Algebraic Methods in Stable Homotopy Theory

DMS-0404651Douglas C. Ravenel这个建议有两个部分：椭圆上同调的高色类似物和寻找更稳定的球面同伦群。 第一部分是关于Ravenel最近的一个发现，即特征P中某些代数曲线的Jacobian有有趣的一维形式群作为形式和数。更准确地说，他有例子，高度可以是p-1的任何倍数。 这可能导致类似的椭圆上同调，这使我们更深入的色塔，这意味着超越v2-周期现象。 这是已知的第一个例子拓扑正式组法律的高度大于2发生在一个几何设置。这里使用的方法来自代数几何和数论。 第二部分涉及球面稳定同伦群的确定和Adams-Novikov谱序列。 这个问题一直是一个中心和最困难的代数拓扑结构，因为集团在1935年被定义。自拉文埃尔1986年出版的《复协ordism and the Stable Homotopy Groups of Spheres》一书（2003年再版）以来，这一领域的进展甚微，该书描述了当时的技术水平。 在它，他确定了稳定的同伦群领域通过尺寸108为p=3和999为p=5。 后一种计算方法是对先验知识的一个实质性改进，而且从那以后也没有改进过。 同伦理论家们普遍认为，试图通过几个词干来提高我们对稳定同伦群的认识是不值得的，但是将已知范围提高p倍的前景是值得追求的。 这种可能性可能是触手可及的现在，由于更好地了解以前使用的方法和改进的计算机技术。这个项目将有助于一般的进步代数拓扑学，一个主题一直是中央纯数学成立以来的庞加莱世纪前。 它一直是代数几何新思想的持续来源，如莱夫谢茨在30年代的工作、60年代和70年代领导韦尔猜想证明的努力，以及最近成功地将动机上同调应用于沃沃茨基的米尔诺猜想。 它在微分几何和理论物理中也有许多应用。 罗切斯特大学是世界上代数拓扑学的主要研究中心之一。

DMS-0404651Douglas C. RavenelThis proposal has two parts: Higher chromatic analogs of elliptic cohomology and finding more stable homotopy groups of spheres. The first part concerns a recent discovery by Ravenel that the Jacobians of certain algebraic curves in characteristic P have interesting 1-dimensional formal groups as formal summands. More precisely, he has examples where the the height can be any multiple of p-1. This could lead to analogs of elliptic cohomology, which takes us deeper into the chromatic tower, meaning beyond v2-periodic phenomena. It is the first example known to topologists of formal group laws of height greater than two occuring in a geometric setting. The methods used here come from algebraic geometry and number theory. The second part concerns the determination of the stable homotopy groups of spheres and the Adams-Novikov spectral sequence. This problem has been one of the central and most difficult in algebraic topology since the groups were defined in 1935. Little progress has been made in this area since the the publication of Ravenel's 1986 book (republished in 2003) "Complex Cobordism and the Stable Homotopy Groups of Spheres" which described the state of the art at the time. In it he determined the stable homotopy groups of spheres through dimension 108 for p=3 and 999 for p=5. The latter computation was a substantial improvement over prior knowledge, and neither has been improved upon since. It is generally agreed among homotopy theorists that it is not worthwhile to try to improve our knowledge of stable homotopy groups by a few stems, but that the prospect of increasing the know range by a factor of p would be worth pursuing. This possibility may be within reach now, due to a better understanding of the previously used methods of and improved computer technology.This project will help the general advance of algebraic topology, a subject which has been central to pure mathematics since its founding by Poincare a century ago. It has been a continuing source of new ideas in algebraic geometry as seen in the work of Lefschetz in the '30s, the efforts leading the proof of the Weil conjectures in the'60s and '70s, and most recently in the successful application of motivic cohomology to the Milnor conjecture by Voevodsky. It has also found numerous applications in differential geometry and theoretical physics. The University of Rochester is one of the leading centers of algebraic topology in the world.