it has b een con rmed in a wide range of cases it is not yet
mathematically fully understo o d
Remark The manifold X and its mirror X are in general top ologically very di erent
The conjecture mathematically translates as follows Since the lo cal structure of M
X
arises from the splitting
T M H T H
X X X
one exp ects the tangent map M to M to induce isomorphisms
0
H T H
X
X
and
0
H H T
X
X
More generally one exp ects a sequence of isomorphisms
p q p q
0
H T H
X
X
q n q
Since the canonical line bundle K is trivial the holomorphic bundles T and
X X X
are isomorphic using a global section of K and this induces isomorphis ms
X
p q p n q
H T H
X X
p q p q
h X dimH X of X where n is the dimension of X and hence the Betti numbers
n p q
see GH should b e equal to h X of X Hence the top ology of X is determined by
that of X
Mirror symmetry and physics
The N sup ersymmetric mo del
Let X g b e a Riemannian manifold Just as p oint particles are classically describ ed by paths
R
T
d
d
T R evolving in space time which minimise the energy S T k k given in
dt
classically by paths X terms of their length we shall de ne a b osonic mo del
de ned as maps on a Riemann surface with metric with values in M which minimise the
action or energy
Z
kd k S
S is invariant under conformal transformations of i e under di eomorphisms of which
multiply the metric p ointwise by a p ositive factor namely di eomorphisms f such that
h
there is a function h on with f x e x x When X is K ahlerian and is the
K ahlerform on X the action reads
Z Z
k k S
where is the pull back of on by Furthermore one can check that the critical
p oints of S which give the classical paths coincide with the critical p oints of the action
Z Z
kd k S
Claire Voisin written by Sylvie Paycha and Alice Rogers
for any closed two form on X From these two remarks it follows that for a K ahlermanifold
X g with K ahler form the classical mo del can equally well b e describ ed in terms of
the action
Z Z
k k S
where i is now a complexi ed K ahler parameter b eing a closed form In
fact the actual actual parameter is the class of in H X R H X Z since an exact
R
via Stokes formula and form contributes to the action only through a b oundary term
physicists are only interested in the exp onentiated action
The action is invariant under conformal transformations of These transformations are
n
generated by L z
n
z
is classicaly describ ed by elds X where X is as A sup er symmetric mo del
b efore a Riemannian manifold and is a sup er Riemann surface with one commuting
co ordinate z and one anticommuting co ordinate as describ ed in Alice Roger s talk which
minimise an action C invariant under conformal transformations in a similar way to the
b osonic action S as well as one set of sup ersymmetries corresp onding to o dd vector elds
on Alvarez Gaum e and Freedman AF showed that if X is a K ahler manifold there is
an additionnal set of sup ersymmetries so that the theory has what is known as N
sup erconformal symmetry these symmetries are maintained under quantisation provided
that X is Calabi Yau Thus we have a mapping from the set of Calabi Yau manifolds into
the set of N sup erconformal theories this map was a key ingredient in the discovery of
mirror symmetry
Quantisation and the Virasoro algebra
Given a classical eld theory V X where V is a given manifold with action S
R
L dv ol x the space of solutions of the classical Euler equations is equipp ed
i
V
with a Poisson structure No ether s theorem asso ciates functions on the space of solutions to
in nitesimal symmetries of the action in a compatible way with the Lie and Poisson brackets
In nitesimal symmetries then yield evolution equations using the brackets fH g
t
Quantising the classical mo del means nding a representation on a Hilb ert space of states
of a set of functionals called observables equipp ed with the Poisson brackets The quantised
f x f x where theory can then b e reconstructed from the correlation functions
k k
x i k are p oints on V and
i
Z
S
f x f x e d f x f x
k k k k
M ap V M
The integral is to b e interpreted here as a formal in nite dimensional Leb esgue integral on
path space M ap V M
Coming back to the b osonic mo del X let us notice that since the action arises
R
in the path integral as an exp onent and since we have S S i
b eing a form in H X R H X Z the correlation functions are indep endent of the
choice of representative mo dulo Zand are thus well de ned
Quantisation of a conformal eld theory should give a p ossibly pro jective representation
of the Virasoro algebra that is the algebra generated by L and L and hence a theory
satisfying Segal s axioms describ ed in Ga
Claire Voisin written by Sylvie Paycha and Alice Rogers
In a similar manner one exp ects quantisation of an N sup ersymmetric sigma mo del
to lead to an N conformal theory that is essentially a representation of the sup er
Virasoro algebra with central charge This algebra has even generators L L m Z as
m m
for the purely b osonic conformal group and J J m Z together with o dd generators
m m
and an even central charge C Full details of this algebra are G G G G r s Z
r s r s
given in V where it is also shown that there is an involution of this algebra de ned by
G G G G G G J J J J Under this involution two subrings
m m m m
r r r r r r
of the representation known as the chiral antichiral ring R and the chiral chiral ring R
ca cc
are interchanged This interchange of rings is the next step in the physicist s construction
of mirror symmetry the nal step involves reversal of the steps that lead from a Calabi Yau
manifold to the for example chiral chiral ring of the corresp onding sup erconformal eld
theory The chiral antichiral ring of one sup erconformal theory derived from a Calabi Yau
manifold X is the chiral chiral ring of a sup erconformal theory which can b e realised by a
non linear mo del on an di erent manifold X and the idea is that these two manifolds
will b e a mirror pair Now the relation with geometry is given by the description of the
anti chiral rings in terms of the Dolb eault cohomology of the target manifold of the mo del
The relation with mirror symmetry comes from the fact that the rings R and R have
ca cc
p q p q p q
q p
H T bigradations R and R corresp onding to eigenvalues of J and J with R
ca cc c c
X
p q
q p
H and R
c a
X
Witten suggests a more geometric interpretation whereby mirror symmetry would ex
change two sup ersymmetric quantum eld mo dels a mo del A built from X and a mo del B
built from its mirror X thus exchanging sp ecial correlation functions called Yukawa couplings
computed for each of these mo dels which we shall denote by Y for the mo del A and Y for
the mo del B As we shall see b elow the normalised Yukawa coupling Y is an n symmetric
0
form on H T and Y is an n symmetric form on H
X
X
Mirror Symmetry and Mathematics
The construction of the mirror pair of a Calabi Yau manifold using sup erconformal eld
theory is b oth indirect and in places tenuous It is natural to ask whether there is not some
more direct and rigorous way of obtaining the mirror pair At present no such metho d is
known but considerable progress has b een made towards this goal
Mathematical evidence for mirror symmetry
Prop erties of Yukawa couplings give a hint towards mirror symmetry
Let X b e a Calabi Yau manifold of dimension such that h X For a given
H X where H X enters in the Ho dge decomp osition H X C H X
H X H X H X one de nes a Yukawa coupling Y on X as a symmetric
three form on H T
X
Y u u u u u u
where corresp onds to Serre duality see e g GH chap u u u H T
X
where u u u denotes the cup pro duct GH of u u u
Claire Voisin written by Sylvie Paycha and Alice Rogers
These Yukawa couplings have a remarkable prop erty namely that they arise as deriva
tives of a p otential for a natural choice of co ordinates on M In other words there is a sp e
X
cial choice for the parameter and there are sp ecial co ordinates z z N dim H T
N X
on M dep ending on a choice of symplectic basis on H X Z for the intersection form
X
and a function F z z such that see V Prop osition
N
F Y
z z z z z z
i j k i j k
co ordinates de ned up to a ne transformation This construction also yields a at structure
on M which dep ends on the symplectic basis chosen on H X C see Lemma in V
X
Predictions of mirror symmetry
Confronting these results with the results predicted by mirror symmetry gives evidence for
mirror symmetry
Let X b e as b efore a three dimensional Calabi Yau manifold with h X If mirror
symmetry exists there is a lo cal identi cation b etween deformations of complex structures
and deformations of the K ahler parameter on its mirror X Since there is a canonical
at structure on the K ahler deformations of X mirror symmetry predicts the existence of
a at structure on the space of deformations of the complex structures on X
0
which dep ends Let M b e the mirror map Y M Y gives a cubic form on H
X
only on the K ahlerparameter One b eautiful prediction due to the physicists cf Witten is
the description of Y in terms of cubic derivatives of the Gromov Witten p otential G the
de nition of which involves Gromov Witten invariants of Y see A LY V Hence with
the privileged choice of co ordinates on either side as describ ed ab ove the mirror map M has
to b e a ne linear and one exp ects that M F G F b eing the p otential de ned ab ove for
Y in fact up to a quadratic function of the at co ordinates
This identi cation predicts the number of rational curves in a quintic of any degree Pre
dictions have b een checked up to degree see ES LY
These are only a few hints for mirror symmetry There have b een many other investigations
made in that direction such as Batyrev s combinatorial construction of a b eautiful series of
examples for which we refer the reader to V
References
These are only a few references around the topic of mirror symetry We refer the reader to
V for further references
A M Audin Cohomologie quantique Bourbaki vol Exp os e Nov
AF L Alvarez Gaum e D Freedman Geometrical structure and ultraviolet niteness in the
supersymmetric model Comm Math Phys p
D R Dijkgraaf Mirror symmetry and el liptic curves in The Mo duli Space of Curves
R Dijkgraaf C Faber G Van der Geer Eds Birkh auser
Ga K Gawedski Conformal eld theory S eminaireBourbaki exp
Claire Voisin written by Sylvie Paycha and Alice Rogers
G D Gepner exactly solvable string compacti cations on manifolds of SU n holonomy
Physics Letters B Vol n p
GH P Gri ths J Harris Principles of algebraic geometry Wiley Interscience Publication
GP B R Greene M R Plesser An introduction to Mirror Manifolds in Essays on Mirror
Manifolds Ed S T Yau International Press
ES G Ellingsrud S A Stromme The number of twisted cubics on the general quintic three
fold in Essays on Mirror Manifolds Ed S T Yau International Press
LY B H Lian S T Yau Mirror Symmetry Rational curves on algebraic manifolds and hy
p ergeometric series Pro ceedings of the XIth International Congress of Mathematical
Physics Ed D Iaglnitzer
M D Morrison Mirror symmetry and rational curves on quintic threefolds a guide for
mathematicians Journ Amer Math So c p
V C Voisin Sym etrie Miroir Panoramas et synth eses So ci et e Math ematiquede France
to app ear
W E Witten Mirror manifolds and topological eld theory in Essays on Mirror Manifolds
Ed S T Yau International Press p
th
Proc of the EWM
meeting Madrid
Mo delling the randomness in physics
Flora Koukiou
Universit ede Cergy Pontoise France
koukiou u cergy fr
Randomness is one of the paradigms of mo dern physics During the last years a wide
variety of mo dels of disordered statistical mechanics have b een introduced and studied By
disordered system we mean a system with quenched frozen in randomness which vary from
sample to sample For mo delling randomness one can either introduce random p ertubations
in an ordered system or consider random interactions b etween the di erent comp onents of
the system In the case of p erturbations an imp ortant question to address is ab out the
stability of phase transitions under random p erturbations The case of random interactions
spin glasses is much more di cult and still heavily debated We can however use a mean
eld theory to study the b ehaviour of the system by neglecting the e ects of uctuations
In many cases uctuations are irrelevant systems in su ciently many spatial dimensions
or with long range interactions each comp onent interacts with each other comp onent In
mathematical physics mean eld mo dels are usually provided by systems de ned on complete
graphs or trees
In the following we present a simple application of the theory of branching random walks to
the mean eld theory of a random systems de ned on regular trees In a tree of co ordination
number d one can study mean eld mo dels of spin glasses or directed p olymers using
the theory of branching random walks
We can de ne a simple mo del on a d tree i e each vertex has d edges for instance the
dyadic tree is a tree as following
Let T b e a nite d tree If v is a vertex we denote by jv j the number of edges or steps
n
n
of the path going from the ro ot to v We have thus d paths p of jv j n steps On this
d tree we can do simple random walks starting at the ro ot and choose any of the d edges
coming out indep endently Let w b e a random variable de ned on the probability space
F P dep ending on a real variable called temp erature We assume moreover that the
variable w has moments of all orders and that w is almost surely b ounded To each
edge b T we assign the random energy w having common distribution with w We
n b
have thus a family of indep endent identically distributed random variables fw b T g
b n
indexed by the edges of the tree In mo dels of statistical physics the variables fw g
b
are asso ciated with the Boltzmann factors In the case of random interactions b etween the
comp onents of a system each path p from the ro ot to the vertex v corresp onds to a spin
con guration whose Gibbs weight is given by the sum of energies over the jv j edges In the
case of directed p olymers the energy of each walk is given by the pro duct of the energies of
the visited edges
Work partially supp orted by the EU grant CHRX CT
Flora Koukiou
Having this in mind we can de ne on T the partition function of the mo dels by
n
X Y
Z w
n b
p
b p
the sp eci c free energy by
F n log Z
n n
Moreover the Gibbs distributions can b e de ned by the following random measures
Q
w
b
b p
n
Z
n
The main ob ject of interest is the b ehaviour of the previous quantities at the macroscopic
or thermo dynamic n limit as function of In order to study this b ehaviour we
can express the previous de ned thermo dynamic quantities as random measures Remarking
that the d adic partition of the unit interval corresp onds to the d tree we can de ne and
study random measures on the unit interval which are related to Z and
n n
In many cases there exists a value of called critical temp erature such that for
c
c
Z go es to a non zero limit as n
n
the limit
F lim n log Z
n
n
exists almost surely and is given by n log EZ E means the exp ectation This
n
result gives the existence and the so called self averaging prop erty of the free energy
i e the coincidence of the annealed
log EZ lim
n
n
n
and quenched limits
E log Z lim
n
n
n
n has a unique weak limit as n
On the other hand for we have
c
Z go es to zero as n
n
the limit F lim n log Z exists almost surely and it is a non random
n n
quantity Using large deviation techniques we can explicitely calculate this limit
in general there are many limits of
n
The phenomenon of phase transition is expressed by the previous setting The region
of is called the high temp erature region and de nes the low temp erature
c c
region This simple mo del provides a general framework for all mean eld mo dels studied
by physicists and gives some insight to our understanding of phases transitions in random
systems The interest reader can found the detailed de nitions and pro ofs in The mean eld
theory of directed p olymers in random media and spin glass mo dels Rev Math Phys
th
Proc of the EWM
meeting Madrid
Sup er mo duli space
Alice Rogers
King s College London England
f a rogers kcl ac uk
Introduction
Mo duli spaces have b een one of the themes of this meeting in this pap er my aim is to show
how mo duli spaces may b e used in the functional integration approach to the quantisation of
systems with symmetry with particular reference to the sup er mo duli space of sup er Riemann
surfaces and the spinning string At the outset it should b e made clear that the term sup er
implies an extension of some standard ob ject to include anticommuting elements in some
sense the term deriving its name from the notion of sup ersymmetry in physics
The Feynman path integral was initially developed as a metho d for determining the time
evolution of a quantum system for some systems the path integral formulation may b e
derived rigorously from canonical quantisation A generic feature of this approach even
when extended either rigorously or heuristically to quantum eld theory in which case path
integrals b ecome functional integrals is the emergence of the term exp iS where S is the
action of the theory so that covariance absent in the canonical approach is restored in the
path integral This suggests that the path integral may b e a more fundamental starting p oint
than the canonical approach and such integrals are widely used to de ne the quantisation of
a theory when there is no direct derivation of the functional integral by canonical metho ds
There remain many features of functional integrals which are not well understo o d but formal
manipulations have led to remarkable insights in b oth mathematics and physics so that the
pursuit of a prop er understanding of these integrals seems highly desirable
In this pap er the basic idea of the Feynman path integral in quantum mechanics is de
scrib ed and a very heuristic description given of the extension of these idea to functional
integrals in quantum eld theory and the mo di cation needed when a system has a gauge
symmetry In Section these ideas are applied to the b osonic string and it is shown that the
functional integral reduces to an integral over the mo duli spaces of Riemann surfaces In the
nal section we see that the spinning string leads to an integral over a sup er mo duli space
The Feynman path integral in quantum mechanics
In quantum mechanics a key equation is the Schr odingerequation
f
iH f
t
which determines the time evolution of the wave function f of a system whose Hamiltonian
is H The Hamiltonian is a sef adjoint op erator on the Hilb ert space H of wave functions
Alice Rogers
For the case of a single particle of unit mass moving in one dimension under a eld of force
derived from a p otential V x wave functions are square integrable functions on the real line
with di erentiable dep endence on time as well and the Hamiltonian takes the form
H H V x
d
The system is solved if the op erator exp iH t is known since this where H
dx
leads to solutions of the Schr odingerequation It is su cient to determine the kernel of
this op erator that is the function exp iH t q q which satis es
I F
Z
exp iH t f q dq exp iH t q q f q
F F
Here we are b eing vague ab out analytic details and cheerfully assuming kernels exist this of
course dep ends on the nature of V all the analytic questions b ecome much more tractable if
one considers exp H t A go o d entry p oint to the literature on these matters is Since
exp iH t exp iH s exp iH s t the kernels of these three op erators are related by
the involution
Z
