Commutative Algebra, Algebraic Geometry, and Computational Aspects

提议者将工作在交换代数，代数几何和符号计算的问题。这些问题的共同主题是它们都与自由解决方案有关。首先，提出者将尝试理解向量丛在除射影空间之外的其他簇上的上同调表的锥。即使是空间是两条投影线的乘积的情况也是具有挑战性的。在这种情况下，提出者和弗兰克·施赖耶已经确定了圆锥的一些极值射线，并将试图证明它们都是极值射线。该提议者将继续与Persi Diaconis合作，试图为对应于没有庞加莱-伯克霍夫-维特基的Koszul代数的1-依赖过程建立有用的概率模型。最后，提出者将开发新的算法，用于交换代数，代数几何的计算，以及它们可以应用的领域。在许多数学领域及其应用中的一个基本工具涉及到寻找随某些参数变化的线性方程组的解，这些解本身以简单的方式变化-例如，作为相同参数的多项式函数。提议者的工作扩展了使用此工具的方法（计算和理论）及其适用范围。这种技术的一种较新的应用的一个例子是研究所谓的"单依赖“统计过程，它可以被认为是连续实验的结果，其中下一个实验的结果取决于当前实验的结果，但是除了知道过去的结果之外，并没有提供任何进一步的信息。（想象一下，反复投掷一枚硬币，当正面朝上时，有2/3的机会落在“正面”上，当反面朝上时，有1/2的机会落在“正面”上。抛硬币多次，但总是从硬币落地的位置（正面朝上或反面朝上）抛硬币，所产生的正面和反面的序列是一个依赖于一的过程。）在与Persi Diaconis的合作中，提议者希望构建有趣的新1-依赖过程的概率定律。

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.