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

本文主要研究平方和公式类型ır，s,n！除以整数和实数。在整数情况下，首席研究员将分析r = s规定z1=x1y1+…+xryr的最短公式。他还将研究通过修改已知的Hurwitz-Radon型公式来构造新公式的可能性。在实系数情况下，他将尝试通过改进当前下界来验证Adem猜想。在欧几里得空间等斜平面上的相关几何和拓扑问题，球之间的多项式映射，以及非奇异双线性映射的构造也将被研究。所支持的研究涉及二次型理论。它最简单的形式，是对二阶多项式形式的研究。等价地，它是对可以定义n维向量空间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.