Conference Proposal in Support of Young Investigators

University Of Nebraska-Lincoln, Lincoln NE

Investigators

Abstract

The investigator and his colleagues study problems in the overlap of algebraic geometry and commutative algebra. A major goal is to understand the structure of minimal free resolutions of homogeneous ideals, particularly those defining zero dimensional schemes. Ideals of interest range from ideals of fat points (motivated by their intimate connections to questions about linear systems on projective varieties), to ideals of generic forms (work on which is closely connected to work on the Weak and Strong Lefschetz properties for Artinian algebras, and, via Matlis duality, to work on fat points), to ideals whose quotients are Gorenstein rings (and hence involve problems on arithmetically Gorenstein subschemes of projective space, and Gorenstein liaison). This broad range of methods and motivations is a significant hurdle that young and future researchers must overcome to do productive work in this area. This grant aims to advance the training and career development of young investigators and graduate students by providing travel support to attend and interact with leaders in the field at an international conference addressing the research cited above. Computational issues underlie much of this research, and machine computation is both an important tool in the research described above and the object of some of this research. However, work in this area can easily outrun the capability of any conceivable computer, so theoretical studies are essential not only for understanding the results of machine computations, but to achieve results beyond the reach of brute force computation. This is, for example, a significant issue for applications of the research described here to interpolation. Large data sets are a common feature of modern life, whether in science, technology or business. Such data sets often involve functional relationships among various variables. Among the most tractable functions are the polynomials. If one wants to model relationships with polynomials it is helpful to know theoretically how complicated the worst case such polynomial might be (as measured, say, by degree), whereas it might be expensive or impossible to determine this directly. Such theoretical results about polynomials are what researchers aim for in studying minimal free resolutions of homogeneous ideals.