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On GAeL: Geometrie Algebrique en Liberte XVIII
Margarida Melo
Department of MathematicsUniversity of Coimbra, Portugal
Abstract
Geometrie Algebrique en Liberte is a school and conference organized by and foralgebraic geometers in the beginning of their scientific careers. The 18th editionof GAeL took place in the Mathematics Department of the University of Coimbra,Portugal, on June 7-11 2010. It gathered together about 70 participants comingfrom whole parts of the world who got the opportunity to learn and discuss togetherand “en Liberte” the most recent developments in this area of research. The seniorspeakers for this year were the Professors Olivier Debarre (C.N.R.S.-Paris), Gerardvan der Geer (Amsterdam) and Bernd Sturmfels (Berkeley).
Algebraic Geometry
Algebraic Geometry is the branch of mathematics thatconsists of the study of algebraic varieties: geometricincarnations of solutions of systems of polynomial equa-tions. The simplicity and generality of this idea, thatinvolves a big number of mathematical objects, allowedthe concept to grow in many different directions deve-loping an amazingly rich theory. It is nowadays a widearea of mathematics that combines tools from manydifferent disciplines as Abstract Commutative Algebra,Number Theory, Complex Analysis, Differential andComplex Geometry, Algebraic Topology, Category The-ory, Homological Algebra, Algebraic Combinatorics andRepresentation Theory to study problems arising fromGeometry. The following words by the fields medalistDavid Mumford reflect how rich and complex are theideas and problems that appear in this area of puremathematics.
Algebraic geometry seems to have acquired the reputa-tion of being esoteric, exclusive, and very abstract, withadherents who are secretly plotting to take over all therest of mathematics. In one respect this last point isaccurate. David Mumford
Even if it is one of the most classical areas of puremathematics, it is certainly one of the most active aswell. The importance of this area of research in theglobal context of mathematical sciences is clear, for in-stance, by the strong presence of algebraic geometersamong the speakers of the next ICM conference (http://www.icm2010.org.in/about-icm-2010/) that willtake place in Hyderabad, India on August 19-27 2010.
The great developments of Algebraic Geometry in thelast decades are also clear by the list of the field medal-ist winners: about a quarter of the total number of fieldmedals so far was delivered to algebraic geometers. Thefield medalists Kodaira, Serre, Atiyah, Grothendieck,Hironaka, Bombieri, Mumford, Deligne, Yau, Donald-son, Faltings, Drinfel’d, Mori, Witten, Kontsevich, Laf-forgue and Okounkov got their award by their contri-butions in algebraic geometry or closely related areas.To these we should also add the name of Andrew Wiles,who got a “Special Tribute” of the Fields Institute forhis proof of the famous “Fermat’s last theorem” whichrelies upon methods from Algebraic Geometry, namelyelliptic curves and modular forms.
One can say that the roots of Algebraic Geometry dateback to the arabian and greek mathematicians, whooften used the geometry of plane curves and their in-tersection properties to solve algebraic equations. Thesame kind of ideas were also used much later by somemathematicians in the renaissance period like Cardanoand Tartaglia while studying the cubic equation. How-ever, after a long period of great development of ana-lytical methods in Geometry, the systematic use ofalgebraic methods in Geometry was established onlyat the end of the 19th century by the italian schoolof Algebraic Geometry composed by mathematicianslike Enriques, Chisini, Castelnuovo and Segre. Thefoundations of Algebraic Geometry using notions fromCommutative Algebra like the theory of ideals wereestablished in the beginning of the 20th century bymathematicians like van der Waerden, Oscar Zariskiand Andre Weil. In the 1950’s and 1960’s Serre and
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Grothendieck used sheaf theory and techniques fromHomological Algebra to introduce in Algebraic Geom-etry the notion of scheme. This led to an enormoustransformation in the whole theory: classical ProjectiveGeometry was more concerned with the geometric no-tion of point while the later emphasizes the concepts ofregular function and regular map. Moreover, this newpoint of view provided Algebraic Geometry with toolsto treat a wide number of problems from other areas ofmathematics like Commutative Algebra and AlgebraicNumber Theory (recall for instance elliptic curve cryp-tography and Wiles’ celebrated proof of Fermat’s lasttheorem). It also allowed to solve classical problems onalgebraic varieties like moduli problems or resolution ofsingularities. Nowadays, Algebraic Geometry is againin great transformation after the introduction of stacksby Grotendieck, Deligne, Mumford and Artin and, morerecently, by the development of the theory of DerivedAlgebraic Geometry.
Even if Algebraic Geometry is fundamentally a ratherabstract discipline, recently it has been heavily used inother rather applied areas of mathematics like statistics,control theory, robotics and also in other sciences. Forinstance, it is fundamental in the development of thephysics’ theory of strings and it has deep connectionswith the Phylogenetics theory of biology.
