## TOPICS IN ALGEBRAIC GEOMETRY

### Bath, 15th-19th September 2003

#### Organiser: G.K. Sankaran

The LMS/EPSRC Short Course entitled *Topics in Algebraic Geometry*
took place at the University of Bath from Monday September 15th to
Friday September 19th 2003.

There were three courses of lectures:

**Vector bundles**: Professor Peter Newstead (Liverpool)

Examples sheets:
Postscript or PDF
**Abelian varieties**: Dr Gregory Sankaran (Bath)

Examples sheets:
Postscript or PDF
**Higher-dimensional geometry**: Dr Alessio Corti
(Cambridge)

Examples sheets:
Postscript or PDF

Tutorial support for the courses was given by Dr Florin Ambro
(Cambridge), Dr Giovanna Scataligni (Durham/Oxford), and Dr Michael
Fryers (GCHQ).

A small temporary library was available.
Further material relating to Professor Newstead's course may be found in
Paris and
Warsaw.

#### Course description

Algebraic geometry occupies a central place in modern pure
mathematics, with connections to number theory, theoretical physics
and differential geometry in particular. For example, elliptic curves
and modular curves play vital roles in arithmetic; startling advances
in the theory of higher-dimensional varieties and moduli spaces have
emerged from, and contributed to, physics; and the theory of real
4-manifolds has similarly interacted with complex algebraic
surfaces. One of the most influential problems for computer algebra
has been to carry out explicit calculations in algebraic geometry.

Within algebraic geometry, there has been great progress over the last
few years. The study of algebraic varieties of dimension three and
more, initiated by Mori and others in the 1970s, has reached an
advanced stage. Major results have been proved in enumerative
geometry, especially on moduli spaces. The geometric meanings
contained in resolutions of ideals (syzygies) have been much better
explained and can be applied very directly, often with computer
assistance.

In part because of its many connections, algebraic geometry is often seen as
being hard to learn, and is left in the hands of specialists. This
course will try to broaden the appeal of the subject by presenting three
different topics at a level suitable to graduate students in algebraic
geometry but in a style accessible to those working in related fields.