exp iH s t q q dq exp iH s q q exp iH t q q
I F I F
The trick required to derive the path integral expression for the kernel of exp iH t is to
use the fact that
N N N
iH t iV q t iH t
exp iH t exp exp exp
N N N
for large N Then the involution rep eated N times gives
Z
N N
Y Y
exp iH t q q dq exp iH t q q
i i i I F
i i
t
where t q q and q q and thus if N is su ciently large
I N F
N
Z
N N
Y Y
exp iV q t exp iH t q q dq exp iH t q q
i i i i I F
i i
Z
N N
Y Y
i q q
i i
exp iV q t exp dq
i i
t
i i
Z
N N
X Y
q q
i i
t V q t dq exp i
i i
t
i i
Z Z
t
V q s q s ds D q exp i
where D q denotes that the integral is to b e taken over all paths q s satisfying q q
I
and q t q The expression in the nal line is essentially de ned by the line ab ove The
F
argument presented here roughly follows the ground breaking work of Feynman and Hibbs
Alice Rogers
The nal expression has the classic form
Z
D q exp iS q
R
t
q s V q s ds is the action of the system As mentioned b efore where S
physicists pro ceed to quantise a vast range of systems by investigating this integral for the
appropriate action For instance in quantum eld theory where instead of a single con gu
ration variable q t there is one for each p oint x in space the integrals b ecome sums over all
functions or elds x t on spacetime and the path integral is written
Z
D exp iS
where S denotes the action of the system
When the system has a symmetry that is some group G acts on the space of elds in
such a way that the action S is invariant the functional integral is taken of the space
of elds mo dulo the action of the group G This space is in general smaller than the full
space of functions but may b e more complicated However in the case of some very highly
symmetric theories particularly string theories and top ological eld theories the space is
nite dimensional and has an interesting geometrical interpretation
The mo duli space for string path integrals
The rst example I shall consider of a theory where the function space is reduced to a nite
dimensional mo duli space is closed b osonic string theory following the approach of Polyakov
which is explained together with many mathematical developments by Bost In this
case the classical action is
Z
q
ij a b
d x S g X det g x g x X X
ij i j ab
where is a dimensional surface g are the comp onents of a Riemannian metric g on
ij
d
X is a mapping of into R a consistent quantum theory b eing obtained when d the
d
dimension of the space time R in which the string moves is equal to and is the
ab
d
Minkowski metric in R To quantise the theory the functional integral
Z
D X D g exp iS X g
must b e evaluated over the space of all elds X g mo dulo the symmetries of the theory It
is su cient to consider only which are compact surfaces without b oundary summing the
results for each p ossible genus The action of the theory is symmetric b oth under the action of
x
the di eomorphism group of the surface and under conformal transformations g e g
It is p ossible to carry out the X integration explicitly since this integral is a Gaussian so
that it remains to integrate a function of the metric g over the space of all p ossible g mo dulo
the symmetries It turns out that this space is simply the mo duli space of all p ossible
complex structures on To see this supp ose that g has comp onents g with resp ect to
ij
i
lo cal co ordinates x i on then it is always p ossible to choose a di eomorphism and
i
hence new lo cal co ordinates y i on such that the new comp onents have the form
Alice Rogers
x
g e Supp ose thaty y are some other co ordinates where the comp onents of g are
ij ij
also diagonal and that z and z are de ned as z y iy z y iy Then by standard
arguments in complex function theory z is an analytic function of z Thus each conformal
and di eomorphism class of metrics determines a complex structure on and the space over
which the functional integral for the closed b osonic string must b e carried out is the mo duli
space of complex structures on the surface This space has b een much studied and is a
nite dimensional manifold with singularities In the next section it will b e seen that the
analogous space for the spinning string is a sup er mo duli space
Sup er mo duli space and the spinning string
To incorp orate fermions that is particles with half integral spin into string theory the
spinning string is introduced The geometric theory of the spinning string can b e formu
lated using anticommuting variables as rst describ ed by Howe The ingredients are
complicated but have b ecome standard in sup ergravity theory They involve the notion of
sup ermanifold which can b e de ned in a b ewildering number of di erent ways however a
quite naive approach is su cient for this lecture The action of the spinning string is
Z
A A
d x d sup erdet E D VD V S E
M M
where x x are even commuting co ordinates and are o dd anticommuting co ordinates
A
on the dimensional sup ermanifold E is a generalisation of a metric on describ ed
M
b elow V is a function on and D is a di erential op erator whose precise de nition is not
imp ortant here
A
is known as a vielb ein and generalises the zweibein version of a metric on The ob ject E
M
a standard dimensional manifold a metric is a symmetric non degenerate quadratic form on
the tangent space thus there exist orthonormal bases e a of the tangent space such
a
a basis is not unique there is an SO bundle of orthonormal frames This bundle contains
a
all the data of the metric With resp ect to lo cal co ordinates the dual basis e of one forms
may b e expanded as
a a m
e e dx
m
A basis e e is known as a zweibein
Returning to the sup ermanifold we consider a reduction of the frame bundle which is in
fact a sup er group bundle to an SO bundle with action on the preferred frames taking
the form
E R E
a a
R E E
where E and E are resp ectively the two o dd and two even elements of the preferred basis
a
of the tangent space and R is an element of SO The allowed vielb ein are constrained so
that the bundle omits a connection with some comp onents of the torsion taking a prescrib ed
form This is a physical requirement
a
Expanding the dual basis E E of the cotangent space in terms of the co ordinate basis
m
x we have comp onents of the vielb ein forming an invertible matrix
a
E E
m m
a
E E
Alice Rogers
The sup erdeterminant of a matrix
A B
C D
of this nature is de ned to b e det A BD C det D It is a non trivial prop erty of the
sup erdeterminant that it ob eys the multiplicative rule
The nal geometrical ingredient needed is a notion of integration this is de ned by
Z Z
d x d f x f x f x f x d x f x
The sup er conformal geometry emerges from the symmetries of the theory these are
sup erdi eomorphisms of together with so called sup er Weyl transformations
a a a
E E D E E E
a
m m m m m
A
a
E E
a a
E E E D
a
Here are the Dirac matrices which represent the dimensional Cli ord algebra asso
a
ciated with spin representations of SO
These are the simplest transformations which preserve the constraints on the connection
without restricting the parameter x except by requiring that it is invertible everywhere
The sup erdifeomorphism symmetry is used to choose a co ordinate system where the vielb ein
is sup er Weyl at that is obtainable from the matrix
a
m
a
i
by a sup er Weyl transformation If one now chooses complex co ordinates z with z x ix
and i then changes of co ordinates z z which preserve sup er Weyl atness
of the vielb ein are what is known as sup erconformal z and are b oth sup eranalytic functions
of z and so that
z f z z
z g z
with all four functions f z z z g z analytic and also the di erential op erator D
transforms mutiplicatively with
z
D D D
The nature of the transition functions from z to z demonstrates that the surface has
the structure of a particular kind of dimensional complex sup ermanifold known as a
super Riemann surface The extra condition means that the change of co ordinates takes
the form
z f z z
p
f z z z z
For future reference it may b e observed that by setting z to zero and ignoring a con
ventional Riemann surface known as the b o dy of the sup er Riemann surface is obtained
Alice Rogers
A sup er Riemann surface with simply connected b o dy is said to b e simply connected and
similarly for any other top ological attribute
From the analysis ab ove we see that the functional integral for the spinning string equation
reduces to an integral over the sup er mo duli space of all p ossible sup er Riemann surfaces
and so one is led to a study of the nature of this space Following Crane and Rabin the
following picture emerges three simply connected sup er Riemann surfaces can b e found
which are sup er extensions of the three simply connected Riemann surfaces and then a
uniformisation theorem established to show that any other compact sup er Riemann surface
without b oundary is a quotient of one of these three simply connected sup er Riemann surfaces
by a discrete subgroup of the sup erconformal automorphism group The starting p oint is the
observation that corresp onding to any Riemann surface with spin structure a sup er Riemann
surface can b e constructed with transition functions
z f z
p
f z
p
f z is determined where f is the transition function on the Riemann surface and the sign of
by the spin structure Since each of the three simply connected Riemann surfaces the complex
plane C the complex sphere C and the upp er half plane U has a unique spin structure they
each p ossess exactly one sup er extension of this nature denoted S C S C and SU resp ectively
Crane and Rabin use cohomological arguments to show that these are the only simply
connected sup er Riemann surfaces and hence establish their uniformisation theorem The
mo duli space corresp onding to each genus and spin structure can then b e investigated At
genus the only p ossible sup er Riemann surface is S C while at genus various toroidal
compacti cations are p ossible for even spin structures on the b o dy these compacti cations
are parametrised by a single even parameter for even spin structures so that the mo duli
space is dimensional while for the o dd spin structure the mo duli space has dimension
Sup er Riemann surfaces of higher genus are obtainable as quotients of SU by a discrete
subgroup of the group of conformal automorphisms of the upp er half plane which consists
of transformations of the standard form with
az b
f z
cz d
z
z
cz d
where a b c and d are all real and even and satisfy ad bc while and are real and
o dd Now by the usual arguments must b e isomorphic to the fundamental group of a
surface of genus g where g is the genus of the b o dy of the sup er Riemann surface S U
and this group has g generators and one relation thus allowing for a freedom of overall
conjugation there will b e g even parameters to b e chosen and g o dd parameters
to b e chosen to determine a particular Thus sup ermo duli space is at least to the extent
that the corresp onding result is true for standard Riemann surfaces a nite dimensional
sup ermanifold and we see that the geometrical formulation of the spinning string given by
Howe leads eventually to the reduction of the functional integral for the spinning string to
a nite dimensional integral A further account of the geometry of sup er Riemann surfaces
and its application to string theory may b e found for example in
Alice Rogers
References
B Simon Functional Integration and Quantum mechanics Academic Press
R P Feynman and A Hibbs Quantum Mechanics and Path Integrals McGraw Hill
A M Polyakov Quantum geometry of b osonic strings Pysics Letters B
J B Bost Fibr es d eterminants r egularis es et mesures sur les espaces de mo dules des
courb es complexes S eminaire Bourbaki
P S Howe Sup er Weyl transformations in two dimensions Journal of Physics A
D Friedan Uni ed string theories In M Green and D Gross editors Proceedings of the
Workshop on Uni ed String Theories
L Crane and J M Rabin Sup er Riemann surfaces uniformization and Teichm uller
theory Communications in Mathematical Physics
A A Rosly A S Schwarz and A A Voronov Geometry of sup erconformal manifolds
Communications in Mathematical Physics
A A Rosly A S Schwarz and A A Voronov Sup erconformal geometry and string theory
Communications in Mathematical Physics
th
Proc of the EWM
meeting Madrid
Universal measuring coalgebras
The p oints of the matter
Marjorie Batchelor
Cambridge University England
Introduction
One of the challenges mo dern theoretical physics has put to classical mathematics is the
necessity of nding a satisfactory concept of manifold or p oint set which will accomo date the
increasingly varied requirements Sup ersymmetry necessitated the introduction of sup erman
ifolds String theories and eld theories in general require the in nite dimensional manifolds
of smo oth maps The geometric interpretation of non commutative geometry is still puzzling
The purp ose of this pap er is to introduce the universal measuring coaglebra and to suggest
that many of these challenges can b e met by this construction It provides a solution to
problems of di erential top ology in sup ermanifolds and the spaces of maps There is evidence
that it may give a satisfactory interpretation of quantum groups and similar ob jects Moreover
as its construction is a universal one requiring only that the input ob jects are algebras it ought
to have applications to other categories which dep end on a notion of function algebra Finally
b ecause of the strong built in niteness prop erties p ossessed by coalgebras this construction
in some sense enco des all nite dimensional information
I will introduce measuring coalgebras describ e how they function as surrogate p oint sets
and how they can b e used to get jet bundle information in b oth the classical case of nite
dimensional manifolds and the case of graded manifolds A further slight generalization
allows one to get hold of the jet bundle of the manifold of smo oth maps b etween ordinary
or graded manifolds The nal example I will give is the simplest one which demonstrates
the p ossibility of representing quantum groups as transformation groups
De nitions of Coalgebras
Motivation Every p oint set S comes equipp ed with a diagonal map S S S taking a
p oint s to the pair s s This map is used implicitly in the usual de nition of multiplication
of functions on S if f g are two say real valued functions on S the pro duct f g is de ned
by
f g s f s g s
This de nition dep ends evidently on multiplication in R but equally on the diagonal map
s s s
The view I take is that the prop erties which de ne the notion of p oint are the following
Marjorie Batchelor
Points take values on functions that is they are linear functionals on a given algebra
F of functions
Points are precisely those functionals s which satisfy the pro duct rule
f g s f s g s
determined by multiplication in R and the usual diagonal map
This is a viewp oint which generalizes easily coalgebras providing sets equipp ed with gen
eralizations of the diagonal map and measuring maps providing the idea of a pro duct rule
De nition A coalgebra is a linear space C together with a comultiplication
C C C
and a counit
C R
satisfying the following identities
C C C
C C C
y y
y y
C C C
C C C C C
Examples
i C Rm m m m m This is the coalgebra with p oint like b ehaviour
ii C Rm Rt m m m m t t m m t t Observe that
comultiplication of t resembles the pro duct rule for derivations Elements with that
comultiplication are called primitive
iii C Rk Rh RE The elements h k are p ointlike as m in example i and E
k E E h E This coalgebra is the one which arises in quantum groups
Notation Following Sweedler the comultiplication c will b e written
X
c c c
c
The coasso ciativity diagram allows us to write
X
c c c c c
c
The counit diagram ab ove states that
X X
c c c c c
c c
Pleasing prop erty of coalgebras The de ning prop erty of tensor pro ducts makes it
easy to construct maps from tensor pro ducts not to them The consequence of requiring the
comultiplication map to go the wrong way is that coalgebras have a very strong built in
niteness prop erty every element in a coalgebra C lives in a nite dimensional subcoalgebra
of C This has a number of desirable consequences on which we can capitalize
Marjorie Batchelor
Measuring coalgebras
This is the concept of maps b etween algebras which satisfy a pro duct rule
De nition Let A B b e algebras over R for example and let C b e a coalgebra A linear
map
C Hom A B
R
P
is said to measure if c a a c a c a and c c
A B
c
Examples
i If C is the coalgebra of example i ab ove then a linear map C Hom A B measures
R
if and only if m is an algebra homomorphism
ii If C is the coalgebra of example ii ab ove a linear map C Hom A B measures if
R
and only if m is an algebra homomorphism and t is a derivation with resp ect
to m
Not only are there familiar examples then of measuring coalgebras but also there exists
a maximal measuring coalgebra for a given pair A B of algebras
De nition A measuring coalgebra P Hom A B is a universal measuring coalgebra
if given any other measuring coalgebra C Hom A B there is a unique map of
coalgebras which makes the following diagram commute
P Hom A B
x
C
The existence of universal measuring coalgebras dep ends on the pleasing prop erty of coal
gebras Sp eci cally that niteness prop erty guarantees that in the category of coalgebras
copro ducts and colimits exist This is enough to construct a suitable measuring coalgebra P
with the required prop erties Such ob jects are unique by general categorical principles
Notation The universal measuring algebra for the pair A B will b e denoted P A B
Applications to manifolds and graded manifolds
The starting p oint for this discussion has b een that a p oint set S is a subset of linear func
tionals on a given function algebra F The idea then is to replace S by P F R As a
test piece we can calculate P C M R where M is a smo oth manifold and C M is the
algebra of smo oth functions on M
We know some elements of P C M R already If m is a p oint in M then the as
signment f f m is an algebra homomorphism Thus by example i ab ove Rm
Hom C M R measures and by the universal prop erty of P C M R Rm is included
in P C M R Denote the image of Rm under this inclusion by T
m
Similarly if is a tangent vector to M at m so that is in the tangent space T M the
m
assignment t determines a measuring map from Rm Rt Hom C M R In this
Marjorie Batchelor
way R T M can b e considered as a measuring coalgebra for C M R Denote the
m m
image of this measuring coalgebra in P C M R by T
m
Higher order tangent spaces can b e treated similarly and give an increasing union of
k
sub coalgebras T of P C M R as in Figure
m
Figure
Result
X
k
P C M R T
m
m M
k
k th
Each sub coalgebra T is dual to the bre of the k jet bundle of M at m The sum over
m
p oints in m is direct
Thus in the familiar case of smo oth manifolds this entirely algebraic construction recovers
all the jet bundle information of the manifold M It is therefore an attractive candidate to
use to recover jet bundle information when conventional manifold techniques fail to apply A
rst application is to graded manifolds
Graded manifolds are manifolds whose function algebra includes anticommuting elements
that is functions f g such that f g g f They are de ned as follows
De nition A graded manifold is a pair M A where M is an ordinary smo oth manifold
and A is a sheaf of graded commutative algebras such that there is an atlas of op en charts on
n
M such that over such an op en set the sheaf A is isomorphic to the sheaf C R The
n n
basis f g of R which generates R are refered to as the o dd co ordinate functions
