Mathematical Sciences: The Hurwitz Problem on Sums of Squares and Related Topics

本研究主要研究ır，S，n！在整数和实数上。在整数情况下，首席调查员将分析r=S的最短公式，其中指定的z1=x1y1…Xryr.他还将研究通过修改已知的Hurwitz-Radon型公式来构建新公式的可能性。在实际系数的情况下，他将试图通过改进目前的下限来验证Adem猜想。还将研究欧氏空间中等倾面上的相关几何和拓扑问题，球面之间的多项式映射，以及非奇异双线性映射的构造。所支持的研究涉及二次型理论。它最简单的形式是研究二次多项式形式。等价地，它是对可以定义n维向量空间的度量几何的内积类型的分析。二次型的研究与代数几何和代数k-理论有着密切的联系。

This research focuses on sums of squares formulae type ır,s,n! over the integers and the reals. In the integer case, the principal investigator will analyze for r = s the shortest formulae with prescribed z1=x1y1+...+xryr. He will also investigate the possibility of constructing new formulae by modifying known formulae of Hurwitz-Radon type. In the real coefficients case, he will try to validate the Adem conjecture by improving current lower bounds. Related geometric and topological problems on isoclinic planes in euclidean spaces, polynomial maps between spheres, and the construction of nonsingular bilinear maps will also be studied. The research supported involves the theory of quadratic forms. This, in its simplest form, is the study of polynomial forms of degree two. Equivalently, it is an analysis of the types of inner products that can define the metric geometry of an n-dimensional vector space. The study of quadratic forms has deep interrelations with algebraic geometry and algebraic k-theory.