Commutative Algebra, Algebraic Geometry, and Computational Aspects

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The proposer will work on problems in commutative algebra, algebraic geometry, and symbolic computation. The common theme in these problems is that they are all related to free resolutions. First, the proposer will try to understand the cone of cohomology tables of vector bundles on varieties other than projective spaces. Even the case where the space is the product of two projective lines is challenging. In this case the proposer, with Frank Schreyer, has identified some extremal rays of the cone, and will try to show that they are all the extremal rays. The proposer will continue his collaboration with Persi Diaconis, trying to make useful probabilistic models for the 1-dependent processes corresponding to Koszul algebras that do not have Poincar'e-Birkhoff-Witt bases. Finally, the proposer will develop new algorithms for computation in commutative algebra, algebraic geometry, and the fields to which they can be applied.A fundamental tool in many fields of mathematics and its applications involves finding solutions to systems of linear equations that vary with some parameters, solutions that themselves vary in a simple way--for example, as polynomial functions of the same parameters. The proposer's work extends the methods (both computational and theoretical) for using this tool, and the range of its applicability. One example of a relatively new kind of application of this technique is the study of so-called ``one-dependent'' statistical processes, which may be thought of as the results of successive experiments where the result of the next experiment depends on the result of the current one, but where knowing in addition the results of the past ones does not give any further information. (Imagine repeatedly tossing a coin that has a 2/3 chance of landing "heads" when tossed with heads up, and a 1/2 chance of landing "heads" when tossed with tails up. The sequence of heads and tails that results from tossing the coin many times, but always tossing it from the positions (heads or tails up) where it landed is a one-dependent process.) In collaboration with Persi Diaconis the proposer hopes to construct the probability laws of interesting new 1-dependent processes.