Algebraic Methods in Stable Homotopy Theory

DMS-0404651 Douglas C.Ravenel这个建议包括两个部分：椭圆上同调的高色类比和寻找更稳定的球面同伦群。第一部分涉及Ravenel最近的一个发现，即特征P中某些代数曲线的Jacobian具有有趣的一维形式群作为形式和。更准确地说，他举了一些例子，其中高度可以是p-1的任何倍数。这可能会导致类似的椭圆上同调，这会让我们更深入到色塔，这意味着超越了v2周期现象。这是拓扑学家已知的第一个在几何环境中出现的高度大于两个的形式群律的例子。这里使用的方法来自于代数几何和数论。第二部分是球面稳定同伦群和Adams-Novikov谱序列的确定。自从1935年定义群以来，这个问题一直是代数拓扑学中的中心问题之一，也是最困难的问题之一。自Ravenel 1986年出版的书(2003年重新出版)以来，这一领域几乎没有取得什么进展，这本书描述了当时的艺术状态。在它中，他确定了球面的稳定同伦群，当p=3时，他通过维108确定了稳定同伦群，当p=5时，他通过维999确定了球面的稳定同伦群。后一种计算比先前的知识有了很大的改进，而且从那以后都没有改进过。同伦理论家普遍认为，试图通过几个词干来改进我们对稳定同伦群的了解是不值得的，但将知道范围增加一个p倍的前景将是值得追求的。由于更好地理解了以前使用的方法和改进的计算机技术，这种可能性现在可能是可以实现的。这个项目将有助于代数拓扑学的全面发展，自一个世纪前庞加莱创立以来，代数拓扑学一直是纯数学的核心学科。它一直是代数几何中新思想的源泉，从30年代Lefschetz的工作中可以看到，在60年代和70年代领导Weil猜想证明的努力，以及最近Voevodsky对Milnor猜想的成功应用，它也在微分几何和理论物理中得到了大量的应用。罗切斯特大学是世界上领先的代数拓扑学中心之一。

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.