Applications of Pro-Homotopy Theory to Algebra

原同伦理论在代数中的应用

• 批准号: 0503720
• 负责人: Daniel Isaksen
• 金额: --
• 依托单位: Wayne State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-06-01 至 2009-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0503720&HistoricalAwards=false
• 关键词: Applications; Pro; Homotopy; Theory; Algebra

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.

我在这个项目中的主要目标是利用原空间的同伦理论证明任意域上平方和公式的不存在定理。这是二次型研究中的一个主要兴趣问题，但它也与数学的许多其他领域密切相关，包括有限维除代数的存在性，球面上独立向量场的计数，以及射影空间到欧氏空间的融合。将上同调方法应用于实数上的平方和公式有着悠久的历史。我的目标是调整这些方法，使其适用于其他领域。这涉及到使用代数簇的广义上同调理论(例如但不限于代数簇的广义上同调理论)，而不是拓扑空间的广义上同调。广义上同调的正确定义涉及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。我打算致力于具体的问题，以进一步阐明动机同伦理论和二次型理论之间的关系。上同调是整个二十世纪拓扑学研究中的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的代数问题。许多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对象)，但技术细节往往要复杂得多。由于实数是一个拓扑空间，所以传统的酒类同调可以用来证明关于实数的定理。代数几何中的上同调允许我们将这些定理推广到其他数字系统，例如有限域。我的项目的基本目标是建立其中的一些概括。简而言之，重点是用几何拓扑学的思想以新的方式思考代数中的问题。