n
on M A
By analogy with the previous example the universal measuring coalgebra P A M R
ought to provide a picture of the dual to the jet bundle for the graded manifold M A It
do es The only novelty is that in addition to conventional tangent vectors corresp onding
x
i
to conventional co ordinate functions on M T contains o dd tangent vectors corresp ond
m
j
ing to the o dd co ordinate functions The higher order tangent spaces contain suitable mixed
derivatives of o dd and ordinary co ordinates The result and the picture are still the same
k k k k
The only novelty is that each T comes with a Z grading T T T and contains
m m
m m
derivatives of b oth o dd and even co ordinates See Figure
Marjorie Batchelor
Result
X
k
P A R T
m
m M k
This idea was exploited extensively by Kostant in his work on graded manifolds
Figure
Applications to the manifold of maps b etween manifolds
Let M and X b e manifolds It would seem that the ab ove recip e for obtaining jet bundle
information ab out the manifold M X M of smo oth maps from X to M would b e a very un
promising one to start with M X M form C M X M and then P C M X M R
is not an attractive task Nonetheless measuring coalgebras do provide simple and easy access
to the jet bundle of M X M through the following trick
There are two key observations
If is a p oint then R C
M M M
Thus since the recip e
P C M R P C M C Hom C M C
gives a satisfactory account of jet bundle information for M M M it is reasonable to
p ostulate
P C M C X Hom C M C X
as a candidate for the dual jet bundle of M X M In fact if one de nes the co commutative
part P C M C X of P C M C X
c
X X
P C M C X fp P C M C X p p p p g
c
p p
Marjorie Batchelor
then P C M C X do es very well as a candidate for the dual jet bundle in that
c
the picture is exactly the same as in the case of ordinary manifolds and it contains most
constructions reasonably exp ected to b e elements of the dual jet bundle
Result
i If X M is a smo oth map then R is a sub coalgebra of P C M C X
c
ii If p TM M is the tangent manifold of M with its pro jection onto M let X TM
b e a smo oth map with p Then if R R is the coalgebra of example ii with
p oint like and primitive R R is a measuring sub coalgebra of P C M C X
c
See Figure
iii
X
k
P C M C X T
c
X M smo oth
Here T R and T is the set of all derivations from C M to C X with resp ect
to that is all linear functionals satisfying
f g f g f g
By insp ecting the e ect of comp osing with the evaluation at a p oint x in X it can
b e seen that all such are maps from X to TM such that See Figure
Figure
Prop erty iii is a consequence of the fact that P C M C X is a co commutative
c
coalgebra all of whose simple sub coalgebras are p ointlike For a long time the attractive
p ossibilitie s of includin g the not necessarily co commutative parts of P C M C X were
neglected However using the non co commutative bits gives hop e of representing quantum
groups as genuine transformation groups This brings the story up to my current interests
I will nish by giving one example of the type of non co commutative measuring coalgebra which may b e useful in representing quantum groups and related ob jects
Marjorie Batchelor
Figure
A non co commutative measuring coalgebra
Let A B R z the p olynomial algebra over C on one generator Let C b e the coalgebra
C RE R R of example iii in the rst section
k h
De ne a linear map C Hom R z R z by setting
h z k z z E z
In order for to b e a measuring map it must b e that h and k are algebra homomorphisms
so that
n n n
h z n h k z z for all n
Moreover inductively it can b e shown that
n n
E z z n E
It is curious that the absence of the co e cient one normally exp ects of di erentiation re
sults in the assymetry in the pro duct rule the measuring condition However the b ehaviour
is typical of di erence op erators as opp osed to di erential op erators and there are many
variants
It is also curious that the algebraic prop erties which enable such di erence op erators to
exist in this case are
i The image of k h is the ideal of R z generated by the single generator z and
ii any element of R z z can b e written uniquely in the form p z z
The interrelations b etween conformal eld theories lo op algebras and quantum groups
are an intriquing eld of study It is my b elief that the algebraic prop erties ab ove are at the ro ot of the connection b etween quantum groups and conformal eld theories
Marjorie Batchelor
References
Batchelor M Measuring coalgebras quantum group like ob jects and non commutative ge
ometry In Di erential Geometric Methods in Theoretical Physics C Barto cci U Bruzzo
R Cianci eds Springer Lecture Notes in Physics
Batchelor M In search of the graded manifold of maps b etween graded manifolds In Com
plex Di erential Geometry and Supermanifolds in Strings and Fields P J M Bongaarts and
R Martini eds Springer Lecture Notes in Physics
Exton H q Hypergeometric Functions and Applications Ellis Horwoo d
Kostant B Graded Manifolds Graded Lie Groups and Prequantisation Springer Lecture
Notes in Maths
Lusztig G Quantum deformations of certain simple modules over enveloping algebras
Advances in Math
Sweedler M Hopf Algebras Benjamin
Mo duli Spaces
An Interdisciplinary Workshop organized by Sylvie Paycha
This session was of an exp erimental nature three short twenty minutes talks were to
b e given on the topic of mo duli spaces in relation to the di erent topics of the sessions of
the conference namely one by Caroline Series given by Tan Lei since Caroline Series was
unable to attend the conference on mo duli spaces and dynamical systems one by Rosa Maria
Mir o Roigon mo duli spaces and classi cation problems in algebraic geometry one by Sylvie
Paycha on mo duli spaces and quantum eld theory Laura Fainsilber sp ontaneously gave a
talk on mo duli spaces and number theory of which she kindly gave a brief account for these
pro ceedings
This interdisciplinary session on mo duli space came out to b e a success it lead to lively
discussions and even to a sp ontaneous talk mentioned ab ove It was followed by a discus
sion session on mo duli spaces where we tried to understand b etter the di erent p oints of
views on mo duli spaces presented in the di erent talks There was clearly a need for such
interdisciplinary discussions within this EWM conference in Madrid and we hop e that further
interdisciplinary sessions will b e organised in the future
Sylvie Paycha
th
Proc of the EWM
meeting Madrid
Mo duli spaces and conformal dynamics
Caroline Series
Warwick University England
cms maths warwick ac uk
Introduction
To understand mo duli we rst have to understand the meaning of a complex structure on a
surface A Riemann Surface is a surface i e a manifold S with lo cal charts to C such that
the overlap maps b etween charts are complex analytic Remember that a complex analytic
map lo cally preserves angles and expands or contracts distances In fact using Taylor series
you see that lo cally it lo oks like
z f a f a z a
In other words it is just an a ne map z cz d for some c d This map is a combination
of a translation rotation and similarity A map like this which preserves angles is called
conformal See Sylvie Paycha s pap er In fact a map of a subset of C is conformal if and
only if it is complex analytic Two surfaces S and S are said to b e conformally equivalent
if there is a homeomorphism f S S such that the maps f induces b etween charts are
complex analytic equivalently conformal We can thus talk ab out the conformal or complex
structure of a surface
The problem of mo duli
The problem of mo duli is to describ e the di erent conformal or complex structures there
can b e on a xed top ological surface
Facts which we are not going to prove
The complex structure of a surface of genus g can b e describ ed by g complex parameters
equivalently by g real parameters The set of all p ossible mo duli can b e given roughly
sp eaking the structure of a g dimensional complex manifold
Note
There is often confusion b etween Teichm uller space and Moduli space Both classify the
p ossible complex structures on a surface You get Teichm uller space when you x what
is called a marking on the surface This means that on each of the two surfaces to b e
compared you sp ecify that certain curves are generators of the fundamental group When
you are matching the two surfaces your homeomorphism is required to carry one set of marked
generators into the equivalent set on the other surface You get Moduli space when you ignore
the marking and only ask for a homeomorphism b etween the surfaces not worrying ab out
Caroline Series
what it do es to the generating curves Mo duli space is a quotient of Teichm uller space by
the action of the Mapping Class Group This is the group of di eomorphisms of the surface
mo dulo di eomorphisms isotopic to the identity
Two rst examples
The rst step in classifying conformal structures is the well known but still remarkable Rie
mann Mapping Theorem rst proved fully by Osgo o d in It states
Any simply connected open subset U C which has at least points in its boundary is
conformally equivalent to the unit disc D fz C jz j g
In other words there is an analytic bijection b etween U and D The inverse of an analytic
map is automatically analytic Recently Sullivan and Ro din gave a very interesting and
more constructive pro of of this theorem which allow one given the region U to implement
the map easily by computer
The next simplest regions to classify are annuli By de nition a conformal annulus is a
b ounded op en subset of C whose complement has connected comp onents By metho ds
similar to those used in the Riemann mapping theorem it is proved by Ko eb e that any
conformal annulus can b e mapp ed by an analytic bijection to the region
fz C r jz j g for some r r might b e in nity
Thus to solve the problem of mo duli for annuli we only have to determine when there is
a conformal map b etween two of these standard regions
Theorem If r s there is no complex analytic bijection between the regions fz C
r jz j g and fz C s jz j g
Proof
Supp ose f were such a map We may assume jf z j as jz j Let
g z log r log f z log s log z
Then h z Re g z is harmonic and vanishes on the circles jz j jz j r By the
maximum principle h Therefore g is also constant by the Cauchy Riemann equations
However if we make a circuit round jz j r g changes by
i log r log s
Since g is constant the change is and hence r s
The real number log r asso ciated to an annulus by this result is therefore unique and
is called the modulus of the annulus It completely sp eci es the annulus up to conformal
homeomorphisms The moduli space flog r R r g do es not quite t into our
picture of mo duli space b eing a complex variety b ecause an annulus is not a closed surface neverthless it gives us a feel for what mo duli are like
Caroline Series
Quasi conformal maps
A very imp ortant idea in complex dynamics is the idea of a quasi conformal map This is a
map which is not conformal but which stretches and shrinks with b ounded distortion It is
easy to write down a simple quasi conformal map b etween annuli of di erent mo duli just
map the annuli to their standard p osition and then stretch the smaller one evenly along radii
to t onto the larger It is a fundamental theorem of Teichm uller that for any map b etween
two surfaces there is always a homotopic map of least stretch and that the surface can
b e divided into rectangles or annuli on each of which the map acts like one of these simple
stretches
The general case
The key to solving the mo duli problem for a general top ological surface is the famous Uni
formisation Theorem proved by Ko eb e in It states that any simply connected Riemann
surface is conformally equivalent to exactly one of C C D
We can use it to classify all Riemann surfaces by the following two facts which are not hard
to prove from the de nitions
The universal covering space of a Riemann surface has a complex structure and is again
a Riemann surface
The covering maps are complex analytic homeomorphisms
Thus we have only to list all the p ossible covering maps in the three cases to nd all p ossi
bilities A covering map must b e a xed p oint free analytic homeomorphism of the universal
cover onto itself In the case of C there are no suitable analytic homeomorphisms b e
cause all of them have xed p oints In case of D the maps are exactly the linear fractional
transformations which map the unit disc to itself without xed p oint in the disc These are
the same as the isometries of dimensional hyperb olic geometry Thus it is p ossible to put
a hyperb olic metric on the quotient surface and measure the mo duli in terms of hyperb olic
geometry measurements This works whenever the surface we want has genus If we had
a bit more time we could explain exactly what the g real measurements for a surface
of genus g are Finally in case of C the only xed p oint free conformal homeomorphisms
of C to itself are translations z z b where b is complex The most interesting case is
when the covering group is Z and the quotient C Z is a torus After scaling translating
and rotating which are all analytic homeomorphisms we can assume that the group G of
covering transformations is generated by z z and z z for some complex with
Im We have to nd out when there is an analytic homeomorphism b etween two tori
C G and C G Supp ose f were such a map Let f b e the lift of f to C Then f must b e
p erio dic relative to the lattice p oints m n m n Z It is not hard to show that the only
p ossibilty is that f maps the lattice p oints m n m n Z to the p oints q r q r Z
a b
and hence that for some integers a b c d with ad bc This gives us the well
c d
known picture that the mo duli space for tori is the upp er half plane the plane quotiented
by the action of the group SL Z See Rosa Maria Mir oRoig s pap er
A go o d reference for this material is G Jones and D Singerman Complex functions Cam
bridge University Press
Caroline Series
Mo duli in complex dynamics
How do mo duli arise in complex dynamics In complex dynamics one is studying a rational
function f mapping the Riemann sphere C to itself The Riemann sphere divides into
the Julia set on which the grand orbits of f are dense and the Fatou set on which the
grand orbits are either discrete in attracting or parab olic basins or leaves of dynamically
de ned foliations in Siegel discs Herman rings and sup erattracting basins Pick a connected
comp onent of the Fatou set on which the action of the map is discrete It makes sense to
form the quotient f This is the orbit space in which all p oints in the same f orbit are
identi ed Because the map was acting discretely the quotient is a Riemann sufrace usually
with some branch p oints A lot of use has b een made of the idea that we can relate di erent
rational maps by deforming these quotient surfaces by using quasi conformal maps We can
also investigate the space of all maps f with a given combinatorial structure by lo oking
at the mo duli spaces of the quotients Sullivan proved a very fundamental theorem the
non wandering domain theorem by using the fact that the mo duli spaces for each quotient
are nite dimensional In my own work the rational map f is replaced by a group of
linear fractional transformations Lo oking at where the group orbits are discrete or not the
Riemann sphere still splits into a Julia set and a Fatou set The quotients are
Riemann surfaces We are interested in studying the relation b etween the mo duli of the
surface and the complex parameters which go into de ning the generators of the group This
gives some very interesting and concrete pictures of Teichm uller space
An application to complex dynamics
We end with a nice application of the mo duli of annuli to complex dynamics In studying the
Mandelbrot set for cubic p olynomials Branner and Hubbard had a situation in which they
had an in nite set of nested annuli and they needed a criterion for whether the intersection
consisted of one or many p oints They used basic results ab out annuli
Gr otzsch s inequality Supp ose a sequence A of op en annuli are nested inside an op en
n
annulus A each winding once around the hole in A Then
X
modA modA
n
If A is an op en b ounded annulus of in nite mo dulus the b ounded comp onent of the
complement i e the hole consists of one p oint
Thus if you can nd a sequence A of nested annuli such that the sum of their mo duli
n
is in nite you know that the hole in the middle of the nest consists of just one p oint
th
Proc of the EWM
meeting Madrid
Mo duli spaces and path integrals
in Quantum Field Theory
Sylvie Paycha
Universit eBlaise Pascal Aubiere France
paycha ucfma univ bpclermont fr
Warning These notes are informal notes which are only meant to give a hint as to the
variety ot topics the investigation of moduli spaces in relation to quantum eld theory can
lead to
Mo duli space of a Riemann surface
This section is closely related to Caroline Series paper on moduli spaces and the reader is
referred to the references therin concerning the contents of this section
Let b e a smo oth compact surface without b oundary of genus p The purp ose of this
rst section is to de ne the mo duli space of such a surface We shall need the notions of
Riemannian metric and conformal structure
Riemannian metrics Let p b e a p oint on and let x x b e a system of lo cal co ordinates
at p oint p on A Riemannian metric on at p oint p is de ned lo cally at p oint p by a
symmetric two by two matrix g with strictly p ositive determinant A Riemannian metric
ab
is a global ob ject a covariant symmetric two tensor on but its lo cal description dep ends
on the choice of lo cal co ordinates If f is a di eomorphism of the surface that takes
x x around p to another lo cal system of co ordinates y y around f p then lo cally
g transforms to
a b
f g g
ab
ab
where
d c
X
dx dx
g g
cd
ab
a b
dy dy
c d
This de nes an action of the group D if f of di eomorphisms on the space M et of
Riemannian metrics on
D if f M et M et
f g f g
The metric is a to ol to measure lengths of curves and areas of surfaces on the manifold
In particular since is compact we can de ne the area of for a given metric g by
Z
p
A detg
g
Sylvie Paycha
as well as the area
A x g A
x g
d
of x where x is a map that embeds in R
One can stretch or shrink the metric by multiplying it by a constant factor g k g k
thus multiplying the area A by the same factor k One can also multiply the metric
g
p
p ointwise g p e g p e g p by a strictly p ositive function on Letting M et
denote the space of Riemannian metrics on it is an in nite dimensional space and setting
W fe M ap R g we thus de ne an action of W on M et
W M et M et
e g e g
The conformal class of a given metric g is the set of Riemannian metrics
g fe g M ap R g
A conformal transformation of is a di eomorphism f of which preserves the conformal
class of the metric i e such that f g e g where is a real function on
Given a metric g on and a lo cal co ordinate system x x at p oint p one can
diagonalise the metric and thus nd another system of co ordinates y y in which the
i h
p
e
where R Then any metric e g in the metric matrix takes the form
p
e
conformal class of g is also diagonal in this co ordinate system and its matrix takes the form
h i
e
The system of co ordinates y y is called isothermal for the conformal class
e
g Setting z y iy one can equip with a complex structure There is in fact a one
to one corresp ondence b etween the set C of complex and the set C onf of conformal
structures on
We nally have the following isomorphisms
M et W C C onf
