There has been much activity in the last ten years in creating and implementing algorithms in algebraic geometry. Many of these algorithms were originally designed for abstract algebraic geometry, but now have the potential to be used in applications. A good example is the primary decomposition of an ideal. Several diverse techniques have been used to decompose an ideal, including Groebner bases, resultants, triangular sets, homotopy methods. Recently, mixing symbolic and numeric techniques has become a reality, with much research activity including algorithms for solving systems of polynomial equations. For example, Sommese, Verschelde and Wampler have given algorithms and software for numeric primary decompositions, which is based more on geometry rather than algebra. Many other researchers have done exciting work combining numeric and symbolic methods. Other important areas of current algorithm research especially relevant for applications include resolution of singularities of algebraic varieties and real algebraic geometry.
Our aim is to bring together experts in algebraic geometry, experts on algorithms, both symbolic and numeric, practitioners who have problems which could be solvable with new algorithms, and implementors, who can bring these ideas to fruition. We anticipate that bringing this wide range of expertise together will lead to exciting progress in algorithms and applications of algebraic geometry.