WAG07-08 application
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Prose for the EPSRC form:
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Q1. List the main objectives of the proposed research in order of
priority [up to 4000 chars]

1. To run a research symposium bringing together national, European and
international experts for collaborative research on several currently
active areas of algebraic geometry and applications, and to run a series
of linked instructional conferences, seminars and workshops.

2. To carry out joint research in the named topics: Higher dimensional
minimal model program (HDMMP), Explicit methods in algebraic geometry
(XPL), Geometry of topological quantum field theories (TQFT), Moduli
spaces (MOD), Commutative algebra, complexes and computer algebra (COM),
Rational points on curves and higher dimensional varieties (Arith),
Derived categories (Der) and applications to related areas.

3. Each named topic has its individual research objectives. A principal
aim of the HD MMP component is to disseminate and develop a recent
breakthrough, the proof of the Minimal model conjectures in higher
dimensions, and to start collecting dividends from 30 years of work on
this long-standing problem.

4. The Explicit methods component XPL has many tasks, including
consolidating recent advances in the classification of Godeaux and
Campedelli surfaces. Another aim is to elucidate what it means to embed
a variety into a key variety (such as a weighted homogeneous space), for
example in terms of prescribing a vector bundle with a suitable
collection of sections, and to obtain new examples of surfaces and
3-folds (e.g. Calabi-Yaus) by this kind of technique.

5. Sample aims in the Moduli spaces component MOD include further work
on the determination of effective divisors on the moduli space M_g of
curves of genus g, following Farkas' work on locuses defined by Koszul
cohomology; and the problem of the Hilbert series for the plurigenera of
A_g, the moduli space of principally polarised Abelian varieties,
following Shepherd-Barron's recent determination of the canonical model.
With the completion of the higher dimensional minimal model program, the
same kind of problems can now be posed for many more moduli spaces.




Q2. Describe the proposed research in simple terms in a way that could
be publicised to a general audience [up to 4000 chars]

Algebraic geometry studies the solution sets of systems of polynomial
equations. These solution sets are called algebraic varieties, and are
viewed as geometric locuses, generalising the circle and hyperbola of
analytic geometry. Algebraic geometry is a mature subject, and the
geometric points of view and the extensive toolbox it provides for
studying varieties apply to a great many problem in mathematics and its
applications.

Algebraic curves occur as the locus f(x,y)=0 in the plane, where f is a
polynomial function of x and y; in low degree (conics and cubics) one
gets useful conclusions by explicit manipulations of the equation, but
as the degree of f increases, the kind of conclusions one hopes for are
more abstract, and necessarily involve more theoretical machinery. One
eventually learns to stop worrying that the points of the curve are not
parametrised in terms of anything more elementary, and to accept the
curve as a primary object of nature, possibly complicated, but to be
understood in its own terms and used in subsequent constructions.

Rather than the degree, a better invariant of an algebraic curve is its
genus, that is, the number of handles (donut-like holes) in its
topological model. Especially important is the case division between the
three cases g=0 (a sphere) or g=1 (a donut) or g>=2 (a surface with many
handles); the case g=1 gives the elliptic curves, that played a key role
in Wiles' proof of Fermat's last theorem. For algebraic curves or
Riemann surfaces, this trichotomy was clearly perceived already in the
19th century, together with its interpretation in terms of positively
curved, flat, or hyperbolic non-Euclidean geometry; the picture of the
three cases g=0 or g=1 or g>=2 serves as an icon for the whole subject.

The same trichotomy was a distant model for Mori theory or the
classification of higher dimensional varieties, one of the most
intensively developed area of algebraic geometry from the late 1970s;
this work led to Mori's 1990 Fields medal. This classification is at
present the subject of a major breakthrough, with the recent
announcement of the proof of the minimal model program in all
dimensions. The first component of the Warwick symposium will develop
and disseminate these new result, and exploit its many applications.

Algebraic varieties, the solution sets of simultaneous polynomial
equations, provide examples and techniques in number theory and in
theoretical physics, in algebra and singularity theory and in other
branches of geometry. Even in analysis, which mostly deals in infinite
dimensional spaces, the ultimate aim is frequently a reduction to a
finite dimensional solution set modelled on algebraic geometry. The
Warwick symposium will include components on each of these topics,
together with applications of algebraic geometry to other areas of
mathematics.