The action of the di eomorphism group on M et induces an action on C onf and
hence on C setting for f D if f f g f g In general there are di eomorphisms
which do not preserve a given complex or conformal structure they take a given complex or
conformal structure to another one see Caroline s talk These two complex or conformal
structures are then called equivalent
The moduli space M od of is the set of non equivalent complex or conformal struc
tures on It can b e describ ed as a quotient space in four equivalent ways
C D if f C onf D if f
M et W D if f
M et D if f W
where D if f acts via the action on the space of metrics and W via the action
In what follows we want to interpret the mo duli space M od in the framework of string theory in terms of the quotient space of a space of paths by the action of a symmetry group
Sylvie Paycha
Mo duli space and path integrals
The space of paths
Here we shall see together with a xed Riemannian metric g on and an embedding
d d
x R as a path descib ed by a loop or closed string moving in space time R The
starting p oint and the end p oint of its tra jectory which would introduce b oundaries to the
surface describ ed by the lo op are taken at in nitely distant times so that we can assume
everything happ ens as though the surface were b oundaryless and compact If we consider
several interacting lo ops i e lo ops meeting up together and splitting again the resulting
surface can b e of any genus p
The space P of paths can therefore b e seen as the pro duct space E mb M et of
d
the space E mb of embeddings of into R with the space of Riemannian metrics on a
b oundaryless compact smo oth surface of any genus
Dif f as a symmetry group
A classical string evolves acording to a minimal energy principle and describ es a surface
with minimal area A x g as de ned in This area is also called the classical action or
energy
Since the area do es not dep end on the chosen parametrization of the surface this action
A x g is invariant under the action of the di eomorphism group D if f i e
A x f f g A x g
We see this group as a symmetry group for the classical action or energy
In fact this action is also invariant under the action of W so that the symmetry group
S y m is in fact a larger group the smallest one containing b oth W and D if f which
we shall not describ e in detail here
Mo duli space
Di eomorphisms act trivially on E mb via comp osition x x f f D if f x
E mb and W do es not act on E mb so that the action of S y m on E mb
reduces to that of D if f Combining this action with the action of S y m on the space
of Riemannian metrics one can de ne an action of the symmetry group S y m on the space
of paths P E mb M et
We shall call two paths equivalent when there is an element of the symmetry group that
transforms one into the other The space of non equivalent paths is thus describ ed by
P S y m E mb Dif f M et S y m E mb D if f M od
Since the mo duli space coincides with M et S y m we see how it arises here as a subspace
of the space of non equivalent paths P S y m
Path integrals
In quantum eld theory one cannot see paths but only mean values over the space of
paths namely observables An observable is the mean value O of a function O R
with resp ect to a formal measure on the space of paths
Z
A p
O Z O p e dp P
Sylvie Paycha
where A p is the classical action or energy of the theory in the case of strings it is the area
A x g given in dp is a formal volume measure on the in nite dimensional space of
R
A p
e dp paths P and Z is the normalising constant Z
P
Giving a meaning to such integrals on in nite dimensional manifolds is extremely di cult
and the description of path integrals gives rise to problems of a geometric or top ological
nature which a priori can lo ok very di erent from the original one such as lo oking for
invariants of manifolds D see also the recent work by Seib erg and Witten on the sub ject
reviewed in B
In the case of strings mentioned ab ove an interpretation of the partition function
Z
S x g
e dxdg Z
E mb M et
was rst suggested in P see also Po ab out this interpretation and further investigated
by many authors using algebraic geometric techniques see e g Bo Ph S
Anomalies
Whenever the function O is invariant under the symmetry group one can hop e to reduce
the path integral given by O to an integral on the mo duli space which is sometimes
nite dimensional as in the case of strings see C Series talk when seen as a quotient of
the space of paths via the action of the symmetry group since the latter leaves the classical
energy A p arising in this integral invariant
However there can b e an obstruction to doing this if the formal volume measure denoted
by dp on the path space is not invariant under the action of the symmetry group This
is the case in the example considered ab ove where dg is not invariant under S y m the
symmetry we have at the classical level S x g is invariant under S y m is broken at
the quantised level when integrating w r to the measure dx dg on the space of paths
This gives rise to anomalies which can b e interpreted as top ological and geometric ob
structions on a certain line bundle built on mo duli space namely a determinant bundle see
e g ASZ Bo BF F The fact that there is a determinant line bund le involved is related
to the fact that a transformation of the ab ove path integral via the action of the group
gives rise to a jacobian determinant also called the Faddeev Popov determinant BV In the
case of stings mentioned ab ove the conformal anomaly arises from the non invariance of
the formal measure dg on the space of metrics under the action of W F
References
ASZ O Alvarez I M Singer B Zumino Gravitational anomalies and the family s index theo
rem Comm Math Phys
B D Bennequin Monopoles de Seiberg Witten et conjecture de Thom S eminaire Bour
baki Nov
Bo J B Bost Fibr es d eterminants d eterminants r egularis es et mesures sur les espaces
de modules des courbes complexes S eminaireBourbaki exp os e Ast erisque
BF J M Bismut D Freed The Analysis of el liptic families I Comm Math Phys
Sylvie Paycha
BV O Bab elon C M Viallet The geometrical interpretation of the Faddeev Popov deter
minant Phys Lett B
D S K Donaldson An application of gauge theory to the topology of manifolds Journ
Di geom
F D Freed Determinants torsion and strings Comm Math Phys
P A M Polyakov Quantum geometry of bosonic strings Phys Lett B
Ph D H Phong Complex Geometry and Strings Pro ceedings of Syp osia in Pure Mathe
matics Vol Part
Pol J Polchinski From Polyakov to Moduli in Mathematical asp ects of string theory
S T Yau ed World Scienti c Singap ore
S D J Smit String theory and algebraic geometry of moduli spaces Comm Math Phys
p
th
Proc of the EWM
meeting Madrid
Notes on mo duli spaces in
Algebraic Geometry
Rosa M Mir o Roig
Universitat de Barcelona Spain
miro cerber ub es
Introduction
Mo duli spaces are one of the fundamental constructions of Algebraic Geometry They arise
in connection with classi cation problems and although it is a fairly delicate sub ject I will
try to describ e it in an elementary fashion
Roughly sp eaking a mo duli space for a collection of ob jects A and an equivalence relation
is a classi cation space i e a space in some sense of the word such that each p oint
corresp onds to one and only one equivalence class of ob jects Therefore as a set we de ne
the mo duli space as equivalence classes of ob jects A In our setting the ob jects are
algebraic ob jects and b ecause of this we want an algebraic structure on our classi cation set
Finally we want our mo duli space to b e unique up to isomorphism
So the basic ingredients for a mo duli problem are a collection of ob jects A an equivalence
relation on A and a concept of family of ob jects of A parametrized by an algebraic variety
or a scheme S See the lectures on Algebraic Geometry for the precise de niton of algebraic
variety and scheme Such a family consists of a collection of ob jects X one for each s S
s
which t together in some way corresp onding to the structure of S The precise de nition
of family dep ends on the particular mo duli problem however in all cases it satis es the
following prop erties
A family parametrized by the variety fptg consisting of a single p oint is a single ob ject
of A
There is a notion of equivalence of families parametrized by any given variety S which
gives the equivalence relation on A when S fptg we denote this relation again by
For any morphism S S and any family X parametrized by S there is an
induced family X parametrized by S Furthermore we have the functorial prop erties
identity and it is compatible with i e if X X then
S
X X
These notes are intended to supp ort our cross discipli nary discussion on mo duli spaces Many p eople
have made imp ortant contributions without even b eing mentioned here If I have made wrong attributions I
ap ologize for that In no case do I claim that any idea here at all originates from me
Rosa M Mir o Roig
The mo duli problem consists in giving to A a structure of algebraic variety which
re ects the structure of families of ob jects of A
Fine mo duli spaces coarse mo duli spaces and quotients
We consider the contravariant functor F V ar ieties S ets S F S where F S is
the set of equivalence classes of families parametrized by the variety S Supp ose that M is
an algebraic variety with underlying set is A For any family X parametrized by S we
denote by S M the map given by s X where X is the equivalence class
X X s s
of the ob ject X
s
for a given classi cation problem is a pair M which De nition A ne mo duli space
represents the functor F
Let M b e a variety and F H om M the natural transformation given by
S X for any variety S and any family X parametrized by S By de nition
X
F if is an isomorphism of functors the pair M represents the functor
Remark From the de nition it easily follows that the underlying set of the variety M is
A
Remark If a ne mo duli space exists for a given classi cation problem then it is unique
up to isomorphism
The morphism H om M M determines up to the equivalence relation a family
M
U F M which gives us the following alternative de nition
for a given classi cation problem consists of a variety M De nition A ne mo duli space
and a family U parametrized by M such that for every family X parametrized by S there
is a unique morphism S M with X U The family U is called a universal family
for the given problem
Unfortunately there are very few classi cation problems for which a ne mo duli space
exists and it is necessary to nd some weaker conditions which nevertheless determines a
unique structure of algebraic variety on M
De nition A coarse mo duli space for a given classi cation problem consists of a variety
M together with a natural transformation F H om M verifying
pt is bijective
For any variety N and any natural transformation F H om N there exists a
unique natural transformation H om M H om N such that
Remark From the de niton it easily follows that M is a variety with underlying set A
Remark If a coarse mo duli space exists for a given classi cation problem then it is unique
up to isomorphism
Remark A ne mo duli space for a given classi cation problem is always a coarse mo duli
space for this problem but in general not vice versa In fact there is no a priori reason why
the map S F S H om S M should b e bijective for varieties S other than fptg
Rosa M Mir o Roig
In the last part of this section we introduce the notion of categorial quotient of a variety
by the action of a group and we will explain its connection with mo duli problems
of De nition Let G b e an algebraic group acting on a variety X A categorial quotient
X by G is a pair Y where Y is a variety and X Y is a morphism such that
is constant on the orbits o x fg x g Gg X of the action
For any variety Z and for any morphism X Z which is constant on orbits there
is a unique morphism g Y Z such that g
If in addition y consists of a single orbit for all y Y we call Y an orbit space
Remark A categorial quotient is unique up to isomorphism and it exists in general cir
cumstances For instance if G is a reductive group acting on an a ne variety X or if G is
a reductive group acting on a pro jective variety X and we restrict our attention to the op en
ss
subset X X of semistable p oints of X
To relate categorial quotients to mo duli spaces we need to introduce some extra de nitions
De nition For a given mo duli problem a family X parametrized by a variety S is said
to have the lo cal universal prop erty if for any family X parametrized by S and any p oint
s S there exists a neighbourho o d U of s such that X is equivalent to the family induced
jU
from X by some morphism U S
Prop osition Suppose that for a given moduli problem there exists a family X parametrized
by S having the local universal property Suppose that a group G acts on S and that X X
s t
if and only if s and t belongs to the same orbit of this action Then
Any coarse moduli space is a categorial quotient of S by G
A categorial quotient of S by G is a coarse moduli space if and only if it is an orbit
space
More details on ne mo duli spaces coarse mo duli spaces and quotient can b e found for
instance in MFK MS or N
Examples
As particular examples of mo duli problems we will brie y discuss the three following cases
Hilb ert schemes
The mo duli space for the isomorphism classes of smo oth curves of genus g
The mo duli space for stable vector bundles with given Chern classes on a pro jective
variety X
Rosa M Mir o Roig
Example
r
Let P b e the r dimensional pro jective space over a eld k The rst classi cation problem
r
that we will deal with is the classi cation problem for closed pro jective subschemes X P
and we will see that there exists a ne mo duli space for such a classi cation problem Roughly
r
sp eaking our ob jects will b e closed subschemes of P with given Hilb ert p olynomial p t Q t
the equivalence relation will b e the equality and we have the following notion of family
r
De nition A at family of closed subschemes of P parametrized by a k scheme S is a
r r
closed subscheme X P P x S such that the morphism X S induced by the pro jection
S
r r
P P x S S is at
S
r r
It is an imp ortant fact that at families P P x S X S of closed subschemes
S
r
of P parametrized by a connected k scheme S have all their bres with the same Hilb ert
p olynomial
P
r
t r
Q t where a s We x an integer r and a p olynomial of the form p t a
i i
i
i
are integers We consider the contravariant functor
r
k schemes S ets H il b
p t
r
r
where H il b S f at families of closed subschemes of P with Hilb ert p olynomial p t
p t
parametrized by S g In A Grothendieck proved See G or M
r r r
There is a unique pro jective scheme H il b which parametrizes a at family P x H il b
p t p t
r r
of closed subschemes of P with Hilb ert p olynomial p t and having the fol W H il b
p t
f
r r
lowing universal prop erty for every at family P x S X S of closed subschemes of P
r
with Hilb ert p olynomial p t there is a unique morphism g S H il b called the classi
p t
r
cation map for the family f such that induces f by base change i e X S x W
H ilb
p t
Following the de niton of ne mo duli space is called the universal family
r
r
In the usual language of categories the pair H il b represents the functor H il b
p t
p t
and the classi cation problem for pro jective subschemes has a ne mo duli space
t n
r
is a linear subspace Since any closed subscheme of P with Hilb ert p olynomial p t
n
r
of P of dimension n we have
r
Gr ass n r H il b
t n
n
Hence the Hilb ert schemes can b e considered as generalizations of the grassmannians i e
varieties parametrizing all n dimensional vector subspaces of a given r dimensional
vector space V
Remark We know the existence of the Hilb ert scheme but its lo cal and global prop erties
are very far from b eing understo o d even for the case of pro jective space curves I will not
discuss here either recent contributions or op en problems
Example
We consider the set fC g of smo oth curves of genus g and we ask whether the set M of such
g
curves up to isomorphism may b e given the structure of an algebraic variety in a natural
Rosa M Mir o Roig
way such is the case for g where the j invariants form an a ne line To b egin with we
de ne a at f amil y of smo oth curves of genus g with base S to b e a variety V and a at
morphism V S such that for each p oint s S the b er V s is isomorphic to
s
C for some The b est way to sp ecify the algebraic structure on M would b e to require it
g
to b e a universal parameter variety for families of curves of genus g in the following sense
we require that there b e a at family X M of smo oth curves of genus g such that for
g
any other at family f V T of smo oth curves of genus g there is a unique morphism
for the curves g T M such that V g X In this case we call M a ne mo duli space
g g
fC g
Unfortunately there are very few classi cation problems for which a ne mo duli space
exists One of the reasons why the universal family X M do es not exist is that there are
g
nontrivial at families of smo oth curves of genus g all whose bres are isomorphic to each
other Hence it is necessary to nd some weaker condition which nevertheless determines a
unique algebraic structure on M This problem was settled by D Mumford He proved that
g
for g there is a coarse mo duli space M which has the following prop erties See M
g
Theorem
The set of closed p oints of M is in one to one corresp ondence with the set of isomor
g
phism classes of smo oth curves of genus g
if f V T is a at family of smo oth curves of genus g then there is a morphism
g T M such that for each closed p oint t T V is in the isomorphism class of
g t
curves determined by the p oint g t M
g
Since all smo oth connected curves of genus g are isomorphic to P we have M fptg
In case g the j invariants of elliptic curves de ne an a ne line which is a coarse mo duli
space M for the family of elliptic curves In P Deligne and D Mumford proved that
M for g is an irreducible quasi pro jective variety of dimension g See DM An
g
excellent discussion of this construction is given in MFK which includes references to other
examples of the applications of geometric invariant theory as well
Remark The dimension of M was already stated by Riemann in his celebrated pap er
g
Theorie der Ab el schen Functionen of See R The word mo duli is due to him
and the sub ject has its origins in the theory of elliptic functions The irreducibili ty was
already observed by Klein See K and follows from results of L urothand Clebsch and it
was only much later that Baily showed that M has a natural structure of quasi pro jective
g
normal variety of dimension g
Remark Nowadays there are three principal approaches to constructing M
g
In a transcendental setting it can b e constructed as a quotient of the Teichm ullerspace
As a subvariety of a quotient of the Siegel upp er half space or
Using Mumfords s geometric invariant theory it can b e constructed as a quotient of a
Hilb ert scheme
The mo duli space M is not compact i e it is a quasi pro jective variety but not a pro jective
g
variety The right choice of b oundary p oints for M was discovered by Mayer and Mumford g
Rosa M Mir o Roig
in in an unpublished work For an excellent discussion of M and its compacti cation
g
the reader could lo ok at MFK
Example
As last example we will consider the mo duli problem for vector bundles on a smo oth pro jective
variety X We consider the set A of isomorphism classes of rank r vector bundles on X
with xed Hilb ert p olynomial H m Q m and we would like to endow A with a natural
structure of scheme To this end we de ne a family of vector bundles of rank r and Hilb ert
