Conference Proposal in Support of Young Investigators

调查员和他的同事们研究问题的代数几何和交换代数。一个主要目标是了解齐次理想的极小自由解的结构，特别是那些定义零维方案的结构。理想的兴趣范围从理想的脂肪点（动机是他们的密切联系的问题，线性系统的投影品种），理想的一般形式（工作是密切相关的工作弱和强Lefschetz性质的阿廷代数，并通过Matlis对偶，工作的脂肪点），理想的同功是Gorenstein环（因此涉及问题的算术Gorenstein子计划的射影空间，和Gorenstein联络）。这种广泛的方法和动机是一个重要的障碍，年轻和未来的研究人员必须克服在这一领域做富有成效的工作。该基金旨在通过提供旅行支持来促进年轻研究人员和研究生的培训和职业发展，以便参加解决上述研究问题的国际会议并与该领域的领导者互动。计算问题是这项研究的基础，机器计算既是上述研究的重要工具，也是一些研究的对象。然而，在这一领域的工作可以很容易地超越任何可以想象的计算机的能力，因此理论研究不仅是理解机器计算的结果，而且是实现暴力计算无法达到的结果。这是，例如，一个重要的问题，这里所描述的研究应用插值。大型数据集是现代生活中的一个共同特征，无论是在科学、技术还是商业领域，这类数据集往往涉及各种变量之间的函数关系。多项式是最容易处理的函数之一。如果想用多项式来建模关系，那么从理论上知道这种多项式的最坏情况有多复杂（比如用次数来衡量）是很有帮助的，而直接确定这一点可能成本很高，或者是不可能的。这些关于多项式的理论结果是研究齐次理想的最小自由分解的目标。

The investigator and his colleagues study problems in the overlapof algebraic geometry and commutative algebra. A major goalis to understand the structure of minimal free resolutionsof 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 towork 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 workin this area. This grant aims to advance the training and career development of young investigators and graduate students by providingtravel 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 acommon feature of modern life, whether in science, technology or business.Such data sets often involve functional relationships among variousvariables. 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 todetermine this directly. Such theoretical results about polynomialsare what researchers aim for in studying minimal free resolutionsof homogeneous ideals.