Q3. Describe who will benefit from the research [up to 4000 chars].


Algebraic geometry is a major component of current research in pure
mathematics, and is targetted for development in many UK and European
universities. WAG07-08 will make an important contribution to raising
the profile of the subject in the UK and Europe, attracting many
employable young mathematicians from overseas to take part in our
seminars and conferences and to initiate collaboration with UK algebraic
geometers. The previous symposia we ran (WAG82-83 and WAG 95-96 and the
Newton Institute project HDG02) were enormously successful and
influential in this respect.

Algebraic geometry is used as essential background material for work in
many areas of math and its applications. The spread of subjects covered
by the proposed symposium includes topics of interest to many UK and
European mathematicians, not only in geometry but in theoretical
physics, representation theory in algebra, number theory, derived
categories and computer algebra. The boundaries between different
categories of geometry are permeable, with key ideas from one
influencing the other; thus Ricci curvature in differential geometry
motivates the canonical class in algebraic geometry, and moduli spaces
in algebraic geometry obviously influenced Yang-Mills in theoretical
physics and 4-manifold geometry. It is not too far-fetched to see
analogies between the minimal model program in higher dimensional
complex geometry and the Hamilton--Perelman breakthrough in the
Poincar{\'e} conjecture and the Thurston geometrisation conjecture.

As another important recent example, the McKay correspondence,
suggested by calculations from string theory, arose in the study of
resolution of singularities in higher dimensional birational geometry;
but the result of Bridgeland King and Reid, and the geometric and
derived categories ideas behind it, were applied to solve long-standing
problems in pure algebra (namely, in the theory of symmetric
polynomials, or the representation theory of the symmetric group).
There is every chance that results from WAG07-08 will likewise find
future applications in some apparently far-removed field.

We estimate that a dozen Warwick PhD students in algebraic geometry,
number theory, singularity theory and related areas such as
representation theory will benefit from WAG07-08, and a similar number
from other UK universities. We pay considerable attention to providing
introductory workshops and other forms of teach-in for our research
topics, so that anyone interested can benefit from them.



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cribbed from old applications/reports


Algebraic geometry is a central subject in mathematics, and relates to
many other areas of both pure and applied mathematics, as well as
physics and other areas of science; it has played a vital role in many
of the prominent breakthroughs of recent years in mathematics and
theoretical physics, and it will surely be a key to future developments.

The topics described in this proposal cover areas in which we are guided
primarily by geometric ideas. At the same time, these geometric ideas
frequently relate to different areas of current research inside and
outside of mathematics, such as representation theory in pure algebra,
modular forms in number theory, cryptography or error-correcting codes in
computer science or string theory in particle physics. Our proposal covers
all of these areas and many more.

Algebraic geometry has always interacted with other branches of
mathematics and theoretical physics. This intersectorial interaction works
both ways. In algebraic geometry we use techniques and tools developed in
other branches of mathematics such as topology, differential geometry or
number theory. Advances in theoretical physics have provided new
techniques and interesting conjectures in algebraic geometry, sometimes
leading to entirely new areas of research such as mathematical aspects of
mirror symmetry or quantum cohomology. On the other hand, algebraic
geometry helps very much in the further development of the fields
mentioned above. One example is the differential geometry of manifolds
with special holonomy, where most known examples are constructed using
algebraic geometry; or Hamiltonian systems, where algebraic geometry
provides explicit solutions of differential equations. Methods and results
from algebraic geometry provide tools and solutions in areas of research
such as representation theory or noncommutative ring theory. The concepts
of algebraic geometry have been very useful in arithmetic theories, and
algebraic geometry provides many objects and constructions which are
indispensible in quantum field theory, in particular string theory. These
examples show the interdisciplinary and unifying nature of algebraic
geometry within mathematics.

Algebraic geometry also relates in essential ways to subjects outside
theoretical mathematics and physics, for example to computer science (via
coding theory and cryptography), and, especially via computer algebra,
links to a number of industrial applications in areas such as robotics or
computer aided design. In this area we are probably at the beginning of
an important and rapidly developing cooperation.