p olynomial H m parametrized by a k scheme S as a vector bundle E on S x X such that
for all s S E s is a rank r vector bundle on X fsg x X with Hilb ert p olynomial H m
Unfortunately this mo duli problem has no solution and to get at least a coarse mo duli
space we must somehow restrict the class of vector bundles that we consider What kind of
subfamily should b e taken In MA MA M Maruyama found an answer to this question
stable vector bundles He proved Let X b e a smo oth pro jective variety and let A the set
of isomorphism classes of stable vector bundle on X of rank r and Hilb ert p olynomial H m
Then there is a coarse mo duli scheme M which is a separated scheme lo cally of nite type
This means
The closed p oints of M are in one to one corresp ondence with the elements of A
Whenever F is a at family of vector bundles of A parametrized by a scheme T
i e F is a vector bundle on X x T at over T whose bres are in A then there
is a morphism T M such that for each closed p oint t T t is the p oint of M
corresp onding to the class of the vector bundle F which is the bre of F ovet t
t
The morphism can b e assigned functorially and
M is universal with the prop erties and
Remark In spite of the great progress made during the last decades on the mo duli spaces of
vector bundles on smo oth pro jective varieties essentially in the framework of the Geometric
Invariant Theory by Mumford very litle is known ab out their lo cal and global structure
References
B W Baily On the theory of functions the moduli of abelian varieties and the moduli
of curves Ann of Math
DM P Deligne and D Mumford The irreducibility of the space of curves of given genus
IHES Publ Math
G A Grothendick Les sch emasde Hilbert Sem Bourbaki
H J Harris On the Kodaira dimension of the moduli space of curves II Inv Math
HM J Harris and D Mumford On the Kodaira dimension of the moduli space of curves
I Inv Math
Rosa M Mir o Roig
Ha R Hartshorne Algebraic Geometry GTM
K F Klein Uber Riemann s Theorie der Algebraischen Functionen un Ihrer Integrale
Teubner Leipzig
MA M Maruyama Moduli of stable sheaves I J Math Kyoto Univ
MA M Maruyama Moduli of stable sheaves II J Math Kyoto Univ
M D Mumford Lectures on Curves on an Algebraic Surface Princeton Univ Press
M D Mumford Geometric Invariant Theory Springer Verlag
MFK D Mumford J Fogarty and F Kirwan Geometric Invariant Theory Springer Verlag
MS D Mumford and K Suominen Introduction to the Theory of moduli Pro o c fth
Nordic Summer School in Math Oslo
N P E Newstead Introduction to moduli problems and orbit spaces Tata Inst of Fun
damental Research Bombay
R B Riemann Theorie der Abel schen Functionen Crelle J
th
Proc of the EWM
meeting Madrid
An example of mo duli spaces in number theory
Laura Fainsilber
Universit ede Franche Comt e France
laura math univ fcomte fr
I thank Leila Schneps for introducing me to the sub ject of Grothendieck Teichm uller
theory for explaining it to me and for crucial help in preparing this text
The inverse problem in Galois theory
The basic ob ject of interest to number theorists is the eld of rational numbers Q along with
its eld extensions We will rst consider nite Galois extensions of Q given an irreducible
Q of Q we let K b e the smallest p olynomial P Q T with ro ots in an algebraic closure
i
eld containing Q and all the Such a eld is called a Galois extension of Q The Galois
i
group G Gal K Q Aut K is de ned to b e the group of eld automorphisms of
Q
K that x the elements of Q It is a nite group and the fundamental theorem of Galois
theory asserts that there is a bijective corresp ondence b etween sub elds of K which are
Galois extensions of Q and normal subgroups of G Namely a normal subgroup H of G
corresp onds to the sub eld L of elements of K xed by all the automorphisms in H and
Gal L Q G H In that case K is also a Galois extension of L with Galois group H
We express this in the following diagram
K
j H
L G
j G H
Q
The inverse problem in Galois theory consists of nding out which groups can b e Galois
groups of extensions of Q and constructing explicit extensions with such groups as Galois
groups For example all nite ab elian groups the symmetric groups S and the alternating
n
groups A can b e Galois groups as can the sp oradic simple groups except maybe the
n
Mathieu group M With a combination of metho ds many families of groups have b een
shown to b e Galois groups E No ether found algebraic geometric conditions under which
a group is a Galois group More recently Matzat and Mestre among others have worked
on constructive approaches and given explicit p olynomials that de ne Galois extensions for
new families of groups One conjectures and most exp erts b elieve that all nite groups are
Galois groups over Q
Laura Fainsilber
Grothendieck s approach
In the last few years a novel top down approach has b een developed in particular by Drin
fel d Ihara Schneps Lo chak inspired by ideas of Grothendieck It consists of studying
Q Q Aut Q We extend our notion of Galois extensions directly the group G Gal
Q Q
Q to in nite algebraic extensions we see a nite Galois extension K of Q as a sub eld of
and see the Galois group of K as a nite quotient of G The inverse problem in Galois
Q
theory is now to see which nite groups are quotients of G and in general to describ e the
Q
structure of the large and mysterious group G
Q
Q
j G
K G
Q
j G
Q
ab
The ab elian part of G is the in nite quotient of G which corresp onds to Q the
Q Q
Q containing all nite extensions of Q with commutative Galois group minimal sub eld of
It is isomorphic to Z the multiplicative group of the pro nite completion of the integers
ab
Whilst the structure of Q and the action of Z on it are relatively well understo o d we know
ab
very little ab out the subgroup Gal Q Q of G or ab out the way it combines with
Q
Z G to form G
Q Q
Q
j
ab
Q
G
Q
j Z
Q
Grothendieck s dream as he expressed it in L esquisse d un programme is to understand
combinatorial prop erties of G by studying the way it acts on geometric ob jects with the hop e
Q
of obtaining a complete description of G The idea is to reformulate geometric prop erties
Q
of certain varieties and in particular of certain mo duli spaces in terms of their fundamental
groups and of the action of G on these fundamental groups
Q
Geometric action of G
Q
We are now going to describ e some geometric groups on which G acts Let us consider
Q
Galois extensions of the eld C T in one indeterminate over C There is a one to one
corresp ondence b etween extensions of C T unrami ed outside of and meaning the
and unrami ed ideals generated by the p olynomials T T and the rational function
T
n f g the eld extensions coverings of the pro jective line with three p oints removed P
C
are the function elds of the coverings
The Galois groups of the eld extensions are the automorphism groups of the corresp onding
n f g which is coverings they are themselves quotients of the fundamental group of P
C
the free group on two generators F
We now consider unrami ed coverings of P n f g Belyi s theorem asserts that if
C
we have a covering X P where X is a Riemann surface i e a curve de ned over C C
Laura Fainsilber
Q then there is an equation for X with co e cients in such that the critical values of lie in
Q and a new rational function on X with co e cients in Q and critical values in f g
Let X P b e a Belyi cover i e X is a Riemann surface and the critical values of
C
lie in f g We can consider the eld C X comp ositum of the function elds C Y for
all nite coverings Y of X The Galois group Gal C X C X is the pro nite completion
X of the fundamental group of X i e the pro jective limit of the groups X N for
normal subgroups N X of nite index
Q it is in fact de ned over a number eld K i e a nite extension Since X is de ned over
of Q We have K Q and every nite cover Y of X is de ned over a nite extension of K
Q X the eld of functions of Let K X denote the eld of functions on X de ned over K
X de ned over Q and for every nite cover Y of X let Q Y denote the eld of functions on
Q Let Q X denote the comp ositum of the elds Q Y for all nite covers Y de ned over
Y of X A classical theorem states that C X Q X C and C X Q X C so the
Galois group Gal Q X Q X is isomorphic to Gal C X C X which is isomorphic as
Q X is we noted ab ove to the pro nite fundamental group X However now the eld
a eld extension of K X with Galois group G Gal Q K and we have the following
K
diagram
Q X
j X
G
Q X
j G
K
K X
A theorem states that the huge group G can b e written as a semi direct pro duct of X
with G In particular this means that there is an action of G on X in fact there are
K K
many such actions In the case where X is de ned over Q we now have an action of G on
Q
X
The mo duli space M
The mo duli space M of Riemann surfaces of genus with four marked p oints is isomorphic
n n f g which is a variety de ned over Q So when we lo ok at coverings of P to P
C C
f g we actually have coverings of the mo duli space and the arguments ab ove show
that there is an action of G on M This is very interesting b ecause mathematicians
Q
from other areas ranging from mathematical physics to geometry have studied such mo duli
spaces and describ ed many of their prop erties
d
There is a very large group called GT which also acts on the pro nite completion of the
d
fundamental group of the mo duli space Drinfel d has given an explicit description of GT as a
set of elements satisfying some combinatorial conditions so that we can actually make certain
d
computations with elements of GT Drinfel d and Ihara have shown that G is contained
Q
d d
in GT One would now like to know whether G is all of GT or how to identify G as a
Q Q
d
subgroup of GT so as to obtain an explicit description of its action on mo duli spaces and
thus get a b etter grasp of the combinatorial prop erties of G
Q
The philosophy of Grothendieck s approach expresses the idea that the mo duli spaces M
g n
of Riemann surfaces of genus g with n marked p oints or punctures are the varieties de ned
Laura Fainsilber
over Q which contain al l information ab out all the curves de ned over C In particular they
contain the information ab out the curves de ned over Q which in turn give us informations
ab out G Indeed G acts on all the pro nite fundamental groups of the mo duli spaces
Q Q
M resp ecting many natural morphisms b etween these groups which come from geometric
g n
morphisms b etween the spaces The following questions are therefore natural ones to ask
d
when we try to reach a further understanding of G and of its relation to GT
Q
What is the full group of tuples of automorphisms of the M for
g n g n g n
varying g and n resp ecting all the natural morphisms b etween these groups It would
d
seem that the answer is GT as Drinfel d suggests
The group of pairs of automorphisms of the Teichm uller Tower consisting of the two
d
mo duli spaces M and M with natural morphisms b etween them is precisely GT
To what extent do M and M contain all the information ab out all the M
g n
Since the mo duli spaces contain all the information ab out a class of ob jects de ned
Q is it reasonable to think that no greater group than G which is after all a over
Q purely arithmetic group can act on the tower of their pro nite fundamental groups
Short Talks
th
Proc of the EWM
meeting Madrid
The multidimensional Riemann Hilb ert problem
generalized Knizhnik Zamolo dchikov equations and
applications
Valentina A Golub eva
S
m
n n
L b e a reducible algebraic variety in C P and C n L GL n Let L
i
i
a representation The Pfa an system of Fuchsian type with singular set L is the system
P P
m m
dL
i
A The statement of the and A of the form dF F where
i i
i i
L
i
multidimensional Riemann Hilb ert problem is the following for given nd the Pfa an
system of Fuchsian type whose fundamental solution realizes the given representation For
the simplest case of the variety L L is the union of non singular algebraic hypersurfaces
with transversal intersections some conditions of solvability were obtained V Golub eva
A Bolibruch T Otsuki some partial results are known in low dimensions also for more
complex varieties L
The Knizhnik Zamolo dchikov equations asso ciated to the ro ot systems A have their ori
n
gin in the Wess Zumino Witten mo del of quantum eld theory as the equations for the
n p oint correlation function The construction of the di erent generalizations of these equa
tions in particular for the other ro ot systems B C D p ermits to consider these equations
as examples of solvable cases of the Riemann Hilb ert problem Indeed for some ro ot system
n
R we are given the reducible algebraic variety in C P usually it is an arrangement of a
nite number of hyperplanes the known fundamental group of the complement to this variety
n
in C P the last results in this direction b elong to D Markushevich and A Leibman and a
prescrib ed by physical mo del representation The generalized Knizhnik Zamolo dchikov
equations for a variety of the assumptions on was constructed by I Cherednik A Matsuo
A Leibman V Golub eva and V Leksin ect The quantum variant of Knizhnik Zamolo dchikov
equations N Reshetikhin and others is known
The generalized Knizhnik Zamolo dchikov equations have applications in contemporary
quantum eld theory and statistical mechanics in the theory of quantum Hall e ect in the
theory of anyons sup er conductivity etc The KZ theory is closely connected with the theory
of many bo dy systems describ ed by the systems of Calogero Moser Sutherland type
Quadratic and hermitian forms over rings
Laura Fainsilber
Universit ede Franche Comte Besan con France
laura math univ fcomte fr
For quadratic forms over a eld Witt s cancellation theorem asserts that if q q and q
are three non singular forms such that q q q q then q q We want to generalize
this cancellation to the situation of hermitian forms over rings Let A b e a ring with an anti
automorphism A A of order let M b e a re exive nitely generated left A mo dule A
hermitian form is a biadditive map h M M A such that h am bn ah m n b and
h m n for all m n M and a b A A form is said to b e unimo dular if the h n m
adjoint homomorphism it induces from M to its dual is bijective If A is commutative and
is the identity then the hermitian forms are exactly the symmetric bilinear forms We give
counter examples to show that the analog of Witt s cancellation theorem do es not hold for
symmetric bilinear forms over the ring of integers Z However cancellation is p ossible for
hermitian forms unimo dular or not over rings which are nitely generated algebras over
complete discrete valuation rings such as rings of matrices over the p adic integers M Z
n p
or group rings Z G for nite groups G
p
th
Proc of the EWM
meeting Madrid
On Sampling plans for insp ection by variables
Vera I Pagurova
Moscow State University Russia
Designing of optimal sampling plans by variables is considered when the item quality
characteristic follows a distribution b elonging to a two parametric lo cation scale family of
distributions Two class and three class pro cedures of sampling plans by variables are inves
tigated
A random sample of n items is drawn from the lot Insp ection pro cedures are based on
the measurement of an item quality characteristic X An attributes plan bases the decision to
accept or reject the lot only on the number of nonconforming items in the sample Variables
plans are able to achieve the same control with a smaller sample size by making use of the
distribution of the initial measurements
Let X X X b e indep endent identity distributed variables with the common dis
n
tribution density f x a b b jaj b a and b are unknown f x F x
P P
n n
X X X are order statistics X X n S X X n
i i
n
i i
Two class pro cedure An item will b e considered to b e conforming if X u otherwise
nonconforming Set p PfX ug The hypothesis of interest is H p p p
is maximum allowable p ercent of nonconforming items against K p p We denote
d a u b d F p Y X a b i n
i i
Theorem Let f x exp x then the uniformly most powerful invariant
u m p i test at level of H against K rejects H when X u S c with c de ned by
Pf X u S c j d d g
Theorem Let f x exp x x then the u m p i test at level of H against K
rejects H when X u X X c with c de ned by Pf X u X X
c j d d g
Theorem Let f x x then the u m p i test at level of H against K rejects
H when X u X X c with c de ned by Pf X u X X c j d
n n
d g
Now we consider an asymptotic approach Denote EY DY m E Y
j
j
d is some function of d m m z is p quantile of the standard normal
p
distribution
Prop osition If EX then the test of H against K with the rejection region X
p
p
n d z has the true level when n u S d
An asymptotic test with estimators of a and b based on central order statistics is consid
ered
Vera I Pagurova
Three class pro cedure An item will b e considered to b e conforming if X u to b e
marginally conforming if u X u and to b e nonconforming if X u where u u
p PfX u g p PfX u g The probability of acceptance of a lot of arbitrary
quality p p is studied
References
Lehmann E L Testing Statistical Hyp otheses N Y J Wiley
Pagurova V I Nesterova S A Sampling Plans for Insp ection by Variables Moscow
University Comput Math and Cyb ern Evaluation
Evaluation of the mathematical asp ects of the meeting
Isab el Lab ouriau
Universidade do Porto Portugal
At the end of the meeting a discussion to ok place in order to evaluate the di erent math
ematical parts The following is a rep ort of that discussion
The general feeling was that the series of talks and the interdisciplinary discussion on
mo duli spaces were of go o d standard Non trivial mathematics were presented in an un
derstandable way generated interdisciplinary discussions and even some informal workshops
nothing to b e ashamed of was one comment The talks had more participation than
usual and sp eakers felt it was more interesting this way that is with a lot of questions asked
although this sometimes made it di cult for them to reach their goals
The work of the planted idiots members of the audience in charge of asking questions
b esides the sp ontaneous ones a habit started at the EWM meeting in Luminy was imp ortant
to keep the talks at the right level They created an informal atmosphere in the rst talks
that was kept through the end In the last talks there were no planted idiots the only
questions were the sp ontaneous ones but it was p ointed out that we should have kept the
go o d habit to the end For next time it was suggested that sp eakers get more information on
the type of talk exp ected some talks underwent big changes at the last minute
It was agreed that three topics in mathematics plus one interdisciplinary discussion to
gether with the non mathematical discussions was to o much for one week We should b e
less ambitious for the next meeting and have more time for talking ab out maths instead of
listening Too little time was allowed for the discussion on mo duli spaces the organizers
were not sure how it would work and it turned out to b e so interesting it was continued
in the spare time One suggestion is that we have fewer talks towards the end of a meeting
in order to have time for interdisciplinary discussions like the one planned on mo duli spaces
and the sp ontaneous one on renormalization
There were several opinions on the b est way to organize a series of talks next time opinions
varied The choice is not only b etween summer school type or research conference type talks
we may want to do something new talks that generate an interdisciplinary discussion For
