Conference Proposal in Support of Young Investigators

这位研究人员和他的同事研究了代数几何和交换代数的重叠问题。理解齐次理想的最小自由分解的结构，特别是定义零维格式的最小自由分解的一个主要目标。感兴趣的理想的范围从脂肪点的理想(由它们的密切联系到关于射影簇上的线性系统的问题所激励)，到一般形式的理想(关于它的工作与关于Artin代数的弱和强Lefschetz性质的工作密切相关，并且通过Matlis对偶，关于肥点的工作)，到其商是Gorenstein环的理想(因此涉及到射影空间的算术Gorenstein子方案和Gorenstein联络的问题)。这种广泛的方法和动机是年轻的和未来的研究人员在这一领域进行富有成效的工作必须克服的一个重大障碍。这笔赠款旨在通过提供差旅支持，促进年轻研究人员和研究生的培训和职业发展，以便参加关于上述研究的国际会议并与该领域的领导人互动。计算问题是这项研究的大部分基础，而机器计算既是上述研究中的重要工具，也是本研究的一些对象。然而，这一领域的工作很容易超过任何可以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至关重要，而且对于实现暴力计算之外的结果也是必不可少的。例如，对于将这里描述的研究应用于内插而言，这是一个重要的问题。无论是在科学、技术还是商业领域，大数据集都是现代生活中常见的特征，这类数据集往往涉及到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模关系，从理论上知道这种多项式在最坏的情况下可能有多复杂是很有帮助的(比方说，按次数来衡量)，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解的目的。

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.