Commutative Algebra, Algebraic Geometry, Computation and Statistical applications

大卫·艾森巴德将对代数几何与同调交换代数混合的一些领域进行研究。其中之一就是研究高次理想的Castelnuovo-Mumford正则性。对于有限长度的射影簇齐次坐标环中由一次形式生成的理想，这种“渐近正则性”与相应的簇映射的纤维的性质有关，而艾森巴德将利用这种联系来研究一般的投影.第二个领域是研究2-正则的方案。这项工作建立在Eisenbud，Green，Hulek和Popescu最近完成的对简化方案的分类的基础上。第三个领域与使用外代数方法计算高次直射图像有关。艾森巴德将以这种方式研究以这种方式自然出现在丛形变空间中的某些品种。所有这些方面都得到了基于Groebner基方法的计算的支持。艾森巴德将与斯蒂尔曼和格雷森合作开发Macaulay2程序，这是目前射影几何计算的最佳工具。最后，艾森巴德将与Diaconis、Holmes以及他们的学生合作一个项目，该项目涉及将代数几何的技术应用于统计学。复n-空间中的代数簇是由n个变量的多项式函数的集合同时消失所定义的集合。许多出现在数学中的重要几何对象都可以这样定义。研究代数簇的一种方法是求在簇上消失的至多d次多项式的向量空间的维度，作为d的函数。这称为簇的Hilbert函数。大卫·希尔伯特展示了如何计算这个函数，并通过计算一种称为变种分辨率的东西来改进它。我的作品在许多背景下使用这种自由的分辨率来研究多样性。例如，如果在n+1维向量空间中嵌入多种维度n，那么这种多样性只用一个方程来定义，这是一个相对简单的情况。大多数维度n的变体不能以这种方式嵌入，但是任何种类的维度n都可以线性地映射到n+1维空间中的相同维度的变体上。我要研究的问题之一是原始品种和它的形象之间的区别，以获得尽可能好的嵌入。

David Eisenbud will perform research on a number ofareas in which algebraic geometry mixes with homological commutativealgebra. One of these is the study of the Castelnuovo-Mumford regularityof high powers of ideals. For an ideal in the homogeneous coordinate ringof a projective variety that is of finite colength and is generated by formsof a single degree, this "assymptotic regularity" is connected with theproperties of fibers of the corresponding map of varieties, and Eisenbudwill use this connection to investigate generic projections. A secondarea is the study of schemes that are 2-regular. This work builds on the classificationfor reduced schemes that was recently completed by Eisenbud, Green, Hulek andPopescu. A third area has to do with the computation of higher direct images usingexterior algebra methods. Eisenbud will study certain varieties that appear naturallyin the deformation spaces of bundles in this way. All these areas are supportedby computations based on Groebner basis methods. Eisenbud will collaboratewith Stillman and Grayson in the development of the Macaulay2 program, whichis currently the best tool for computations in projective geometry. Finally, Eisenbudwill collaborate on a project with Diaconis, Holmes, and their students involving theapplication of techniques in algebraic geometry to statistics.An algebraic variety in complex n-space is a set defined bythe simultaneous vanishing of a collection of polynomial functionsof n variables. Many of the important geometric objects that appearin mathematics can be defined this way. One way to study an algebraicvariety is to ask for the dimension of the vector space of polynomialsof degree at most d that vanish on the variety, as a function of d. Thisis called the Hilbert function of the variety. David Hilbert showed how tocompute this function, and refine it, by computing something called afree resolution of the variety. My work uses such free resolutions to studyvarieties in many contexts. For example, if a variety of dimension n is embedded ina vector space of dimension n+1,then the variety is defined by just one equation, a relatively simple case.Most varieties of dimension n can not be embedded in this way, but anyvariety of dimension n can be mapped linearly onto a variety of thesame dimension in an n+1 dimensional space. One of the questions I willstudy is the difference between the original variety and its image, for thebest possible embeddings.