each series we may need an introductory session to introduce the language and basic results
This is hard on the rst sp eaker and where do we stop when going backwards The goal of
the series is not to learn the language but to transfer ideas with a minimum of language Two
of the series in Madrid had an introduction that was necessary but not the main goal Some
elds may b e naturally more technical and need more introduction some areas also have a
tradition on non technical talks Maybe we should not sp end to o much time on elementary
things after all one may understand a lecture without the details and appreciate it we are
used to that With such a wide audience it is di cult to avoid some form of introduction
the question is how much
The role of the organizer of a series was also discussed and how much she should interfere
either directing the sp eakers and choosing the topics so as to reach a certain goal or just
trying to get the b est maths we can even at the cost of some coherence Again some felt that
it dep ended on the sub ject as it is di cult to give a go o d idea of the eld with homogeneous talks
The talks that were not part of a series had a small audience partly b ecause the program
was already heavy and partly for the lack of advertising those interested could not always
attend a talk Parallel sessions were not thought to b e a go o d solution it makes talks more
di cult to attend and somehow tend to b ecome a secondary hierarchy without any scienti c
basis Many of us prefer p osters as they can b e read at one s own sp eed and it is easier to
ask questions The problem is that p eople usually do not make go o d p osters maybe the rst
time we need guidelines It might work if everybo dy had a p oster with a photo on it and
maybe some historical background so the p osters would b e less linked with hierarchy and
take the place of introductions It would b e go o d to leave them on all the time and have a
so cial event near them to break the ice
OTHER TOPICS
In b etween meetings the everyday life of EWM
Rep ort by Sylvie Paycha
International EWM meetings are organised every two years they o er an op ortunity for
women mathematicians from all over Europ e to meet together exchange ideas on mathemat
ics as well as on women and mathematics However once every other year is seldom and
we feel EWM should go on living in b etween meetings What could the everyday life of
EWM b e E mail o ers many p ossibilities such as sending the Newsletters to members of
EWM and thereby keeping them informed of what is going on in Europ e in the way of women
and mathematics discussing plans for the future among members of the standing committee
organising the pro ceedings of the last conference However exchanges via e mail is a bit
immaterial and maybe not fully satisfactory as the everyday life of an organisation
We have thought of trying to organise concrete pro jects b etween the meetings of EWM
which could b e smaller scale meetings around a sp eci c topic involving a few members of
EWM interested in the topic
A rst attempt in that direction is an interdisciplinary workshop on Renormalisation in
Mathematics and Physics of which you will nd an announcement here which will take
place in Paris The idea of the topic of this workshop came up during the EWM meeting
in Madrid where sp ontaneous discussions arose as to the di erent interpretations of the
notion of renormalisation a concept which was mentioned in various talks in the eld of
statistical physics of complex dynamical systems of quantum eld theory in the course of
the conference
Such meetings b etween the general international meetings of EWM are a step towards a more concrete everyday life of the organisation
Renormalisation in Mathematics and Physics
Paris June and
Preliminary announcement
The purp ose of this workshop is to study the various interpretations of renormalisation
in dynamical systems statistical physics and quantum eld theory and to explore the con
nections among them Preliminary discussions on this sub ject to ok place at the last EWM
meeting in Madrid in September
The workshop a small scale two day meeting will consist of four sessions two each day
each session having at most two talks Each session will include ample time for discussion by
the participants of the various manifestations of renormalisation
Abstracts of the talks should b e available at the workshop and more detailed pro ceedings
of the workshop including the discussions and additional comments by the sp eakers will b e
available afterwards
Lo cation Institut Henri Poincar e Paris France where femmes et math ematiques has its
o ce Accomo dations will b e organised dep ending on the number of participants either in
private homes or in a student hall
Programme
Friday session
morning session Renormalisation in dynamical systems I
sp eakers Laura Tedeschini Lalli Rome Betta Scopp ola Rome
afternoon session Renormalisation in dynamical systems I I
sp eaker N uriaFagella Barcelona
Saturday session
morning session Renormalisation in statistical physics and quantum eld theory
sp eaker Annick Lesne Paris
afternoon session Extension of the notion of renormalisation to in nite dimensional ge
ometry
sp eaker Sylvie Paycha Clermont Ferrand
Other speakers wil l be announced later on
Scienti c committee
Bro dil Branner Lyngby Denmark
Laura Tedeschini Lalli Rome Italy
Sylvie Paycha Clermont Ferrand France
Organising committee
Colette Guillop e Cr eteil chairperson of femmes et math ematiques
Sylvie Paycha Clermont Ferrand convenor for EWM
For further information please contact Sylvie Paycha at
paycha ucfma univ bpclermont fr tel
Fe mail E mail discussion
Rep ort by Sylvie Paycha and Capi Corrales
Based on notes taken by Ute B urger and Antje Petersen
The use of e mail as a mean of communication b etween mathematicians is now wide spread
around the world As a new way of exchanging information and ideas it can have some impact
on how the mathematical community functions We felt it was time for us to talk ab out the
assets and draw backs of such a mean of communication and more sp eci cally for women
mathematicians
Time for discussion was short and at this stage we can only formulate a few questions
around this topic which we hop e will lead to further discussions on and use of e fe mail
What is the status of the language used in an e mail message
It seems to b e neither sp oken nor written but something in b etween This in b etween
status has the draw back that it can lead to very abrupt messages like No as an
answer to a question if the p erson who sends the message do es not feel the need to
write a few words of introduction as one otherwise would in written form In this sense
it can sometimes b e felt as an unpleasant although fast mean of communication On
the other hand p eople sometimes feel free to suggest an idea in an e mail message that
is not yet rip e enough to b e formulated in usual written form but which is still worth
sending by e mail This leads to the second question
Do es e mail mo dify the way mathematics are done
New ideas can b e exchanged rapidly by e mail even if they are still in the pro cess of
b eing elab orated They can therefore b e elab orated in collab oration a deep idea can
come from many ideas even if sup er cial suggested by many p eople The notion of
authorship which was and still is so imp ortant for recognition in the mathematical
community for one idea might gradually lo ose its meaning b ecause of this sup erp osi
tion of ideas coming from di erent p eople But what if one is not in a discussion or
information network and cannot participate in this common elab oration of ideas
What ab out exclusion phenomena via e mail
An e mail discussion or information net grows fast but it can easily happ en that one
do es not b ene t from it by lack of information or b ecause one do es not have access to
computers on which one can enter these nets Since a lot of information mathemati
cians use nowadays transits through these means of communication there is a clear
segregation b etween mathematicians who use these nets and the others who for some
reason or other are excluded from them
What ab out women mathematicians and e mail
Which among the advantages draw backs mentioned ab ove of this new mean of commu
nication b ene t to disadvantage women mathematicians Do women mathematicians
have in actual facts less access to it do they use it less and why Some concrete
suggestions came up during this informal discussion such as
to organise a technical workshop during the next general EWM meeting to make
women more familiar with the p ossibilitie s of the internet
that Riitta Ulmanen could write a manual ab out how to make the b est use of e mail within EWM
Mathematical studies at Europ ean universities
Magdalena Jaroszewska
Adam Mickiewsicz University Poland
Data enclosed in this pap er have come partly from information b o oklets and partly from
direct communication with teachers and students of a number of Europ ean universities
The Europ ean universities b egan to develop in the th century the rst ones stem
ming from monastery schools In spite of numerous transformations they still preserve their
traditional medieval organization and status including divisions into faculties structure of
authorities autonomous management Some of the oldest universities established in th
th centuries are Bologna Paris Oxford Cambridge Parma Salamanca Valencia Padoa
Naples Rome Prague Cracow Vienna Heidelb erg K oln and Barcelona The Sorb onne in
Paris provided the university organizational pattern for other universities
In our days there are numerous new scienti c disciplines under development which in
creases demand for highly quali ed sp ecialists The university graduate should have broad
general education and also b e well prepared for highly sp ecialized professional career How
to balance the two aims i e broad horizons and go o d professional training in education of
students This goal p oses still di cult problems for p eople and institutions resp onsible for
education
Quite recently extensive p olitical and so cial changes have b een taking place in Europ e
Students of to day may choose place for studies practically all over Europ e Also the univer
sities are changing in many ways resp onding to market rules and economic pressures of the
new Europ e The academic world tends to show a more elastic approach b oth in admission
of students and in programs o ering the students the chance to choose more freely their own
line of studies
The system of university studies di ers quite distinctly in di erent countries and even
b etween universities in the same country
Most often the students start the studies at the age of years The studies are
p erformed at most of the universities in three stages corresp onding to the degrees of Bachelor
of Science Master of Science and Ph D resp ectively The duration corresp onding to these
stages di er quite signi cantly
The rst stage of mathematical studies lasts two to four years during which the student
acquires basic education The principal sub ject mathematics is frequently accompanied by
another one like physics informatics economy psychology
The second stage lasts one to three years The studies are more sp ecialized and the
individual work of the students is emphasized Most students end their formal education
after this stage
The third stage Ph D studies is undertaken in general by few students Each of the
students remain under care of a promotor of his her choice and is given a sub ject for scienti c
work usually within a sp ecialized branch As a rule the students have to pass some exams
at this stage but their principal aim is to present a pap er containing some new own results
Let us lo ok at some of the countries
BELGIUM Mathematical studies take years and the rst diploma of the Candidature
or Baccalaureate is granted after the rst two years The diploma carries no practical
signi cance After years of studies the student receives the Licence Diploma which is
accepted at the lab or market
DENMARK Mathematical studies are often combined with some other sub ject like in
formatics physics statistics After years the students receives B Sci degree and after
another two years the degree of Candidatura Scientiarum corresp onding to Master degree
A Ph D degree consists of a Master degree followed by a year research training program In
Denmark there are several universities with alternative organisation of studies At Roskilde
University for example all degree courses commence with a two year basic studies program
The basic studies program is a broad introduction to the humanities the so cial sciences or
the natural sciences
ENGLAND The rst three years long stage provides the student with the degree of the
Bachelor of Science The second stage of years grants the student the title of a Master
of Science The third stage yields the title of a Philosophy Do ctor Ph D and in conjunction
with the earlier stages completes the years studies During the rst three years frequently
only one branch of science is taught while the remaining sub jects are treated only marginally
Some universities in particular the newer ones o er a variety of Single Honours Joint Hon
ours Combined Honours For example at the University of Birmingham the sp ectrum of
years long studies includes among other the following types Single Honours in mathe
matics pure mathematics applied mathematics mathematics and statistics statistics Joint
Honours in mathematics and computer science mathematics and psychology mathematics
and sp ort science Combined Honours in mathematics and ancient history mathemetics
and French studies mathematics and music etc
FRANCE The rst stage of studies takes years and provides the student with the Diplome
d Etudes Universitaires Generales In the course of the second two years long stage La Licence
is granted after the third year and La Maitrise after the fourth year of studies Ph D title
can b e obtained after the sub ject stage which lasts to years
FINLAND After years the rst diploma Filoso an Kandidaatti and after the next
years Filoso an Tohtori are awarded
GERMANY The studies last to years they lack the rst stage and yield no title
which would corresp ond to the B Sci title After years of studies one can get the title
Diplommathematiker in The curriculum of studies may include two branches of science
the principal one e g mathematics and the accessory one e g chemistry
HOLLAND Similarly as in Germany no title is given which would corresp ond to B Sci
The student is given the title of Do ctorandus an equivalent of M Sci after years of studies
and after the subsequent years obtains Ph D title
ITALY All university studies take usually years After years the student receives the
title of Laurea Dottore which corresp onds more or less to the B Sci degree The next stage
lasts to subsequent years and yields the title of Dottorato di Ricerche Ph D
NORWAY The title of Candidates Magistrates can b e reached after years of studies
Candidates Scientiarum after further years and Do ctor Scientiarum after ab out more years
POLAND Until recently the uniform studies lasted years Now Polish universities o er
to years studies yielding the title of Licentiate with the p ossibility of prolonging them
by another two or three years and obtaining the title of Magister The curriculum of the
years long studies frequently contains the list of sub jects required to obtain in passing the
title of Licentiate After more years of studies the title of do ctor can b e reached
PORTUGAL Four years of studies lead to the Licenciatura After additional years of
studies the student may obtain the title of Mestre and after another to years a Ph D degree
SPAIN The title Licenciado can b e obtained after years and Do ctorado after more years
SWEDEN The rst years of studies lead to the title of Filoso e Kandidat After
more years of studies the degree of Magister and after subsequent to years the Ph D
degree can b e reached
SWITZERLAND At some universities the title Diplomierte Mathematiker in can b e
reached after to years of studies After more years the title Do ctor Philosophiae can
b e received
By the years of studies we mean academic years An academic year for example includes
weeks Poland weeks Sweden or weeks Holland organised in either two semesters
or three trimesters In most countries the program corresp onding to the M Sci degree
includes to thousands hours of classes lectures lab oratory exercises and seminars
Some universities preserves traditional mo del of studies in which the student is confronted
with a curriculum distributing all sub jects to individual years with sp eci ed terms at which
given credits should b e obtained or exams passed In many countries the p oint system of
studies is b eing introduced To b e graduated the student has to collect an adequate number
of credit p oints for obligatory sub jects as well as for optional sub jects For example at the
University of Amsterdam UVA all parts of the year program in M Sci studies form
mo dules of the same size For the course load of a given mo dule the student receives credit
p oints The course load is measured in hours p oint week of studies days x hours
of work h of work including ab out h of classes h of individual work One year
of studies trimesters x weeks of studies p oints Then years of studies
p oints Each sub ject can provide a de ned number of p oints At the rst year of studies
the so called prop edeutic year all sub jects are obligatory At the year the so called
do ctoraal phase some sub jects in individual sp ecialities are obligatory and some are elective
The studies are highly individual the students conciously shap e theirs curriculum of studies
Examination systems di er very strongly Frequently the students pass exams after n
ishing each course or during the year semester at which lectures on the sub ject were given
In some countries the examination is held after two or more years of studies In most of
Europ ean countries the examination system oscillates b etween these two extremes Let us
screen the patterns at some of the countries resp universities
DENMARK Each sub ject culminates in the form of an exam at the end of semester or
academic year Interestingly the examining b o dy includes the lecturer but also an additional
professor
SCOTLAND Universities grant degrees with evaluation of the relevant quali cations by external examiners
ENGLAND Oxford Students are examined at the end of the rst and the third year of
studies Within a week the students pass written exams each lasting hours The exams
test uency in the material of obligatory sub jects b oth in the basic and in highly sp ecialized
branches of science
GERMANY The students have main exams Vordiplom Prufung at the end the second
year of studies and Diplom Prufung at the end of the fth year of studies
PORTUGAL Knowledge of most basic disciplines is tested by a single written test exam
at the end of each semester If the student fails at the exam he she can correct the result
passing the oral exam
According to the available information the rst stage of mathematical studies includes
basic sub jects common to ma jority of universities while curricula of mathematical studies
di er a lot b etween universities at the second stage of the studies
Almost all universities o er the following program
Mathematical analysis sequences limits continuity derivatives inde nite and Rie
mann integrals of functions of one and several variables curvilinear and surface inte
grals ordinary di erential equations
