Applications of Pro-Homotopy Theory to Algebra

在这个项目中，我的主要目标是使用pro-space的同伦理论来证明任意域上平方和公式的不存在性定理。 这是二次型研究中的一个主要兴趣问题，但它也与其他数学领域密切相关，包括有限维可除代数的存在性，球面上独立向量场的计数，以及射影空间到欧氏空间的映射。将上同调方法应用于真实的数的平方和公式有着悠久的历史。我的目标是使这些方法适用于其他领域。这涉及到代数簇的广义余同调理论（例如但不限于余K-理论）的使用，而不是拓扑空间的广义上同调。广义Etale上同调的正确定义涉及到原空间同伦理论的一些微妙方面。动机同伦理论是研究平方和公式和更一般的二次型的另一个有用的工具。 我打算研究具体的问题，进一步阐明动机同伦理论和二次型理论之间的关系。上同调是整个世纪拓扑学研究中一个关键的统一概念。 上同调的基本原理是将拓扑学中的难题转化为可计算代数问题。许多这些上同调方法也适用于代数几何（即，研究由多项式方程定义的几何物体），但技术细节往往要复杂得多。由于真实的数是一个拓扑空间，所以可以用代数上同调来证明关于真实的数的定理。 代数几何中的上同调使我们能够将这些定理推广到其他的数制，如有限域。 我的项目的基本目标是建立一些这样的概括。简而言之，关键是要用几何拓扑的思想来用新的方式思考代数问题。

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.