Applications of Pro-Homotopy Theory to Algebra

点击翻译按钮获取中文摘要

My main goal in this project is to use the homotopy theory of pro-spacesto prove non-existence theorems for sums-of-squares formulas overarbitrary fields. This is a question of primary interest in the study ofquadratic forms, but it is also closely linked to a variety of otherfields of mathematics, including the existence of finite-dimensionaldivision algebras, counting independent vector fields on spheres, andimmersions of projective spaces into euclidean spaces. There is a longhistory of applying cohomological methods to sums-of-squares formulas overthe real numbers. My goal is to adapt these methods so that they work forother fields. This involves the use of generalized etale cohomologytheories (such as but not limited to etale K-theory) of algebraicvarieties instead of generalized cohomology of topological spaces. Thecorrect definition of generalized etale cohomology involves some subtleaspects of the homotopy theory of pro-spaces. Motivic homotopy theory isanother useful tool in studying sums-of-squares formulas and moregenerally quadratic forms. I intend to work on specific problems thatfurther elucidate the relationship between motivic homotopy theory and thetheory of quadratic forms.Cohomology was a key unifying concept in the study of topology throughoutthe twentieth century. The basic principle of cohomology is to turn adifficult problem in topology into a computable algebra problem. Many ofthese cohomological methods work also in algebraic geometry (i.e., thestudy of geometric objects that are defined by polynomial equations), butthe technical details tend to be much more complicated. Traditionalcohomology can be used to prove theorems about the real numbers becausethe real numbers are a topological space. Cohomology in algebraicgeometry allows us to generalize these theorems to other number systems,such as finite fields. The basic goal of my project is to establish someof these generalizations. In short, the point is to use ideas fromalgebraic topology to think about problems in algebra in new ways.