Commutative Algebra, Algebraic Geometry, Computation and Statistical applications

大卫艾森巴德将进行研究的一些领域，其中代数几何混合同调交换代数。其中之一是对理想的高幂的Castelnuovo-Mumford正则性的研究。对于一个理想的齐次坐标环的一个投影品种是有限的collength和所产生的formsof一个单一的程度，这种“渐近正则性”是连接的性质纤维的相应地图的品种，和艾森巴德将使用这种连接调查一般的投影。第二个领域是2-正则概型的研究。这项工作建立在最近完成的艾森巴德，绿色，Hulek和Popescu的classificationfor减少计划。第三个领域与计算更高的直接图像usingexternal代数方法。Eisenbud将用这种方法研究在丛的变形空间中自然出现的某些变种。所有这些区域都用Groebner基方法计算得到。Eisenbud将与Stillman和Grayson合作开发Macaulay 2程序，该程序是目前射影几何计算的最佳工具。最后，Eisenbud将与Diaconis，Holmes和他们的学生合作一个项目，涉及代数几何技术在统计中的应用。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定义的集合。数学中出现的许多重要的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为零的次数至多为d的多项式的向量空间的维数，作为d的函数。这就是所谓的希尔伯特函数的品种。大卫希尔伯特展示了如何计算这个函数，并通过计算变量的自由分解来改进它。我的作品使用这样的自由决议来研究在许多情况下的品种。例如，如果一个维数为n的簇嵌入到维数为n+1的向量空间中，那么这个簇只由一个方程定义，这是一个相对简单的情况。大多数维数为n的簇不能以这种方式嵌入，但任何维数为n的簇都可以线性映射到一个维数为n+1的向量空间中的同一个簇上。我将研究的问题之一是原始变种和它的图像之间的差异，以获得最佳的嵌入。

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.