Commutative algebra, complexes and applications of computer algebra (COM) Organised by Schreyer, Brown, Eisenbud and Reid Complexes are central to algebraic geometry and commutative algebra. Whereas computer algebra systems currently contain packages for syzygy modules and projective resolutions, these are now being complemented by routines for general coherent cohomology, together with homological functors such as Hom and Ext, Castelnuovo--Mumford regularity (as an explicit form of Serre vanishing) and Kustin--Miller unprojection, all subjects with strong theoretical connections. Following Bayer and Mumford's seminal paper `What can be computed in algebraic geometry?', recent work on the asymptotic regularity of ideals relate these to other asymptotic invariants (see [CEL]). Combined with classical methods, theoretical progress and some ingenuity, computer algebra is approaching Mumford's goal of describing classes of surface of general type and their moduli explicitly via computer algebra. Green's conjecture on the syzygies of a canonical curve were an entry point for algebraic geometers into computer algebra. Farkas' recent work introduces divisors on M_g as jumping loci of Koszul cohomology; this suggests generalisations of the Green--Lazarfeld conjecture on syzygies. At the same time, these jumping loci provide estimates for the `slope problem' and hence methods for the Kodaira dimension of moduli spaces. Analysing the precise dividing line between unirational moduli spaces and moduli spaces that are of general type is another challenge for computer algebra methods: Monte Carlo methods (or `needle in a haystack') over finite fields provides a probabilistic substitute for unirational parametrisations of moduli spaces or special loci in them. Although not primarily concerned with implementation in software, we hope that people from computer algebra systems such as Magma, Macaulay2 and Singular will take part in our workshops, and may be inspired to complete their implementations of Eisenbud, Floystad and Schreyer's algorithms for coherent cohomology [EFS], Papadakis' algorithm for Kustin--Miller unprojection and so on. The COM activity of WAG07-08 follows the explicit component XPL: to a large extent, these are different methods to tackle the same subject. COM will run in May and Jun 2008. A workshop is planned for Tue 27th May--Tue 3rd Jun 2008. Confirmed participants: Brown, Eisenbud, Farkas, Reid, Schreyer Targetted: Bayer, Cremona, Decker, Graf v. Bothmer, Fl{\o}ystad, Greuel, Hess, Kreuzer, Lazarsfeld, Papadakis, Popescu, Robbiano, Ryder, Schicho, Steel, Voisin, Walter, [CEL] S.D. Cutkosky, L. Ein and R. Lazarsfeld, Positivity and complexity of ideal sheaves, math.AG/0007023 [EPW] D. Eisenbud, S. Popescu and C. Walter, Lagrangian subbundles and codimension 3 subcanonical subschemes, Duke Math. J. 107 (2001) 427--467 [EPSW] D. Eisenbud, S. Popescu, F. Schreyer and C. Walter, Exterior algebra methods for the minimal resolution conjecture, Duke Math. J., 112 (2002) 379--395 [EFG] D. Eisenbud, G. Fl{\o}ystad and F. Schreyer, Sheaf cohomology and free resolutions over exterior algebras, Trans. Amer. Math. Soc., 355, (2003) 4397--4426 [EGHP] D. Eisenbud, M. Green, K. Hulek and S. Popescu, Restricting linear syzygies: algebra and geometry, Compos. Math., 141 (2005) 1460--1478