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.
