Extending Kervaire invariant methods in stable homotopy theory

这个提议是关于扩展PI和他的合作者（M. a . Hill和M. J. Hopkins）发展的方法，并用于建立具有Kervaire不变量的球体的稳定同伦群中的元素不存在，从而解决了代数拓扑中一个50年的问题。他们的主要定理的结论与该领域的专家们在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。代数拓扑是研究高维几何问题的工具集合。虽然这些想法是抽象的，但它们帮助科学家理解了电子和其他基本粒子的特征，并提供了几年前用于解决数学中最具挑战性的问题之一——费马大定理（Fermat’s Last Theorem）的想法。这对理解结和机器人技术很有用。物理学家利用代数拓扑学试图弄清楚宇宙的形状。它帮助他们理解他们可以观察到的东西，并给他们工具来想象他们无法观察到的东西。它的思想也是“弦理论”的核心，物理学家认为空间不仅有三个维度，而且有10个甚至更多。拓扑学为我们提供了一种处理10维物体的方法，而无需实际观察和触摸它们。

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.