Extending Kervaire invariant methods in stable homotopy theory

推广了Pi和他的合著者(M.A.Hill和M.J.Hopkins)发展并用来证明具有Kervaire不变的球面的稳定同伦群中元素不存在的方法，从而解决了代数拓扑学中一个已有50年历史的问题。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，这个定理提出的问题和它回答的问题一样多。代数拓扑学是研究高维几何问题的工具的集合。虽然这些概念是抽象的，但它们帮助科学家了解了电子和其他基本粒子的特征，并提供了几年前用来解决数学中最具挑战性的问题之一--费马最后定理--的想法。它对理解结和机器人很有用。物理学家使用代数拓扑学试图弄清楚宇宙的形状。它帮助他们了解他们可以观察到的东西，并给他们提供工具来想象他们不能观察到的东西。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是三维的，而且是10维的，甚至更多。拓扑学为我们提供了一种处理10维对象的方法，而不需要实际看到和接触它们。

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.