Finding solutions to polynomials in the integers or rational numbers arises naturally while counting objects subject to various constraints. It is also one of the oldest problems in mathematics. The solutions to the same polynomials over the complex numbers form topological spaces. For example, the complex solutions to the degree n Fermat equation is a torus with (n-1)(n-2)/2 holes. The fact that the shape of the space of complex solutions influences the solutions over rational numbers or integers can be viewed as a first instance of the utility of using methods of homotopy theory to study this problem. Homotopy theory gives machinery to replace a procedure by a derived version which frequently gives more control over the problem. For example, consider the procedure of taking a rotating sphere and returning the points which do not move. This procedure can be derived to produce a space called the homotopy fixed points, which records not only the points which do not move, but also compatible paths between points and where they have traveled. Fixed points and homotopy fixed points are equivalent under some hypotheses. If fixed points and homotopy fixed points are equivalent for a certain analogue of the space of complex solutions of polynomial equations, one can show that the solutions to these polynomials over the rationals are then determined by the loops on the corresponding space of complex solutions, under certain restrictions. This latter prediction is part of Grothendieck's anabelian program, and is unsolved. It produces strong restrictions on the solutions of the corresponding equations. The focus of this proposal is to study solutions to polynomial equations and Grothendieck's anabelian program from this perspective. The project also aims to stimulate research in homotopy theory, and make the tools of this theory available to mathematicians in very different areas, and other scientists more generally.
The projects in this proposal share the approach wherein one views a scheme as a space in the sense of Morel-Voevodsky's A1-homotopy theory, and then applies various realization functors, for instance to Z/2-equivariant spaces by taking C-points of a scheme over R, or to pro-spaces with an action of the absolute Galois group of the base field for schemes over more general fields. The Principal Investigator studies the pro-space maps from the étale homotopy type of a field k to the étale homotopy type of the projective line minus three points using lower central series approximations to the latter. Additionally James-Hopf maps in A1-homotopy theory are used to study the same mapping space. Both have applications to Grothendieck's anabelian program. Running the same methods backwards, produces results on the algebraic topology of schemes starting from information about solutions to polynomial equations. For instance, the Principal Investigator continues a study of the differential graded algebra associated to group cohomology of absolute Galois groups. Information about the unstable category of spaces in the sense of Morel-Voevodsky is sought in conjunction.
