Maria Sk\lodowska-Curie graduate school
Threefolds in algebraic geometry (3-fAG)
reference number MCFH-1999 00687
Research to be supported:
Classification of surfaces and 3-folds. Group actions, singularities and
McKay correspondence. Calabi-Yau 3-folds and mirror symmetry, related
algebra, Kaehler and symplectic geometry and string theory in physics.
Keywords:
1. Classification of varieties
2. Resolution of singularities
3. Mirror symmetry
4. String theory
Fellows requested
Number of researcher-months: 72
Indicative number of fellows: 10--12
Legal conditions: A candidate must be a citizen of the EU or an associated
state, and must be registered for a Ph.D. in an EU country other than the UK.
Studentships can only be awarded for a period between 3 months and 1 year.
The geometry group at Warwick invites applications for postgraduate
fellowships to be held in the EU Framework 5 Maria Sklodowska-Curie
training site "Threefolds in algebraic geometry". The research topics
covered by this program include the following:
3-folds, minimal models and classification, the birational geometry of
Mori fibre spaces; Calabi-Yau 3-folds and mirror symmetry; group actions,
quotient singularities and McKay correspondence; related areas of the
classification of varieties, symplectic geometry, Kaehler geometry,
string theory in theoretical physics, singularity theory and commutative
algebra.
These topics here will be interpreted broadly.
The Warwick geometry group currently includes Mark Gross, John Jones,
Mario Micallef, David Mond, John Moody, John Rawnsley, Miles Reid, Dmitry
Rumynin, Balazs Szendroi, in addition to Marie Curie fellows Adrian Langer
and Nikos Tziolas, and about 12 graduate students.
The Maria Sklodowska-Curie training site program will cover two or three
postgraduate fellowships each year (lasting for 3 to 12 months, with
preference given to longer stays), from a starting date in mid 2000. Those
interested should contact Prof Miles Reid (e-mail Miles@Maths.Warwick.Ac.UK).
For further details, see www.maths.warwick.ac.uk/research.
Recruitment plan:
1st year 2000 3 appointments
2nd year 2001 3--4 appointments
3rd year 2002 3--4 appointments
4th year 2003 1 appointments
Description of research area of the doctoral training:
The geometry group at Warwick offers training to postgraduate students in a
broad area of algebraic geometry centred around three topics in 3-folds:
(1) Mori theory of minimal models and classification of 3-folds,
including the birational geometry of Mori fibre spaces;
(2) Calabi--Yau 3-folds and mirror symmetry, including the Strominger--
Yau--Zaslow approach to mirror symmetry based on special Lagrangian
fibrations and more recent geometric approaches to string dualities in
theoretical physics;
(3) group actions, quotient singularities, Hilbert schemes and the McKay
correspondence.
Mori theory generalises and extends the classification of curves and
surfaces, using a whole range of ideas from other areas such as
singularity theory, commutative algebra, Abelian varieties, toric geometry
derived categories and Hodge theory, in addition to related areas of
symplectic geometry, Kaehler geometry, cohomology theories and string
theory. Graduate students working primarily in these areas will be
encouraged to continue their direction of research in the context of the
wider view of the subject afforded by 3-folds.
Expected benefit to the fellows:
As a deep and well-developed mathematical discipline having fruitful
overlap with many other branches of science, algebraic geometry is a
crucial area for training young mathematicians. The last 20 years have
seen extraordinary developments in threefolds, both internal work such as
Mori theory that build on extensive experience with curves and surfaces,
and new applications to other areas of math and physics, some quite
unexpected. Our program aims to provide European graduates with an
opportunity to learn the methods and results of 3-folds, but just as
important, to view these in the wider context of the rich interplay
between algebraic geometry and other areas of math and physics. This will
add to the intellectual value of their research, and certainly improve
their prospects for a successful research career.
Detailed proposal information
Part I. Description of the doctoral training
1. The research area
The foundations of minimal models of 3-folds (Mori theory) and the
classification of 3-folds were laid during the 1980s by Mori, Reid,
Kawamata, Miyaoka, Koll\'ar and others, the subject obtaining
recognition in 1990 with Mori's Fields medal. This theory builds on work
on algebraic curves and the classification of surfaces going back to the
19th century, but the generalisations to higher dimension involve a
sophisticated new conceptual framework and many technical difficulties.