GAeL origins
The origins of “GAeL: Geometrie Algebrique en Lib-erte” date back to France, as one can guess by itsname. In fact, the first 13th editions of GAeL tookplace at the “CIRM: Centre International des Ren-contres Mathematiques” (Marseille, France). However,nowadays all communication and talks are in English.Most recently, GAeL has taken place in in Bedlewo(Poland), Istanbul (Turkey), Madrid (Spain), and Lei-den (The Netherlands). This year edition was the 18thGAeL conference and it took place in the Mathemat-ics Department of the University of Coimbra, Portugal.GAeL has been, since its beginning, a reference meetingfor young algebraic geometers specially from Europeancountries.
As the name indicates, the aim of GAeL is to giveyoung algebraic geometers the opportunity to learn anddiscuss the most recent developments of Algebraic Ge-ometry in a relaxed atmosphere with no fear to askquestions of any type. Young participants have the op-portunity to lecture, often for the first time, in frontof an international audience. At the same time, se-nior experts deliver courses in selected topics at thecutting edge of modern and classical Algebraic Ge-ometry. Among the speakers of previous editions ofGAeL are Batyrev, Beauville, Bridgeland, Campana,
Ciliberto, Colliot-Thelene, Corti, Demailly, Faber, Fan-techi, Farkas, Hassett, Huybrechts, Itenberg, Izvoskiy,Lehn, Manivel, Mukai, Muller-Stach, Mustata, Okonek,Oort, Oxbury, Peskine, Reid, Siebert, Sottile, Teissier,Thomas, Tyurin, Vakil, Van Straten, Vistoli, Voisin,and Zak.
Between the senior speakers’ mini-courses and the ju-nior speakers talks there is also time for discussions andfor exercise sessions. The junior participants who donot give a talk must present their work on the postersession at the beginning of the conference. Posters re-main available for the rest of the week so that at theend everybody gets the opportunity to learn about eachothers own research. For that reason the number ofparticipants of GAeL is quite limited: each person canparticipate at most in two editions of GAeL.
The organizing committee is also made of young re-searchers in Algebraic Geometry. Each organizer willorganize GAeL for two years: in each edition orga-nizers with some experience help the beginning ones.This year organizing committee was Vıctor GonzalezAlonso (Universitat Politecnica de Catalunya, NathanIlten (Freie Universitat, Berlin), Pedro Macias Marques(Universidade de Evora), Margarida Melo (Universi-dade de Coimbra), Kaisa Taipale (University of Min-nesota) and Filippo Viviani (Universita Roma Tre andUniversidade de Coimbra).
There is also a scientific committee that helps the or-ganizing committee in aspects like the choice of topicsor funding advice. The actual scientific committee forGAeL is composed by Professor Frances Kirwan (Ox-ford), Professor Yuri Manin (Max-Planck Institute furMathematik) and by Professor Farns Oort (Utrecht).
This year’s GAeL
This year edition of GAeL, the XVIII, took place in theMathematics Department of the University of Coimbra,Portugal, on June 7-11. It was possible thanks to thefinancial support of the Foundation Compositio Mathe-matica, the Center for Mathematics of the University ofCoimbra (CMUC), the International Center for Mathe-matics (CIM), the Foundation for Science and Technol-ogy of Portugal (FCT), the FCT project “GeometriaAlgebrica em Portugal” and of the Mathematics De-partment of the University of Coimbra.
As in the previous editions of GAeL, the program ofthe conference consisted of 3 mini-courses of 4 hourseach delivered by selected experts in different areas ofAlgebraic Geometry and of 25 contributed talks by thejunior participants. The poster session was organizedin the first afternoon of the program: we had this yearalmost 30 posters presented by participants at differentstages of their careers. The talks and posters presented
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by the junior participants where quite various: therewere talks on moduli spaces of curves and sheaves, mod-uli spaces of surfaces, moduli spaces of abelian varieties,toric geometry, tropical geometry, derived algebraic ge-ometry, arithmetic geometry, deformation theory, sin-gularity theory, minimal resolutions among others. Theprogram of the conference as well as a list of partici-pants and abstracts can be found in http://severian.
mit.edu/gael/files/quickschedule.pdf. This di-versity of subjects allowed the young participants tolearn from their colleagues several recent achievementsand ongoing projects in this vast discipline and alsofavored discussions and questions in a GAeL flavor: al-ways “en Liberte”!
The senior speakers of this edition of GAeL werethe Professors Olivier Debarre (http://www.math.ens.fr/~debarre/) from the C.N.R.S. -Ecole NormaleSuperieure of Paris, Gerard van der Geer (http://www.science.uva.nl/~geer/) from the University ofAmsterdam, The Netherlands, and Bernd Sturmfels(http://math.berkeley.edu/~bernd/) from the Uni-versity of Berkeley, California.
