Extending Kervaire invariant methods in stable homotopy theory

这个建议是关于扩展PI和他的合著者（M。A. Hill和M. J.霍普金斯）发展并用于建立球的稳定同伦群中元素的不存在性，其中Kervaire不变1，从而解决了代数拓扑学中的一个50年前的问题。 他们的主要定理的结论与该领域的专家在20世纪70年代所希望的相反。 由于这个原因，定理提出了许多问题，因为它的答案。代数拓扑学是一个收集的工具，研究问题，在高维几何。 虽然这些想法很抽象，但它们帮助科学家理解了电子和其他基本粒子的特性，并提供了几年前用来解决数学中最具挑战性的问题之一--费马大定理的想法。这对于理解节点和机器人技术很有用。物理学家使用代数拓扑学试图找出宇宙的形状。它帮助他们理解他们能观察到什么，并给他们工具去想象他们不能观察到什么。它的思想也是“弦理论”的核心，在弦理论中，物理学家认为空间不仅仅是三维的，而是10维甚至更多。 拓扑学给了我们一种方法来处理10维物体，而不必实际看到和触摸它们。

This proposal is about extending the methods that the PI and his coauthors (M. A. Hill and M. J. Hopkins) developed and used to establish the nonexistence of elements in the stable homotopy groups of spheres with Kervaire invariant one, thereby solving a 50 year old problem in algebraic topology. The conclusion of their main theorem is the opposite of what experts in the field were hoping for in 1970s. For this reason the theorem raises as many questions as it answers.Algebraic topology is a collection of tools for studying problems in higher dimensional geometry. While the ideas are abstract, they have helped scientists understand the characteristics of electrons and other elementary particles and provided ideas used a few years ago to solve one of mathematics' most challenging problems, Fermat's Last Theorem. It is useful for understanding knots and robotics. Physicists use algebraic topology to try to figure out the shape of the universe. It helps them understand what they can observe and gives them tools to imagine what they cannot observe. Its ideas are also central to "string theory," where physicists conceive of space as having not just three dimensions but 10 or even more. Topology gives us a way to deal with 10-dimensional objects without actually seeing and touching them.