Mori theory offers rewarding new insights into classical results on
algebraic surfaces, and has been an essential ingredient in advances such
as mirror symmetry and the McKay correspondence. Unfortunately, partly on
account of the difficulty of the subject, the study of 3-folds has been
localised to a few centres, mainly in Japan or the USA. The geometry
group at Warwick is unique in Europe in being among the world leaders in
3-folds and several related areas of geometry; the training site
``Threefolds in algebraic geometry'' aims to share this expertise with
graduate students from many European centres.
While higher dimensional geometry and mirror symmetry certainly contain
problem areas that vary from extremely difficult to completely
intractable, there are also many worthwhile accessible research problems
in the subject. For example, there are still challenging problems in
reinterpretting classical results on algebraic curves and surfaces (say,
Cremona transformations) from the standpoint of Mori theory, and to
develop these further; these types of problems may be particularly
appropriate for graduate students coming from European centres with
interests in traditional areas of algebraic geometry. Members of the
Warwick geometry group have extensive experience of finding good problems
that are do-able while appropriate to the needs of graduate students. We
mention here some areas each containing many problems for which a
successful outcome can be predicted:
i. Graded rings: Classical construction of algebraic surfaces can in
many cases be viewed in terms of the commutative algebra of canonical
rings. We can now frequently go beyond what was known classically, to find
new treatments of algebraic varieties and their moduli spaces in terms of
algebra. This links to methods of projective geometry and to research
problems in commutative algebra. Algebraic methods based on Gorenstein
graded rings are also applicable to the classification of the flips and
divisorial contractions of the 3-fold minimal model program.
ii. Biregular and birational geometry of Fano 3-folds and Mori fibre
spaces: Kawamata has proved that Fano 3-folds form a bounded family, but
there is still a long way to go before we can turn this result of
principle into a concrete finite list; however, the conjectural outlines
of such a classification are slowly coming into focus: in small
codimension they are given by algebraic constructions involving graded
rings; at the other end (in big codimension), they should be obtainable by
projections in the style of Fano and by graded version of the Mukai
symmetric spaces.
iii. Singularity theory: One of the new conceptual tools of Mori theory
is the realisation that nature is singular: Mori theory works with singular
3-folds, and only resolves their singularities as a technical device.
Following Mirel Caib{\u a}r's 1999 Warwick Ph.D. thesis, we expect the
study of 3-fold singularities to become a significant growth area in the
next 5 years; the crepant resolution of a 3-fold canonical singularities
behaves like a local piece of a Calabi--Yau 3-fold, and Caib{\u a}r showed
how to calculate its invariants in singularity theory terms. With his
results, many traditional tricks of singularity theory (such as deformation
theory) become applicable to these models of Calabi--Yaus.
iv. Group actions, quotient singularities and McKay correspondence: The
McKay correspondence relates the geometry of the resolution of
singularities of a quotient singularity to the representation theory of
the group. The two dimensional case is the ``classical'' McKay
correspondence, and there has been considerable progress in understanding
the higher dimensional case, following conjectures by Reid in 1992.
However, the known theorems are rather abstract statements (about derived
categories or motivic integration) giving little information about how the
McKay correspondence works in concrete cases. Since the groups are listed
in the 3-fold case, we obtain a large number of concrete problems that
involve toric geometry, rather easy group theory and representation
theory, and several different categories of geometry. At the same time,
this area contains problems that are subtle and likely to be deep, for
example, on the interpretation of the resolution of singularities in terms
of moduli, and the multiplicative structure of the cohomology.
While developments on the algebraic geometry side of 3-folds continue
apace, one of the substantial current application of Mori theory is to
Calabi--Yau manifolds and mirror symmetry. This idea arose about 10 years
ago in the work of string theorists in theoretical physics, and came as a
big surprise to mathematicians, giving us a whole range of new problems at
many levels. This theory is now in part mathematical, and Mark Gross has
given several lecture courses on it as part of the Warwick M.Sc. and at a
number of summer schools. The special Lagrangian approach to mirror
symmetry of Strominger, Yau and Zaslow present many exciting problems.
Gross' work gives new approaches to special Lagrangian fibrations around a
singular fibre, and many problems in the style of toric geometry can be
derived from this. With Gross and Szendr{\H o}i, the Warwick geometry
group is in a strong position to teach this subject at the research level.
