Topology is a subject of mathematics that studies the shape of spaces, and homotopy theory studies how spaces can be deformed to each other without tearing things up or puncturing holes in the process. Among all spaces, the spheres are the most fundamental and beautiful objects. A central question in homotopy theory and algebraic topology - the computation of homotopy groups of spheres - is to understand how spheres can be mapped to other spheres, in different dimensions, and to understand how different maps can or cannot be deformed to each other. This topological question is not just a fundamental question on fundamental mathematical objects, but it also has deep connections and interactions to other subjects in mathematics. It has been a major research question since 1950's. For example, Kervaire and Milnor established a connection to the question on how many ways one could do calculus on spheres - smooth structures on spheres; Kervaire and Browder built a connection to the question on how one could or could not do surgeries to higher dimensional spaces to turn them into spheres - the Kervaire invariant problem. Moreover, a summand of the stable homotopy groups of spheres, the image of J, is closely related to the Bernoulli numbers; Quillen and Goerss, Hopkins, Miller, Lurie established a connection to the moduli stack of formal groups. More recently, Hill, Hopkins, Ravenel, Voevodsky, Morel, Isaksen and others established connections between equivariant homotopy theory and motivic homotopy theory. The goal of this project is to deepen the existing connections, as well as discovering new connections by pushing further the limit of existing ones.
This research concentrates on computations of stable homotopy groups of spheres, with interactions among motivic, equivariant and chromatic homotopy theory, and applications to problems in differential topology, such as uniqueness of smooth structure on spheres and the Kervaire invariant problem. More specifically, in current and ongoing projects with Isaksen and Wang, the Principle Investigator (PI) develops new computational tools in motivic homotopy theory, with connection to chromatic homotopy theory, which computes 40 more new stems of classical stable homotopy groups of sphere within two years. The PI will deepen the new connection between motivic homotopy theory and chromatic homotopy theory, carry out more computations of stable stems in the next a few years, and use the computations to attack the last unsolved case of the Kervaire invariant problem in dimension 126. The PI will also explore connections between real motivic homotopy theory and C2 equivariant homotopy theory, following ongoing work of Behrens, Dugger, Guillou and Isaksen. The goal is to prove structural theorems in this direction, as well as providing concrete computational results. In ongoing projects with Hill, Shi and Wang, the PI will also apply equivariant techniques, such as the slice spectral sequences that are developed by Hill-Hopkins-Ravenel, to do computations in heights greater than 2 in chromatic homotopy theory and to understand its connection to stable homotopy groups of spheres.
This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.
