Simons Foundation Collaboration Grants for Mathematics

Algebraic Geometry and Commutative Algebra have their common roots in the study of solutions of polynomial equations. The set of common solutions is a geometric object such as an ellipse and called a variety or a subscheme if one also takes the multiplicities of the solutions into account. The polynomials in the equations generate an algebraic object, called ideal. Liaison Theory addresses the question how one can decide if two geometric objects have similar properties of one knows only their defining equations. 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 unifed whole. The new idea of enlarging the space in which links take place has lead to further evidence that this is indeed possible. On the application side, liaison theory has been used to answer question on Hilbert functions and study various determinantal and monomial ideals. Another focal point of my work has been the problem of developing new methods for deciding wheather a graded algebra has the Weak Lefschetz Property. This property has its origins in the Hard Lefschetz Theorem of topology. However, recent results show that it is related to many other areas. For examples, recently new connections to combinatorics were established because the presence of the Weak Lefschetz Property is intimately related to enumerations of lozenge tilings and of families of non-intersecting lattice paths.