Mirror symmetry is not an isolated case in recent interactions between
geometry and physics. What happens typically is that ``string dualities''
in physics predict cross-over between different categories of geometry,
for example, the algebraic geometry of a Calabi--Yau 3-fold and the
special Lagrangian geometry of its mirror. The insights of the
physicists teach us the importance of the broader view of the mathematical
subject, including surprising new relations between algebraic 3-folds and
singularity theory, PDEs and geometry, gauge theory, symplectic
geometry, geometric quantisation, elliptic cohomology, Lie theory etc.
The Warwick geometry group has specialists in all these disciplines
(David Mond, Mario Micallef, Peter Topping, John Rawnsley, John Jones,
John Moody, Dmitry Rumynin, etc.)
2. Benefits to the fellows
Marie Curie postgraduate fellows visiting the Warwick Math Institute under
3-fAG will be allocated a shared office, and the same library, computer,
and other facilities as other postgraduate students. They will be assigned
a research tutor chosen from the Warwick geometry group, and will get
expert advice on which of our regular M.Sc. and M.Math. lecture courses
and seminars to attend; they will be encouraged to discuss math problems
with our existing graduate students. It should be mentioned that in recent
years we have had several visiting international students each year (for
example, in addition to many short term visitors, Hirokazu Nasu from
Nagoya visited for 6 months in 1997--98, and Thomas Fangel from Aarhus for
6 months in 1998); our visitors quickly fit into a lively and friendly
group of fellow students with related research interests.
Incoming graduate students are welcome to take part in all our other
research activities, including our symposia, conferences and workshops.
They may also visit Cambridge, Oxford, London and other British
universities for meetings of the COW (Cambridge Oxford Warwick algebraic
geometry symposium) and Calf (Junior COW, organised by graduate
students).
The members of the Warwick geometry group will provide a broad range of
advice on research problems. Depending on the individual postgraduate,
this may take the form of help with an existing problem, or a completely
new project, or some combination of the two; a completely new research
problem will mainly only be appropriate for those making longer term
visits. We should again mention that, in recent years, several
postgraduate students who have visited us obtained substantial help in
starting up on a research problem, or in developing their ideas into a
thesis: as examples, Yukari Ito (Tokyo University), Adrian Langer
(Warszawa) and Massimiliano Mella (Trento) wrote their theses in part on
ideas suggested during visits to Warwick. It is likely that visiting
students will in some cases also indulge in joint projects with Warwick
students, leading to joint publications and reciprocal visits.
In addition to study and research, Warwick offers visiting students many
facilities for self-improvement. Visitors will be persuaded to give
seminars at a basic and a research level. Most of our graduate students
have experience of mathematical presentation, classroom teaching, computer
typesetting, preparing material for publication, use of the web, etc., and
are happy to give advice on these things. Our students regularly comment on
each others' presentation at seminars, and proofread each others' papers
to improve the English and the math presentation. International students
spending an entire 10 week term at Warwick will be given the opportunity to
take part in undergraduate teaching for a few hours a week. International
students may use the facilities of the Language Lab to brush up their
English or to embark on an introductory course in Japanese.
Finally, our collective experience and contacts in the world of research
mathematics puts the Warwick geometry group in a strong position to provide
visiting students with advice on further study, the opportunities for
postdoc placing, or for permanent employment.
Part II. Description of the training site
1. The Warwick Math Institute
Warwick is one of the UK's most successful universities. It obtains high
ratings in the various periodic assessments of teaching and research, and
is commonly ranked 4th for quality of research among the UK's 100
universities. The University attracts a large number of international
visitors at the undergraduate, M.Sc., postgraduate and postdoc level, and
has excellent facilities for handling them.
The Warwick Math Institute, founded by Professor Sir Christopher Zeeman in
1965, is one of the world's leading centres of teaching and research in
math. It was one of only 3 UK Math departments awarded the 5* rating in the
UK Funding Council's 1996 Research Assessment Exercise. The Institute has
39 permanent academic staff -- all active research mathematicians of
international stature -- plus research fellows and junior staff. Our
visitor program attracts an average of 200 visits per year, including
leading mathematicians from all over the world. The Math Institute runs an
annual research symposium, usually funded by the British Engineering and
Physical Science Research Council (EPSRC), and runs high level workshops
and conferences on a regular basis (listed on our website
www.maths.warwick.ac.uk/research.)
