Chromatic stable homotopy theory

色稳定同伦理论

• 批准号: 0905160
• 负责人: Douglas Ravenel
• 金额: $ 17.49万
• 依托单位: University of Rochester
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-09-01 至 2013-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0905160&HistoricalAwards=false
• 关键词: Chromatic; stable; homotopy; theory

This research concerns two approaches to the stable homotopy groups of spheres. The first is joint work with Hirofumi Nakai of Japan on extending the methods described in the last Chapter of the PI's book "Complex Cobordism and Stable Homotopy Groups of Spheres." The second is joint work with Mike Hill of the University of Virginia and Mike Hopkins of Harvard University. It involves studying the homotopy fixed point sets of finite subgroups of the Morava stabilizer groups.Since the proposal was written, the PI and his two American collaborators have used the method in it (and a new one) to solve the 45 year old Arf-Kervaire invariant problem, which was one of the biggest questions in our field. In the 1970s there were several unsuccessful attempts to solve it by showing that there are infinitely many dimensions in which the invariant of a framed manifold (a certain kind of higher dimensional shape) can be nonzero. Our result is that there are at most six (five are currently known) such dimensions. This statement is so surprising that it is now known as the Doomsday Theorem. The techniques we developed to prove it are quite different from previous approaches and could be applicable to other problems in topology and mathematical physics.

这项研究涉及稳定的球体组稳定均值组的两种方法。 第一个是与日本的Hirofumi Nakai的联合合作，以扩展PI的书《复杂的恢复和稳定的球形群体》的最后一章中描述的方法。第二个是与弗吉尼亚大学的迈克·希尔（Mike Hill）和哈佛大学的迈克·霍普金斯（Mike Hopkins）的联合合作。 它涉及研究Morava稳定器组的有限亚组的同置固定点集。从提案中，PI和他的两个美国合作者使用了其中的方法（以及一种新方法）来解决45岁的ARF-KERVAIR不变性问题，这是我们领域中最大的问题之一。 在1970年代，通过证明有许多维度的尺寸（一种较高的较高形状）可能是非零的，有几项未成功解决它的尝试来解决它。 我们的结果是，此类维度最多有六个（目前为五个）。 这种说法是如此令人惊讶，以至于现在被称为世界末日定理。我们开发的技术证明它与以前的方法完全不同，并且可能适用于拓扑和数学物理学的其他问题。