Commutative Algebra, Algebraic Geometry, Computation and Statistical applications

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David Eisenbud will perform research on a number ofareas in which algebraic geometry mixes with homological commutativealgebra. One of these is the study of the Castelnuovo-Mumford regularityof high powers of ideals. For an ideal in the homogeneous coordinate ringof a projective variety that is of finite colength and is generated by formsof a single degree, this "assymptotic regularity" is connected with theproperties of fibers of the corresponding map of varieties, and Eisenbudwill use this connection to investigate generic projections. A secondarea is the study of schemes that are 2-regular. This work builds on the classificationfor reduced schemes that was recently completed by Eisenbud, Green, Hulek andPopescu. A third area has to do with the computation of higher direct images usingexterior algebra methods. Eisenbud will study certain varieties that appear naturallyin the deformation spaces of bundles in this way. All these areas are supportedby computations based on Groebner basis methods. Eisenbud will collaboratewith Stillman and Grayson in the development of the Macaulay2 program, whichis currently the best tool for computations in projective geometry. Finally, Eisenbudwill collaborate on a project with Diaconis, Holmes, and their students involving theapplication of techniques in algebraic geometry to statistics.An algebraic variety in complex n-space is a set defined bythe simultaneous vanishing of a collection of polynomial functionsof n variables. Many of the important geometric objects that appearin mathematics can be defined this way. One way to study an algebraicvariety is to ask for the dimension of the vector space of polynomialsof degree at most d that vanish on the variety, as a function of d. Thisis called the Hilbert function of the variety. David Hilbert showed how tocompute this function, and refine it, by computing something called afree resolution of the variety. My work uses such free resolutions to studyvarieties in many contexts. For example, if a variety of dimension n is embedded ina vector space of dimension n+1,then the variety is defined by just one equation, a relatively simple case.Most varieties of dimension n can not be embedded in this way, but anyvariety of dimension n can be mapped linearly onto a variety of thesame dimension in an n+1 dimensional space. One of the questions I willstudy is the difference between the original variety and its image, for thebest possible embeddings.