Linear algebra and algebra of basic algebraic structures
Euclidean and analytic geometry
Principles of informatics numerical analysis probability and statistics
As evident from the ab ove Europ ean universities di er from each other in mathematical
studies by the system of studies the ways in which sub jects are taught and the exams are
conducted they grant distinct titles and grades Only the p ortion of basic mathematical
knowledge is common to ma jority of studies curricula at the preliminary years of studies
The following questions arise
Is it purp oseful and p ossible to harmonize or standardize curricula of the rst years
of studies and to do the same with the granted degrees in the contemporary Europ e
with op en b orders and p ossibilities of free choice and change of place of studies
Is it p ossible to de ne a certain Europ ean standard in the range
These problems have b een discussed at the Round Table of Europ ean Congress of Math
ematics
Suggestions The rst step towards uni cation is comprehensive information It would
certainly b e very useful if mathematical institutes faculties could publish information b o ok
lets in English The b o oklets should contain curriculum of mathematical studies This would
greatly facilitate work and decisions of b oth students in mathematics and the p ersons in the
institutes faculties resp onsible for education and for student transfers
References
Europ ean International Education Yearbo ok Nexus Business Communication LTD
The annual to study abroad
Mathematics and Statistics School of Mathematics and Statistics The University of Birm
ingham
H J Munkholm J Netuka V Soucek Degrees harmonization and student exchange pro
grams preliminary rep ort for A Round Table at the Europ ean Congress of Mathematics
Paris July
Notiziaro Facolta di Science Matematiche Fisich e Naturali Universite degli studi Perugia
Programme des cours Facult edes Sciences Universit ede Mons Hainaut
Roskilde University Denmark Information O ce
Stokholm Universitet Information ab out students
Faculteit der Wiskunde en Informatica Universitet van Amsterdam Studiegids
Universit ede Paris Sorbonne Paris VI I Livret de l etudiant
Family versus Career
The activity around the discussion on Family versus Career was organized by science
historian Eulalia P erezSede no from Universidad Complutense de Madrid In the spring of
she formulated a questionnaire which was distributed by EWM The pap er which follows
is her record of the answers which she received
Eulalia presented the results of her studies at the EWM meeting following which a lively
discussion to ok place Participants emphasized that only a tiny amount of material was
gathered that no statistical conclusions could b e drawn and that those who answered mainly
represented survivors in the mathematical community
We would like to make quite clear that opinions and conclusions expressed in this pap er
do not necessarily agree with those of EWM as an organization or those of its members
Family versus Career in Women Mathematicians
Eulalia P erezSede no
Universidad Complutense de Madrid Spain
For a long time it was claimed that the fact that girls chose less scienti c sub jects than
b oys was re ected in the number of female scientists and as the number of women studying
sciences is smaller smaller to o is the number of women reaching professional success in
that area However to day we know that in some countries the global number of women
in university is greater than that of men but there are fewer women studying physics or
mathematics the distribution of women in the di erent scienti c sp ecialities is unequal
When women surpass men as well as when this is not the case neither the quantity of
professionals corresp onds to the quantity of women prepared for this nor is the distribution
in the di erent knowledge areas equal and there is no equality in the p ositions that men and
women o ccupy Furthermore a few so ciometrical studies seem to show that the scienti c
careers developed by men and women are di erent Cole and Zuckerman et al
Why this is the case is something so ciologists and psychologists have attempted to explain
in many various ways
Many scholars tend to blame partnership and maternity for the fact that women do not
reach in great quantities the working market in general and do not access the highest
p ositions in all the scienti c areas The underlying argument pro ceeds more or less in
the following way the pursuit of a scienti c career is a full time job women carry the
resp onsibiliti es of the household they take care of the house children sick or old p eople
etc Performing the housewife functions takes a lot of time For that reason women either
decide for the family and thus there are less women than male in these professions or their
work is resented in terms of their scienti c p erformance and e ciency That would mean for
example that single women would have to progress just as single males In the same way
if it is said that marriage and children reduce the pro ductivity of women we would have to
examine whether married women with children pro duce less than men in the same situation
There would have to b e di erences b etween married and unmarried women among women
with children and those without and in other words men and women with a similar quantity
and quality of publications would have to have equal status In fact few empirical studies
fo cus on these problems The few existing studies deal with a few North American women
b elonging to various scienti c disciplines and they suggest the contrary Zuckerman et
al However it is continuously asserted that the family is an obstacle to reaching full
equality in science
The present pap er presents the results of a study accomplished through the asso ciation
Europ ean Women in Mathematics A questionnaire was sent out through its network It
was intended to analyse educational professional and economic status of the resp ondents and
their households of origin the development of their professional career year of completion
This research has b een partially supp orted by Spanish DGICYT pro ject number PB C and
by the Researchers Temporary Mobility Program funded by the Spanish Government
That is the case in Spain see P erezSede no
See for instance Cole Rossiter and P erezSede no
On territorial and hierarchical discriminati on s see Rossiter and P erezSede no
See Zuckerman Cole and Bruer and Cole The study made by Jaiswal is more extensive
so it is based in women and man But it do es not distinguis h b etween di erent sciences and it is made in India a country with a di erent structure
of studies rst job Ph D and publications if any the time they devote to domestic tasks
that is supp osedly subtracted from study time and research and it was intended to examine
to o if di erences in the previous factors result in di erent levels of p erformance but also it
claimed to grasp how these women p erceive other women mathematicians how they p erceive
themselves what qualities they consider women mathematicians should have in order to
p erform their work if they feel discriminated against etc In short it was intended to throw
light on some asp ects of life of a handful of female mathematicians so we could understand
as much as p ossible ab out their situation and reality with no vague sp eculations
We claim to have obtained informative and qualitative data We must take into account the
fact that the sample is extremely small Accordingly anytime p ercentages are o ered they
must b e understo o d as merely informative of the situation opinion etc of the resp ondents
At no time is it intended to make any extrap olation nor generalization ab out the joint total
of all the Europ ean women mathematicians And moreover it would b e ridiculous to do so
Furthermore many of the analysed asp ects are very di cult to quantify just as what happ ens
in the case of sex based discrimination
I also have to indicate that from the p oint of view of so cial science it is very imp ortant for
the results that the interviewee do es not know what working hypotheses guide the researcher
or what he she is lo oking for Otherwise their answers can b e guided so they can think
of answers that they would not have considered or they can conceal other ones For that
reason I attempted to hide the pursued ob jective In any case the particular nature of the
asso ciation and the questionnaire didn t leave a very wide margin to sp eculation
The questionnaire had b oth closed and op en ended questions and it was very long it was
claimed it was directed to women from di erent countries at di erent ages with di erent
professional economical and family situations and even taking this into account many
questions had to b e obliterated
We do not know the exact number of women the questionnaire reached but we estimate
that it was transmitted to some two hundred women answers were received The re
sp ondents b elong to almost all the Europ ean countries In this sense there are only two
notable things the absence of women mathematicians from the eastern countries and from
Ireland Switzerland Austria and Greece furthermore nine resp ondents live in the U S And
in connection with mobility in work only eight of the resp ondents work in a country which
is di erent to their country of origin
The o ccupations of these women are in the academic world as I assumed originally and
students answered to o Also there is one technical writer and one housewife They describ e
their work in a variety of ways sometimes they are very sp eci c and sometimes they are
very general and vague The most prevalent activity is the teaching of mathematics in
compulsory six to sixteen years old or upp er secondary schools sixteen to eighteen for
undergraduates graduates or Ph D students this is closely followed by research But only
resp ondents say they are p erforming administrative tasks All of them b elong to several
scienti c so cieties except the average is to b elong to so cieties or working groups
These women were aged from to years old and they were distributed in the following
way women were aged from to years old Group The so called Group was
formed by women from to years old The greatest number of women Group
were from to years old women that is to say and only women were
But some surprising reactions were pro duced Many resp ondents said their partners were mathematicians
to o but a few said that they didn t b elieve their partners were interested in answering the questionaire they were asked to pas the questionaire to their partners in the case were they were mathematicians to o
from to years old In an academic context and generally in the lab ouring world the
age is extremely relevant With rare exceptions there is a minimum age at which to b egin
university studies to b egin to work and a maximum age to retire The age is also a frame
of reference with resp ect to maturity and exp erience in work Generally sp eaking a greater
service time gives as a result a promotion a b etter p osition and b etter salaries This remains
fully con rmed in the case of our resp ondents since the prop ortion of p ermanent p ositions
diminishes as age decreases But it must b e emphasized that the global p ercentage of women
that have obtained p ermanent or tenured p ositions in their workplace is very high
The so cio economic background of the resp ondents results is relevant in order to under
stand their place in the so cial structure It also must b e taken into account that in traditional
so cieties individual s form a part of the family and this shap es them Because of this it do es
not app ear strange to supp ose that the fact that women b ecome a member of professions
dominated by men and mathematics is one of them is in uenced by the sexual role the
so cialization and the upbringing of girls in their families The education and profession of
the parents as well as the family income facilitates or hinders the education and profession
of women Ab ove all the parents educational level is a very imp ortant factor in order to
provide a b etter education and to select a suitable o ccupation
The so cio economic status of the original households of all these women was quite uniform
it was middle class though said they pro ceeded from a lower middle class home and
pro ceeded from a higher middle class household
The educational status of fathers in Group is quite uniform to o all of them p ossess a
high educational level that moreover is sup erior to the mothers one of fathers had
university degrees and a third of them had a Ph D concerning the mothers only had
a university degree and another had attended high school The fathers professions
are varied but were university teachers also there were attorneys engineers etc
Curiously in this group only two mothers did not work and the professions of the rest of
the mothers were varied although a third of them are teachers Fathers from Group had
a high educational status only Group fathers did not p ossess university studies half of
the mothers did not have university degrees or similar education In relation to fathers
professions the academic ones were numerous and the rest of the fathers were business men
farmers or architects Four mothers out of the total did not work the professions of
the others were architects psychoanalysts business or mathematics teachers In Group
just three fathers didn t have university education and seven fathers had a Ph D engineers
architects and teachers with scienti c training and esp ecially mathematical knowledge were
numerous Regarding the mothers were housewives the rest were distributed among
di erent professions but teaching was the most common Finally parents in Group
were the most assorted economically educationally and professionally Although just
women make up this group all the economic situations of the questionnaire app ear and in
connection with the educational status three fathers have university education and there is
just one mathematics teacher the rest of the professions varies The educational status
of the mothers in this group is slightly sup erior to the fathers one obtained university
degrees and two high school degrees There are three teachers two administrative o cers a
secretary a banker and only one housewife
Marriage or partnership and the children if any and the time these women devote to
the household are very imp ortant in this research as we can learn how their family situation
Men who answered the questionaire are not included here
a ects or has a ected their professional p erformance or e ciency In order to measure profes
sional p erformance four factors have b een used the year of completion of the undergraduate
degree the year they obtained their Ph D degree the year they b egan to work remunerated
work and the number of publications
Marriage or partnership provides a new status to individuals It usually confers new
roles and p ositions on one hand so ciety exp ects as a minimum that households as well
as the upbringing of children and care of older family members if any shall b e the domain
of women additionally the p osition the household o ccupies in so ciety tends to sum up the
status of b oth members of the couple All women in Group have or have had a partner
Generally they obtained a partner late supp osedly after having established their professional
career in a certain way Just two of them obtained a partner b efore the age of and
just one was a student All the partners of these women have obtained a university degree
in science for the most part mathematics or related sub jects we do not know ab out two
of them since this information was not given to us In Group all women have or have
had partners Four women had their rst partner when they were less than years old
Mathematicians or p ersons with strong mathematical training such as engineers economists
and so on are numerous among the partners of this group In any case all of them have a
university education except one where information was not given Group was comp osed
of women and just ve of them have not or have not had partners Nine women
obtained partners b efore they were years old All the women with partner except two had
formal relationships with scienti cally trained p eople mathematicians engineers computer
scientists etc Finally in Group only four women have a partner of them obtained
partner b efore the age of the other two women later Regarding the educational or
professional status three partners in Group have scienti c training and is a travel agent
Previous data show almost total congruency in educational and professional status among
the couples Such a fact supp orts the conjecture that in our current so ciety such congruency
is a very imp ortant variable in the vital situation of an individual That congruency a ects the
lifestyle the couples b ehaviour and adjustment it also provides stability and psychological
reward Nonetheless it is worth emphasizing that some women state that their partner has
b een one of the most principal obstacles they have found in the development of their career
We can examine to what extent family situation a ects the professional and scienti c
p erformance or e ciency in these women In Group women nished their graduate studies
from to years old All these women b egan to work from to years except one
b egan when she was thirty years old b efore she was married and had children and the other
when she was thirty seven years old years after her marriage and after having her last
child Just two women did not have children and four had the rst b efore they were thirty
years old That is to say in this group of women with children were older mothers
that is when they were thirty years old or more These children went to school when they
were six or seven years old just four children b egan school b efore
However neither partnership nor maternity have a ected the p erformance or e ciency of
these women against what is many times claimed Just one woman resigned from a tenured
p osition to b e devoted to her children but the rest continued working after marriage and
I am aware of problems implied by using these factors For instance some resp ondents have not nished
their graduate studies yet others sp ent so much time teaching that they can scarcely publish And concerning
motivation I am aware to o that some questions would have to b e asked
I am referring to heterosexual partnership the homosexual one varies as it do es so cial p ermissiveness
We must take into account that in most countries women lo ose their maiden name and they go on to b e called by their husbands names that is to say in a sense their previous identity disapp ears
children Seven obtained their Ph D after having their rst child and when at least one
child was not going to school yet Just in one case do es the rst childbirth and do ctorate
coincide In relation to women with no children just two it do es not seem they have had
b etter p erformance than the other ones Even though one of them obtained her Ph D and
published her rst pap er by the time she was years old the other obtained her Ph D
at twenty nine and published her rst pap er at thirty seven this is not meaningful since
there are Ph D women at twenty four twenty eight and twenty nine when they already
had children Concerning publications of women with children and the subsequent pace of
publication they do not seem to b e a ected negatively by marriage or children only one of
them published for the rst time when her youngest child was years old
In Group all women nished their graduate studies b etween the ages and just
