Algebraic geometry is a central area of research in Pure Mathematics. Because of the simplicity of the basic objects with which it deals (polynomials), it has many interactions with other areas of mathematics and science; this makes it a particularly interesting and productive field. Brill-Noether theory originates from 19th century work on algebraic curves; in modern terms, it is concerned with line bundles and their sections. This expanded into vector bundles in the late 1980s. Progress has been made, but there are many interesting problems to be solved, some of a general theoretical nature, others much more specific and detailed. It will remain a fruitful area of investigation for many years and in 2003 I launched the Brill-Noether project, aimed at solving the basic problems within 10 years. Of course, this was not achieved and there is a great deal still to be done. Many problems are best expressed in terms of coherent systems; a coherent system is a vector bundle together with a subspace of its space of sections. These generalise linear systems, which have been extensively studied since the 19th century.
The VBAC group has a programme which includes Brill-Noether theory and many other topics. It runs annual workshops which are especially aimed at young researchers and research students. The first workshop was held in Liverpool in 1994 and the second in Miraflores de la Sierra, Spain, in 1997. From 1998 to 2000 these were funded by a EuroConference grant from the EC; in 2001 the group won a new EuroConference grant (FP5 Contract no. HPCF-CT-2001-00248) for the period 2001-2003. The 2001, 2002 and 2003 workshops were in Rome, Luminy and Oporto. In 2004 a school and workshop was held at SISSA, Trieste. Due to lack of funding, there was no full-scale workshop in 2005, but there were mini-workshops in Madrid, Liverpool and Budapest. In 2006 there was a conference in Guanajuato, Mexico (11--15 Dec) in honour of my 65th birthday and in 2007 a workshop in Bad Honnef, Germany (18--22 June). In 2008 a conference was held in Miraflores (16--20 June) in honour of the 70th birthday of S. Ramanan, in 2009, there was a workshop at Freie Universitaet Berlin (15--19 June) and, in 2010, a workshop at the Instituto Superior Tecnico, Lisbon, Portugal (14-18 June). The 2011 workshop was held at the Newton Institute in Cambridge (27 June - 1 July) in conjunction with the closing conference for a programme on Moduli Spaces of which I was the principal organiser. There was a similar arrangement in 2012 at CRM, Barcelona (18-22 June) in connection with the programme "Geometry and Quantization of Moduli Spaces". The 2013 workshop was held at SISSA, Trieste (17-21 June) and the 20th anniversary meeting in 2014 at Freie Universitaet Berlin (1 - 5 September). The 2015 workshop was held at the University of Warwick (15-19 June) with the title "Fourier-Mukai: 34 years on" and the 2016 workshop was at the Centre Interfacultaire Bernoulli, Lausanne (11 - 15 January) with the title "Higgs bundles and Hitchin system". The 2017 meeting will be on "Motivic methods and derived categories" and will be held in Essen (4 - 10 September), while that of 2018 will be in Luminy (18 - 22 June) on "Gauge theory and complex geometry".
VBAC was affiliated to the European Algebraic Geometry Network (EAGER), which was funded by the EC (FP5 Contract no. HPRN-CT-2000-00099), but this funding ended some years ago.
I retired from my position as Professor of Algebraic Geometry at the end of September 2006, but remain active in research.
Information on the COW seminar and other events in Algebraic Geometry may be found here.