This research focuses on sums of squares formulae type ?r,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.