The geometry group has been particularly active in this respect, running
an algebraic geometry symposium in 1995--96 funded by British EPSRC,
featuring 4 or 5 workshops and conferences, including a highly successful
concluding EuroConference. The current 1999--2000 EPSRC symposium in
geometry and topology looks like being equally strong. Algebraic
geometers at Warwick takes an active part in European research activities:
we have hosted many meetings as part of the EU Science, HCM and TMR
programs, and expects to be a major component of the EU research training
network EAGER (currently under application, reference number
RTN1-1999-00202). There are currently 10 graduate students in algebraic
geometry, 5 of whom are cross-border Europeans, and another 10 in related
areas of geometry.
The Math Institute's graduate program trains around 70 Ph.D. students at
any time, plus 20 M.Sc. students. Just over half these are cross-border
European or overseas students. In the last 5 years, we have awarded 75
Ph.D. degrees, of which 36 to international Ph.D. students. The Math
Institute is a popular destination for international visiting students at
all levels: we take 12 undergraduate students each year under the
Erasmus/Socrates program, most of whom take M.Sc. courses and participate
actively in graduate seminars; European graduate students frequently visit
conferences, workshops and seminars, or make short stays to collaborate
with our research staff.
2. The training site ``Threefolds in algebraic geometry''
Current research staff
Permanent appointments: Mark Gross, John Jones, Mario Micallef, David
Mond, John Moody, John Rawnsley, Miles Reid, Peter Topping
Warwick Zeeman lecturers (on 3-year contracts): Dmitry Rumynin,
Bal\'azs Szendr{\H o}i
Marie Curie fellows: Adrian Langer, Nikos Tziolas
Several new appointments in geometry are expected during the lifetime of
the proposed training site, one this year to replace Victor Pidstrigatch
who moves to Goettingen. The training site will function against the
background of a comprehensive M.Math. and M.Sc. teaching program,
involving many specialists not listed here in algebra, topology,
hyperbolic geometry, computation, dynamical systems, etc.
Current doctoral students:
Rui de Albuquerque (Portugal), Ricardo Castano Bernard (Mexico), Alastair
Craw, Rita Gaio (Portugal), Rebecca Leng, Diego Matessi (Italy), Anita
Mo{\v c}nik (Slovenia), Jonathan Munn, Elisabeta Nedita (Romania), Jorge
Neves (Portugal), Stavros Papadakis (Greece), Simone Pavanelli (Italy),
Daniel Ryder, Marc Sommers (Germany), Andrew Stacey, Andrei Caldar{\u a}ru
(USA, visiting from Cornell)
Approximately 6 M.Sc. students each year; in addition, visiting
postgraduate students (for example, Hokuto Uehara from Tokyo University
will visit for 12 months from April 2000).
3. Research quality of the training site
The main researchers in the geometry group at Warwick and their research
interest are as follows:
Mark Gross: algebraic and differential geometry, Calabi--Yau
manifolds, mirror symmetry, geometry and string theory, classification of
algebraic varieties
John Jones: geometry, topology, gauge theory, K-theory, cyclic
homology, index theorems, elliptic genus
Mario Micallef: partial differential equations, differential geometry,
K\"ahler geometry
David Mond: algebraic geometry, singularity theory
John Moody: algebra, algebraic geometry, resolution of
singularities
John Rawnsley: symplectic geometry, quantisation, differential
geometry
Miles Reid: algebraic geometry and commutative algebra,
classification of surfaces and 3-folds, Mori theory, canonical
singularities, flips, McKay correspondence
Peter Topping: analysis, partial differential equations and geometry,
K\"ahler geometry
Bal{\'a}zs Szendr{\H o}i: algebraic geometry, Calabi--Yau 3-folds,
Torelli problems, toric geometry and mirror symmetry, string theory
Of these, Reid, Gross and Szendr{\H o}i work directly in 3-folds, and the
others in areas of geometry having important relations with algebraic
geometry. We give a small number of references directly relevant to the
work of the training site.
