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:
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.
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.