Algebraic geometry is the geometrical study of solutions of systems of
polynomial equations. This programme was centred around 3-folds, that
is, solution sets of 3 dimensions. The subject of 3-folds has received
much attention in the past 20 years because of significant progress in
the classification programme. The aim is to classify solution sets into
three broad classes of geometry: positive, zero and negative curvature.
This idea goes back to the treatment of conic sections by the Greeks and
non-Euclidean geometries in the 19th century, but it is only in the past
20 years that a general picture has emerged in the context of
higher-dimensional algebraic geometry. The main features of this picture
have been established for 3-folds but remain conjectural for higher
dimensional algebraic varieties. Today the field is moving in two main
directions. One is to work out explicit consequences of the general
theory for special classes of 3-folds. The other direction is to
establish the general features of the theory in higher dimensions.

Algebraic geometry is a central subject in mathematics today. A trend
has been established over the past 50 years where key ideas of 19th
century algebraic geometry, such as moduli spaces, deformations,
enumerative problems and motives, have been exported to other styles of
geometry (differential, symplectic, analytic, special), PDEs, and
mathematical physics, where they have been the basis for the development
of major new theories. Algebraic geometry is in turn cross-fertilized by
development in all these fields.

Geometry, and especially algebraic geometry, is increasingly the
language of theoretical physics and string theory. A recent area of
interaction and enormous activity over the past 10 years has been the
field of mirror symmetry which, for mathematicians, started with some
enumerations by physicists of rationally parametrised curves on
Calabi-Yau 3-folds (a special class of 3-folds of zero curvature), and
has been studied subsequently from many different points of view. A more
recent area of interaction started with the realisation by some
physicists, among them Douglas, that

D-branes in type IIa string theory form a derived category, a concept
first discovered by algebraic geometers in the 1960s. This trend
constituted a key area of focus for the programme.

This was not the first major international event to concentrate on
3-folds: it came after symposia and meetings in Warwick (1982), Utah
(1988 and 1992), Warwick again (1995) and RIMS Kyoto (1997). We can now
see that these events have been instrumental in shaping the field as we
know it today, each stimulating progress and anticipating some cultural
change. We can hope that our programme will be recognised in the future
as having been equally important.


Mathematical Themes

We describe the outcomes of the programme in more detail in the final
section below. Here we sketch the main themes and how they fit in to the
general advancement of the subject. We see clearly how new ideas and
approaches, many significant simplifications in the theory and an
increased relevance of explicit methods have arisen right across our
spectrum of interests.

The programme had two central themes: the classification of all
varieties, in particular the Minimal Model Programme (MMP) in 3 and 4
dimensions; and the detailed study of special classes of varieties such
as Fano 3-folds, but especially Calabi-Yau 3-folds and their many
relations with physics.

A famous bottleneck in the proof of MMP is the proof that flips exist. A
flip is a surgery operation on varieties that occurs in codimension 2:
for 3-folds this means cutting out a finite collection of curves and
stitching up the variety with new curves in their place. In particular,
flips are an ingredient of classification not needed in the surface
case. The programme was a platform for understanding new work by
Shokurov that has provided far simpler proofs in the case of 3-folds,
and furnished the first proof for 4-folds. We worked through these new
ideas, revisiting the foundations of the subject in this new light. In a
series of lectures given in tandem by members of the programme, we
worked through each part of Shokurov’s manuscript. Already Corti was
able to give a short course, suitable for motivated graduate students,
that presented an outline of every part of a proof of 3-fold flips.

There was also a great deal of activity surrounding the study of
Calabi-Yau manifolds and mirror symmetry. Calabi-Yau 3-folds occupy one
slot in the classification of 3-folds, analogous to that of elliptic
curves or K3 surfaces in lower dimensions. Since the late 1980s they
have attracted interest because of their deep connections with string
theory in physics. This programme ran in parallel with the M-Theory
programme at the Newton Institute, which was of great benefit to us, and
in fact crucial for this point of view. Gross led the work in this area,
and gave a number of lectures from the mathematical point of view. There
is an array of related fronts, most prominently work on derived
categories, equivalences between them and the McKay correspondence, but
also work on birational geometry and the Kähler cone, etc. Much of the
work done in this area has been inspired by work done by physicists,
many of whom attended for some part of the programme.