Professor Olivier Debarre lectured on “Rational curveson algebraic varieties”, a very classical argument thathad important developments on the last few years yield-ing important contributions to the recent spectacularadvances towards the proof of the minimal model pro-gram (see [1]). One of the most classical arguments ofalgebraic geometry concerns the classification of alge-braic varieties. The most classical case is the classifica-tion of algebraic curves, which was already understoodby Riemann in the 19th century. The classification ofalgebraic surfaces is more intricate and was one of thebiggest achievements of the italian school of the begin-ning of the 20th century. After several decades with-out significant progress towards the higher dimensionalcase, S. Mori proposed a program, the so-called “min-imal model program”, which would lead to the clas-sification of algebraic varieties in any dimension. Thepresence of rational curves on algebraic varieties detectsseveral important information on birational invariantsof the same varieties (see [4]). Professor Debarre lec-tures were mainly on the contributions of S. Mori him-self and later of J. Kollar on this part of the program.Professor Debarre notes for this course are available athttp://www.math.ens.fr/~debarre/NotesGAEL.pdf.
Professor van der Geer delivered a course on “Alge-braic cycles on abelian varieties”. Abelian varieties areamong the most studied objects in Algebraic Geome-try due to their extremely interesting and rich prop-erties: they combine the structure of a projective va-riety with the structure of a compact algebraic group(see, for instance, [2] or [3]). The study of algebraic cy-cles on varieties is closely related to the famous “Hodgeconjecture”, which is one of the millennium problems.Moreover, the case of abelian varieties is commonly con-
sidered to be a crucial test case towards the validity ordisproval of this famous conjecture. Among abelian va-rieties the so called jacobian varieties are of particularinterest. Recent conjectures relate classical aspects ofthe geometry of algebraic curves, namely the study oftheir linear series which is the subject of Brill-Noethertheory, with the existence of cycles on their associatedjacobian varieties. Professor van der Geer lectures cul-minated in this important aspect of this theory.
Professor Sturmfels’ lectures were on a new branch ofalgebraic geometry that lies on the border line betweenAlgebraic Geometry and Optimization Theory: “Con-vex algebraic geometry”. This new area of mathemat-ics arises from the necessity of studying certain con-vex objects arising from linear and/or semidefinite pro-graming. These objects seem particularly featured tobe studied with tools from algebraic geometry. Pro-fessor Sturmfels lectures were centered on several ex-amples that are crucial to understand the foundationsof this new discipline in an attempt to systematicallystudy such convex objects. References as well as theslides of Professor Sturmfels lectures are available athttp://severian.mit.edu/gael/sturmfels.html.
To relax a bit of such an intense program there werealso some social activities scheduled. Monday June 7ththere was a visit to “Patio das Escolas” and the his-torical buildings of the University of Coimbra. Therewas also a small reception while the poster session washeld. On June 10th afternoon a visit to “Conimbriga”was organized and, after that, the participants weretransported to Figueira da Foz were the social dinnerwas held.
Even if algebraic geometry is a very classical sub-ject of mathematics there were not many portuguesemathematicians working on it until recently. How-ever things are changing rapidly and a proof of thisis the fact that there were more then 10 portuguese ju-nior participants. Most of these obtained their PhDin Algebraic Geometry either in Portugal or abroadin the last 3 years. Most algebraic geometers in Por-tugal participate in the project “Algebraic Geometryin Portugal”, run by Margarida Mendes Lopes, thataims to promote a strong portuguese Algebraic Geom-etry community. Already in this century, there havebeen several other events in Portugal related to al-gebraic geometry. among which we recall the “ISTcourses on algebraic geometry” that are regularly or-ganized by Margarida Mendes Lopes: there were al-ready 5 editions, the “Oporto meeting on Geometry,Topology and Physics”, the “Geometry in Lisbon Sum-mer school”, the 2010 edition of “VBAC: Vector bun-dles on algebraic curves”, the “Lisbon Summer Lec-tures in Geometry” in IST (Lisbon) and the “Coimbra-Salamanca Algebraic Geometry seminar”, whose firstedition was held in Coimbra in February 2010. Onthe “GA-P: Geometria Algebrica em Portugal” web-
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page, http://home.utad.pt/~ga-p/index.html, runby Carlos Rito from UTAD, is possible to find updatedinformation on people working in Algebraic Geometryin Portugal, on events in Algebraic Geometry either inPortugal and abroad as well as several useful links andother information concerning this beautiful discipline,that is creating solid roots in Portugal as well.
Bibliography
[1] C. Birkar, P. Cascini, C. D. Hacon, J. McKernan:Existence of minimal models for varieties of loggeneral type. J. Amer. Math. Soc. 23 (2010), 405-468.
[2] G. van der Geer, B. Moonen: Abelian varieties,Book in preparation, http://staff.science.
uva.nl/~bmoonen/boek/BookAV.html
[3] C. Birkenhake, H. Lange: Complex abelian vari-eties. Second edition. Grundlehren der Mathema-tischen Wissenschaften [Fundamental Principlesof Mathematical Sciences], 302. Springer-Verlag,Berlin, 2004.
[4] J. Kollar: Rational curves on algebraic varieties.Ergebnisse der Mathematik und ihrer Grenzgebi-ete. 3. Folge. A Series of Modern Surveys in Mathe-matics [Results in Mathematics and Related Areas.3rd Series. A Series of Modern Surveys in Mathe-matics], 32. Springer-Verlag, Berlin, 1996.
Figure 12: Groupe Picture, On GAeL: Geometrie Algebrique en Liberte XVIII.
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