Paul Aspinwall and Mark Gross, Heterotic--heterotic string duality and
multiple K3 fibrations, Phys. Lett. B 382 (1996), 81--88
Paul Aspinwall and Mark Gross, The SO(32) heterotic string on a K3
surface, Phys. Lett. B 387 (1996), 735--742
Bal{\'a}zs Szendr{\H o}i, Calabi--Yau threefolds with a curve of
singularities and counterexamples to the Torelli problem. I, to appear in
Int. J. Math. II, to appear in Math. Proc. Camb. Phil. Soc.
Bal{\'a}zs Szendr{\H o}i, Some finiteness results for Calabi--Yau
threefolds, to appear in J. London Math. Soc.
Tom Bridgeland, Alastair King and M. Reid, Mukai implies McKay, preprint:
math/9908027, 17~pp.
A. Corti, A. Pukhlikov and M. Reid, Birationally rigid Fano hypersurfaces
(May 1999 draft pp. 84), to appear in book ``3-folds at Warwick'', A.
Corti and M. Reid (eds.), CUP 2000
Alastair Craw and M. Reid, How to calculate A-Hilb C^3, preprint:
math/9909085, 29~pp.
Mark Gross, Topological Mirror Symmetry, preprint: math/9909015, 61~pp.
Mark Gross and Sorin Popescu, The moduli space of (1,11)-polarized Abelian
surfaces is unirational, preprint: math/9902017, 27~pp.
Mark Gross, Special Lagrangian Fibrations, I. Topology, Integrable systems
and algebraic geometry (Kobe/Kyoto, 1997), 156--193, World Sci.
Publishing, River Edge, NJ, 1998. II. Geometry, preprint: math/9809072,
72~pp.
Mark Gross and P.M.H. Wilson, Mirror symmetry via 3-tori for a class of
Calabi--Yau threefolds, Math. Ann. 309 (1997), 505--531.
Mark Gross and Sorin Popescu, Equations of (1,d)-polarized Abelian
surfaces, Math. Ann. 310 (1998), 333--377
J.D.S. Jones and John Rawnsley, Hamiltonian circle actions on symplectic
manifolds and the signature, J. Geom. Phys. 23 (1997), 301--307
John Moody, On resolving vector fields. J. Algebra 189 (1997), 90--100
Miles Reid, Young person's guide to canonical singularities, in Algebraic
geometry (Bowdoin, 1985), Part 1, 345--414, Proc. Sympos. Pure Math., 46,
A.M.S., 1987
Miles Reid, Canonical 3-folds. Journ\'ees de G\'eometrie Alg\'ebrique
d'Angers, 1979, 273--310, Sijthoff \& Noordhoff, Alphen aan den Rijn 1980
M. Reid and Y. Ito, The McKay correspondence for finite subgroups of
SL(3,C), in Higher Dimensional Complex Varieties (Trento, Jun 1994), M.
Andreatta and others Eds., de Gruyter, Mar 1996, 221--240
M. Reid Chapters on algebraic surfaces, in Complex algebraic varieties,
J. Koll\'ar (Ed.), IAS/Park City lecture notes series (1993 volume), AMS,
Providence R.I., 1997, 1--154. Warwick preprint 9/1996, alg-geom/9602006
M. Reid, McKay correspondence, in Proc. of algebraic geometry symposium
(Kinosaki, Nov 1996), T. Katsura (Ed.), 14--41, alg-geom 9702016, 30 pp.
4. Facilities
See under Part~1, ``Benefit to the fellows'' above.
5. Evidence of past success
Warwick has awarded 75 Ph.D.s in math since 1994, of which 36 to EU and
overseas students. Succesful Ph.D.s trained by the geometry group include
the following
Selma Alt{\i}nok (Turkey), Santos Asin Lares (Spain), Claudio Arezzo
(Italy), Mohan Bhupal (India), Mirel Caib{\u a}r (Romania), Jose
Cisneros-Molina (Mexico), Marcello Felisatti (Italy), Peter Gothen
(Denmark), Laurent Lazzarini (France), Andrea Loi (Italy), Sergio Santa
Cruz (Spain), Ioannis Sardis (Greece), Yorgos Terizakis (Greece)
Over the same period, a similar number of international postgraduate
students visited the Warwick geometry group for extended periods as part of
their studies towards a Ph.D. at their home university.
P.S. The EU commission scheme to which we are applying is named after the
Polish lady scientist Maria Sk{\l}odowska-Curie. In discussing and
advertising our grant, we prefer to use her full name, which seems better
to reflect the internationalist and non-discriminatory spirit of the times.