This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).
The Principal Investigator will work on problems relating arithmetic properties of classical objects of study in algebraic geometry, i.e., so-called rational and nearly rational varieties, to their geometry by means of extremely refined invariants rooted in topology. Together with F. Morel, he will investigate the problem of classifying such varieties using recently introduced techniques inspired by the celebrated Browder-Novikov-Sullivan-Wall classification of manifolds. Together with B. Doran, he will continue to investigate invariant theory for unipotent group actions, its relationship to construction of A^1-contractible varieties, and the problem of characterizing affine space as an algebraic variety. By their very nature, these problems draw together several branches of mathematics and thus illustrate the fundamental unity of the subject.
Algebraic geometry, one of the oldest branches of mathematics, is at its core concerned with the study of algebraic varieties, i.e., solutions to systems of polynomial equations in many variables. Algebraic topology studies the problem of attaching invariants, e.g., numbers or, more generally, so-called abstract algebraic structures to spaces; the most important property of such invariants is that they do not change as the underlying geometric object is pulled and twisted, so long as it is not torn in the process. A fundamental problem in algebraic geometry is classification of algebraic varieties, i.e., explicit determination and taxonomy of the possible configurations of solutions. When solutions to systems of equations have topological structure (think of the equation defining a sphere), one can try to distinguish them by means of invariants. However, solutions to systems of polynomial equations arising in arithmetic do not always have obvious topological structure (think of the integer solutions to the equation defining a unit sphere). Nevertheless, the relatively recently introduced subject of A^1-homotopy theory provides a framework in which one may apply the full power of techniques of algebraic topology to algebraic varieties--spaces having deep arithmetic (but a priori limited geometric) structure are, from the standpoint of invariants, put on equal footing with those that are more inherently geometric. The aim of this project is to further amalgamate ideas of arithmetic, topology, and algebraic geometry by transplanting the fantastically successful method of classifying topological spaces via surgery, a cutting and pasting procedure very carefully controlled by appropriate invariants, into algebraic geometry by means of A^1-homotopy theory.
