Collaborations in Cummutative Algebra

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. Given a finite set of points, an Hermite interpolation polynomial of degree $d$ exists regardless of prescribed values of derivatives if and only if $d$ is at least the Castelnuovo]Mumford regularity of a corresponding fat point scheme. Confirming a 2001 conjecture, upper regularity bounds have been established. The arguments rely on a new partition result for matroids. Deciding whether the ideal of a fat point scheme contains a polynomial of specified degree is another challenging problem. Unexpected hypersurfaces arise if such a polynomial exists despite the fact that an elementary dimension count suggests otherwise. A structure of unexpected curves has been established. Their existence is related to subtle properties of hyperplane arrangements. Another focal point of my work has been the problem of developing new methods for deciding whether a graded algebra has a Lefschetz property. The strong Lefschetz property has its origins in the Hard Lefschetz Theorem of topology. However, recent results show that Lefschetz properties are related to many other areas. For examples, 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. Failure of a Lefschetz property is also interesting as it may, for example, indicate the presence of an unexpected curve. Recently I embarked on a program to develop asymptotic commutative algebra in the presence of symmetry. Informally, one studies ascending sequences of symmetric ideals in polynomial rings with more and more variables. The growth of invariants of such ideals along a sequence appears eventually predictable. This has been established for invariants such as the dimension and multiplicity and conjectured in other cases. Generalizations to modules require a more abstract functorial approach. Finiteness results for the resulting modules have recently been established, begging for further applications.