one of them interrupted her university studies b ecause of maternity and she nished some
years later They started to work at a slightly di erent age just one at seven from
to and two when they were more than years old The seven women with a Ph D
obtained the do ctorate at similar ages from to Although all had children much after
their do ctorate their motherho o d did not in uence their scienti c p erformance all of them
continued publishin g at the same pace and even to a greater pace after having children
Just one published at forty two for the rst time much after her youngest child had b egun
to go to school There do es not seem to b e a meaningful di erence b etween those women
and those who do not have children three in relation to their rst publications one of them
started to publish at one at and the third at
The tra jectory in the university of women in Group is standard to o Most of them
started to work when they were b etween and years old but one b egan at and from
to years and two of these do not have children Thirteen women did not have children
and twelve of them obtained their Ph D From nine women with children eight have a Ph D
and six of these mothers obtained their Ph D after having had their rst baby and when at
least one child was not going to school yet The quantity and pace of publications do es not
vary b ecause of marriage or maternity in most cases as the average of publications of women
with children is sup erior to that of women without children Of course this must b e taken
with caution since some resp ondents have not answered this part of the questionnaire
or they have answered partially The analysis of the situation of the last group is not very
informative as no one has published just two have obtained a Ph D and just one has a son
The dedication to the tasks of the household varies among the p eople p olled from
hours p er week that some women reveal until hours that other ones assert There is no
di erence b etween women living alone and women with a partner or family hours as a
weekly average Most interesting is how much time each member of the couple devotes to
domestic work in Group ve women say they sp end the same amount of time as their
partners but three claim they sp end more time than them on these tasks In Group just
one sp ends the same amount of time as her partner another one sp ends less time than her
partner and in the rest of the cases they sp end more time on household tasks than their
partners In Group one woman asserts her partner sp ends more time on housework than
she do es but the quantity of couples in which women do more household work than their
partners is equal to couples in which the work is distributed in a similar way Finally in
Group all women answered they sp ent more time on household work than their partners
All these women work in a masculine context that is to say in their workplace more men
than women work and those male colleagues exercise control over greater number of p ersons
usually students Mainly they feel they are equally treated to their male colleagues in terms
of salary and promotion and resp onsibiliti es But ve women have a higher
quali cation needed in order to p erform their work and they could o ccupy sup erior p ositions
that is to say they are underemployed And the number of women that feel undervalued is
very similar to those which feel themselves to b e equally valued In relation to the rest of the
women of the resp ondents think there are more women undervalued than men and
of the resp ondents think that there are more overvalued men than women
We can classify the supp ort and the opp ortunities they have found in their career in
two types p ersonal intellectual and economic In the rst kind it should b e emphasized
that twenty resp ondents admitted having had a male counsellor a female counsellor and
one resp ondent had b oth male and female counsellors And some of the answers were Meet
interesting p eople enjoy mathematics invitation to collab orate on a problem supp ort
from colleagues and husband general interest ability to talk to others meeting p eople
who have encouraged me many excellent programs to develop teaching metho ds In
relation to the second kind scholarships or grants are the most cited items The fact that
their advisor help ed them to nd their rst job opp ortunity to work as an assistant
sabbatical p erio ds etc are quoted to o It must b e emphasized that only two women refer
to programs for promoting women
The answers concerning the obstacles they met in their careers are also varied but family is
the most quoted When they say family they mean balancing b etween family and profession
or to have to decide b etween family and career And obstacles directly related to their
sex are numerous Sexual and sex based harassment most of the time the fact I am a
woman some male colleagues resent my e orts to hire and retain women in the faculty
bad rumours men were preferred in p ositions family problems partner s comp etitive
relation to my career women who discourage other women having to justify how much
I contributed in joint publications Very rare opp ortunities for promotion and blatant
discrimination can b e understo o d as sex related to o However there are many more lack
of p eople to talk to for research loneliness isolation in my department which it is not
delib erated to you by the men one adds incomp etent advisors or lack of strong counsellors
lack of supp ort from pure mathematics colleagues in the department academic attitude
towards teaching versus research teaching lo oked down up on time not available p ositions
p olitical corruption favoritism and incomp etent decisions at departmental level and so on
These few pages express a brief summary of the results of the questionnaire I am aware
of many things that are obliterated but other asp ects have come to the light Some of them
can b e ob ject of re ection by the asso ciation itself But many can b e a matter of general
thinking
So cial scientists don t agree on a unique de nition of discrimination The variance in their
de nitions reveals the di erent p oints of view in discriminatory b ehaviour scholars Cole
And on sex discrimination the di erences are great The traditional concept of
women the so cialization pro cess of the girls and the so cial accepted p osition of women in
family life seem indicate that women p erformance would have to go in a certain direction But
the results of the questionnaire show that pregnancies upbringing babies publications Ph D
p ositions and so on are interwoven And the result of the questionnaire says to o that there are
not meaningful di erences with resp ect to the age at which single and married women women
Just one resp ondent said that there are more overvalued women
Many of these obstacles could b e adduced by other academicians or scientists
For instance there are few young female mathematicians These can b e b ecause the foundational charac
teristics of the asso ciation But it can b e b ecause the educational status of the household origin
with children and without obtain their Ph D and their p ositions or when they publish
Usually these women have to adjust their career and their family and nevertheless they
obtain their Ph D p ositions and publish So we can conclude that the unequal participation
of women in mathematics has to pro ceed from cultural and so cial structured norms and values
not b ecause they decide for family nor by their intellectual and academic achievements I
know that many of us presumed this assertion But now we have these data on our hands
And p erhaps this pap er can illustrate how science works as a so cial system and how science
rewards women s participation in a male dominated world Perhaps we can learn from it that
values and interests form part of science to o
Cole J R Fair science Women in the scienti c community New York The Free
Press
Jaiswal R P Professional Status of Women Jaipur New Delhi Rawat Publications
Oakes J Lost talent the underparticipation of women minorities and disabled p ersons
in science Santa Monica Ca Rand
Ogilvie M B Women in Science Antiquity through the Nineteenth Century Cam
bridge Mass Londres The M I T Press
Osen L Women in Mathematics Cambridge MA MIT Press
P erez Sede no E Scienti c Academic Careers of Women in Spain History and
Facts in press
Ramaley J A ed Cover discrimination and women in the sciences AAAS selected
symp osium Westview Press
Rossiter M Sexual Segregation in the Sciences Some Data and a Mo del Signs
vol n Women Scientists in America Struggles and Strategies to The
John Hopkins University Press
Zuckerman H Cole J and Bruer J T The outer circle New Haven Yale Univ
Press
Eulalia P erezSede no
Universidad Complutense Madrid
Spain eulalia eucmax sim ucm es
COMMITTEES COORDINATORS
AND PARTICIPANTS
Committee Members
STANDING COMMITTEE
Sylvie Paycha convenor e mail paycha ucfma univ bpclermont fr
Polyna Agranovich e mail agranovich ilt kharkov ua
Valentina Barucci e mail barucci gpxrme sci uniroma it
Bo dil Branner e mail branner mat dtu dk
Capi Corrales deputy convenor e mail capi sunal mat ucm es
Marie Demlova e mail demlova math feld cvut cz
Laura Fainsilber e mail laura math univ fcomte fr
Emilia Mezzetti e mail mezzette univ trieste it
Rosa Maria Mir o Roig e mail miro cerber ub es
Marjatta N a at anen treasurer e mail mnaatanen cc helsinki fi
Ragni Piene e mail ragnip math uio no
Caroline Series e mail cms maths warwick ac uk
Names of women who might b e nominated for prizes should b e sent to
Magdalena Jaroszewska e mail magjar math amu edu pl
Link with Asso ciation for Women in Mathematics AWM
Caroline Series e mail cms maths warwick ac uk
Link with Europ ean Mathematical So ciety EMS
Bo dil Branner e mail branner mat dtu dk
Statistics ab out women mathematicians on di erent Europ ean countries
Kari Hag e mail kari imf unit no
e mail EWM network
to subscrib e contact Sarah Rees e mail sarah rees newcastle ac uk
Co ordinators
Technicka International Co ordinators
Praha Czech
East
e mail demlova math feld cvut cz
Inna Berezowskaya
Center for Forest Ecology RAS
Denmark
Arh Vlasova st
Bo dil Branner
Moscow Russia
Department of Mathematics
Building
Marie Demlova
The Technical University of Denmark
Department of Mathematics
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Faculty of Electrical Engineering
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Czech Technical University
Technicka
Praha Czech
Estonia
e mail demlova math feld cvut cz
Helle Hein
Institute of Applied Mathematics
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Marketa Novak
Liivi Str
Department of Computer Sciences
EE Tartu Estonia
Chalmers University of Technology
e mail helle h vask ut ee
S G oteborg Sweden
e mail novak cs chalmers se
Finland
West
Marja Kankaanrinta
Capi Corrales
Department of Mathematics
Departemento de Algebra
P O Box Hallituskatu
Facultad de Matematicas
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Universidad Complutense de Madrid
e mail mkankaanrint hylk helsinki
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France
Colette Guillop e
Regional Co ordinators
Universite Paris Sud
Bat
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Isab el Pays
e mail guillop e math u psud fr
Universite de Mons Hairaut
Section de Mathematiques
Germany
Avenue Maistriau
Christine Bessenro dt
B Mons Belgium
Institut fuer Algebra und Geometrie
e mail spays bmsuem binet
Fakult atf urMathematik
Otto von Guericke Universit at Czech
PSF Marie Demlova
D Magdeburg Germany Department of Mathematics
e mail Christine Bessenro dt Mathematik Faculty of Electrical Engineering
Uni Magdeburg DE Czech Technical University
Norway Great Britain
Ragni Piene Amanda Chetwynd
Department of Mathematics Mathematics Department
University of Oslo Lancaster University U K
P O Box Blindern Lancaster LA YF Great Britain
N Oslo Norway e mail maa central lancaster ac uk
e mail ragnip math uio no
Greece
Vassiliki Farmaki
Poland
Department of Mathematics
Magdalena Jaroszewska
University of Athens
Faculty of Mathematics
Panepistimiopolis Athens Greece
and Computer Science
e mail sma grathun bitnet
Adam Mickiewicz University
Matejki
Maria Leftaki
Poznan Poland
Department of Mathematics
e mail mag jar math amu edu pl
University of Patras
Patras Greece
Agata Pilitowska
e mail leftaki math upatras gr
Warsaw Technical University
Mathematics Institute
Italy
Plac Politechni
Emilia Mezzetti
Warsaw Poland
Dipartimento di Scienze Matematiche
e mail agapil plwatu bitnet
Universita di Trieste
Piazzale Europa
Portugal
I Trieste Italia
Isab el Lab ouriau
e mail mezzette univ trieste it
Grup o de Matematica Aplicada
FCUP
Latvia
Rua das Taipas
Daina Taimina
Porto Portugal
The University of Latvia
e mail islab our ncc up pt
Rainis Blvd
Riga LV Latvia
Romania
e mail dtaimina lanet lv
Ro dica Tomescu
Str Transilvania Nr
Lithuania
Sect Bucuresti Romania
Nerute Kligiene
e mail tomescu vala elia pub ro
A Gostauto
Vilnius Lithuania
Russia
Institute of Mathematics and Informatics
Inna Berezowskaya
e mail nerute kligiene mji lt
Center for Forest Ecology RAS
Arh Vlasova st
The Netherlands
Moscow Russia
Coby Geijsel
Inna Yemelyanova Fac der Wiskunde en Informatica
Gagarin Ave Universiteit van Amsterdam
Nizhny Novgorod Plantage Muidergracht
GSP Russia TV Amsterdam Netherlands
e mail yemel nnucnit unn ac ru e mail coby fwi uva nl
Spain
Capi Corrales
Departemento de Algebra
Facultad de Matematicas
Universidad Complutense de Madrid
E Madrid Spain
e mail capi sunal math ucm es
Sweden
Marketa Novak
Department of Computer Sciences
Chalmers University of Technology
S G oteborg Sweden
e mail novak cs chalmers se
Switzerland
Francine Meylan
rue du lievre
Case p ostale
CH Geneve Switzerland
e mail francine meylan ima unil ch
Turkey
Betul Tanbay
Department of Mathematics
Bogazici University
Istanbul Turkey
e mail tanbay boun edu tr
Ukraine
Polyna Agranovich
Lenin Avenue
Kharkov
Ukraina e mail agranovich ilt kharkov ua
List of Participants in EWM
Riitta Ulmanen Denmark
Secretary O cce of EWM
Bo dil Branner
Department of Mathematics
Department of Mathematics
PO Box Yliopistonkatu
The Technical University of Denmark
FIN University of Helsinki
Building
email ulmanen sophie helsinki fi
DK Lyngby
email branner mat dtu dk
Pia Willumsen
France
Institut des Hautes Etudes Scienti ques
Laura Fainsilber
F Bure sur Yvette France
Lab oratoire de Math ematiques
email pia mat dtu dk
Universit ede Franche Comt e
route de Gray
England
F Besan con
Marjorie Batchelor
email laura math univ fcomte fr
Department of Pure Mathematics
Flora Koukiou
and Mathematical Statistics
Universit ede Cergy Pontoise
Mill Lane
Department of Theoretical Physics
Cambridge CB SB
Cergy Pontoise
Catherine Hobbs
email koukiou u cergy fr
School of Computing and
mathematical sciences
Tan Lei
Oxford Bro oks University
Mathematics Institute
Gipsy Lane
University of Warwick
Headington Oxford OX PB
Coventry CV AL
email cahobbs brookes ac uk
email tanlei maths warwick ac uk
Alice Rogers
Mireille Martin Deschamps
Department of Mathematis
Departement de Math ematiques
King s College
Ecole Normale sup erieure
Strand London WC R LS
rue d Ulm
email f a rogers kcl ac uk
F Paris Cedex
e mail deschamp dmi ens fr
Finland
Sylvie Paycha
Marjatta N a at anen
Department de Math ematiquesApliqu ees
Department of Mathematics
Universit eBlaise Pascal
PO Box Yliopistonkatu
F Aubiere Cedex
FIN University of Helsinki
e mail paycha ucfma univ bpclermont fr
email mnaatanen cc helsinki fi
Marie Fran coiseCoste Roy Marja Kankaanrinta
Universit ede Rennes Department of Mathematics
Rue de la Rabine PO Box Yliopistonkatu
F Cesson Sevigne FIN University of Helsinki
email costeroy univ rennes fr email mkankaanrint cc helsinki fi
Italy Claire Voisin
Universit ede Paris XI
Valentina Barucci
D epartement de Math ematiques
Dipartimento di Matematica G Casteln
F Orsay
uovo
email voisin diam ens cachan fr
Universit aLa Sapienza
Piazzale Aldo Moro
I Roma
Germany
email barucci gpxrme sci uniroma it
Ute B urger
Marina Bertolini
Sielwall
Dipartimento di Matematica F Enriques
D Bremen
via C Saldini
email a r hatnix statistik
I Milano
uni bremen de
email bertoli vmimat mat unimi it
Marlen Fritzsche
Emilia Mezzetti
Universit atPotsdam
Department of Mathematics
Institut f urMathematik
University of Irieste
PF
Piazzale Europa
D Potsdam
I Trieste
email mfritz rz uni potsdam de
e mail mezzette univ trieste it
Priska Jahnke
Enza Orlandi
Salegrund
Dipartimento di Matematica
D Marburg
Terza Universit a
email radloff mailer uni marburg de
Via Segre
I Roma
Mara D Neusel
email orlandi matrm mat uniroma it
Institut f urAlgebra und Geometrie
Fakult atf urMathematik
Laura Tedeschini Lalli
Otto von Guericke Universit at
Dipartimento di Matematica
D Magdeburg
III Universit adi Roma
email mara neusel mathematik
Via Corrado Segre
uni magdeburg de
I Roma
email tedeschini axcasp caspur it
Antje Petersen
Gr Johannisstra e
Cristina Turrini
D Bremen
Dipartimento di Matematica F Enriques
email antje mathematik uni bremen de
via C Saldini
I Milano
Christina Schmerling
email turrini vmimat mat unimi it
Mauerstra e
D Bremen
email tina mathematik uni bremen de
Norway
Ragni Piene Sandra Hayes
Department of Mathematics Mathematisches Institut
University of Oslo Technische Universit at
P O Box Blindern Arcisstra e
N Oslo D M unchen
email ragnip math uio no email hayes mathematik tu muenchen de
Spain Poland
Mariemi Alonso Magdalena Jaroszewska
Departamento de Algebra Faculty of Mathematics and
Facultad de matem aticas Computer Science
Universidad Complutense de Madrid Adam Mickiewsicz University
E Madrid Matejki
email alonso sunal mat ucm es Poznan
email magjar math amu edu pl
Capi Corrales Ro drig a nez
Portugal
Departamento de Algebra
Facultad de Matem aticas
Isab el Lab ouriau
Universidad Complutense de Madrid
Grup o de Matem aticaAplicada FCUP
E Madrid
Rua das Taipas
email capi sunal mat ucm es
Porto
email islabour ncc up pt
N uriaFagella
Margarida Mendes Lop es
Centre de Recerca Matem atica
Depto de Matem atica
Universidad Aut onomade Barcelona
Facultade de Ciencias de Lisb oa
Apartat
R Ernesto de Vasconcelos
E Bellaterra Barcelona
Lisb oa
email nuria manwe mat uab es
e mail mmlopes ptmat lmc fc ul pt
Raquel Mallavibarrena
Romania
Departamento de Algebra
Daniela Florenta Ijacu
Facultad de Matem aticas
Bd Cuza Voda apt Sect
Universidad Complutense de Madrid
Bucuresti
E Madrid
Ro dica Tomescu
Transilvaniei
Rosa Mar aMir o Roig
Bucarest
Facultad de Matem atiques
email tomescu vala elia pub ro
Departament d algebray geometria
Universitat de Barcelona
Russia
Gran Via de les Corts Catalanes
Valentina Goloub eva
E Barcelona
A Usiyevich str
e mail miro cerb er ub es
Department of Mathematics
All Russian Institut of Scienti c and Tech
Isab el Paniagua Garc a Calder on
nical Information VINITI
Departamento de matem aticaaplicada
Moscow
Facultad de matem aticas
Vera Pagurova
Universidad Complutense de Madrid
Department of Computational Mathemat
E Madrid
ics and Cyb ernetics
e mail Paniagua mat ucm es
Moscow University
Vorobievy Gory
Mar aVictoria Ro dr guez Ur a
Moscow
Universidad de Oviedo
Elvira Perekhotseva
Departamento de matem aticas
Hydrometeorological centre of Russia
Facultad de Econ omicas
email kohan glas ape org Oviedo
Sweden
Charlotte Lundvall
c o Falck
Brahev agen
S Sto cksund
email f clu nada kth se
Marketa Novak
Departent of Mathematics
Chalmers University of Technology
S G oteborg
email novak cs chalmers se
Switzerland
Karin Baur
Imfeldstrasse
CH Z urich
email mikado amath unizh ch