Problems in Commutative Algebra

Liaison theory is an important area in the overlap between algebraic geometry and commutative algebra. Its history begins in the 1800’s with the idea of producing interesting examples of space curves by means of looking at the residuals of known curves inside complete intersections. In the middle of the 20th century it was realized that much more important information is passed from curve to curve in this process, and that in fact it is far from being restricted to curves! The activity in this field has exploded in the last 35 years. Important contributions have come both from the algebraic side and from the geometric side, and equally important aspects are the theoretical results and the open questions on one hand, and the myriad ways that liaison can contribute to seemingly unrelated topics on the other hand. On the theoretical side, the known picture reveals a beautiful panorama of complete results and highly suggestive partial results, begging for a resolution of the conjectures which, if true, would make an incredibly elegant unified whole. On the application side, it is proposed to study problems that are related to liaison theory and illustrate its impact on commutative algebra, algebraic geometry, and combinatorics. For example, this includes questions about Hilbert functions of Buchsbaum or Gorenstein rings and studies of various determinantal and monomial ideals. A