Commutative Algebra and Algebraic Geometry

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Algebraic geometry is the study of the qualitative properties of forms described by polynomial equations. The questions it addresses arise in many areas of science and mathematics, from statistical sampling to robotics and mathematical physics. This research project concerns the study of aspects of algebraic geometry related to syzygy theory, a sort of microscope that allows one to see details of polynomial equations that are not apparent from the equations themselves.The investigator will work on problems in commutative algebra, algebraic geometry and computational methods for these fields. He will focus on four areas: 1) Cohen-Macaulay and Ulrich complexity of hypersurfaces: A central problem is to describe the minimal size of a non-free maximal Cohen-Macaulay module over a hypersurface, in particular over the permanent hypersurfaces. 2) The structure of high syzygies over complete intersections: Central problems are to give criteria for a module over a complete intersection to be a "high syzygy" in the sense of Eisenbud and Peeva; and to understand the relations between the even and odd parts of the infinite minimal free resolutions of modules over a complete intersection. 3) Duality for residual intersections: The central remaining open problem is to prove and extend a conjecture of van Straten and Warmt on the socle of a particular representation of the canonical module of a 0-dimensional residual intersection. 4) Tate resolutions for complexes of sheaves on a toric variety: The central problem is to define and compute a good analogue for toric varieties of the Tate resolution of a sheaf or a complex of sheaves on a projective space.