My main goal in this project is to use the homotopy theory of pro-spaces to prove non-existence theorems for sums-of-squares formulas over arbitrary fields. This is a question of primary interest in the study of quadratic forms, but it is also closely linked to a variety of other fields of mathematics, including the existence of finite-dimensional division algebras, counting independent vector fields on spheres, and immersions of projective spaces into euclidean spaces. There is a long history of applying cohomological methods to sums-of-squares formulas over the real numbers. My goal is to adapt these methods so that they work for other fields. This involves the use of generalized etale cohomology theories (such as but not limited to etale K-theory) of algebraic varieties instead of generalized cohomology of topological spaces. The correct definition of generalized etale cohomology involves some subtle aspects of the homotopy theory of pro-spaces. Motivic homotopy theory is another useful tool in studying sums-of-squares formulas and more generally quadratic forms. I intend to work on specific problems that further elucidate the relationship between motivic homotopy theory and the theory of quadratic forms.
Cohomology was a key unifying concept in the study of topology throughout the twentieth century. The basic principle of cohomology is to turn a difficult problem in topology into a computable algebra problem. Many of these cohomological methods work also in algebraic geometry (i.e., the study of geometric objects that are defined by polynomial equations), but the technical details tend to be much more complicated. Traditional cohomology can be used to prove theorems about the real numbers because the real numbers are a topological space. Cohomology in algebraic geometry allows us to generalize these theorems to other number systems, such as finite fields. The basic goal of my project is to establish some of these generalizations. In short, the point is to use ideas from algebraic topology to think about problems in algebra in new ways.
