Conference Proposal in Support of Young Investigators

研究者和他的同事们研究代数几何和交换代数的重叠问题。一个主要目标是理解齐次理想的最小自由分辨率结构，特别是那些定义零维方案的结构。感兴趣的理想范围从肥胖点的理想（由于它们与射影变上的线性系统问题的密切联系），到一般形式的理想（其工作与阿提尼代数的弱和强Lefschetz性质的工作密切相关，并通过矩阵对偶性，对肥胖点的工作），到商数为Gorenstein环的理想（因此涉及射影空间的算术Gorenstein子方案和Gorenstein连接的问题）。这种广泛的方法和动机是年轻和未来的研究人员必须克服的一个重大障碍，以便在这一领域开展富有成效的工作。该资助旨在通过提供旅费支持，促进年轻研究人员和研究生的培训和职业发展，以参加针对上述研究的国际会议，并与该领域的领导者进行互动。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，也是某些研究的对象。然而，这一领域的工作很容易超出任何可想象的计算机的能力，因此理论研究不仅对于理解机器计算的结果，而且对于实现暴力计算无法达到的结果至关重要。例如，这是将本文描述的研究应用于插值的一个重要问题。无论是在科学、技术还是商业领域，大数据集都是现代生活的一个普遍特征。这类数据集通常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。如果一个人想用多项式来建立关系模型，从理论上知道这种多项式最坏的情况可能有多复杂是有帮助的（比如，用程度来衡量），而直接确定它可能是昂贵的或不可能的。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辨率的研究者所追求的。

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.