In this paper, we prove that a map u between two polynomial rings, each with an associated Adem relation, is injective. We prove injectivity of u, by first finding formulas for elements within each ring polynomial, and then by computing the map with our associated formulas. After having computed the mapping of u, we then use our computations to show that the kernel of $u$ only contains the zero vector, which proves that the map u is injective. Then having proved that the map u is injective, we then use it to find a basis for u*, the dual map of u.

Author Bio

Louis Atsaves is from Mansfield, Texas. He attended the Oakridge School in Arlington, Texas. While a highschool student in Texas, Louis attended many math competitions which sparked an interest in mathematics. Louis graduated Oakridge and then entered MIT as an undergrad in 2008. At MIT, Louis majored in mathematics and also founded the MIT Math Modeling Club, which served as a training course for the annual MCM (Mathematics Contest in Modeling). As an undergrad at MIT, Louis participated in six research projects and published two papers. His research projects were in the fields of mathematical biology, quantum computation, civil engineering, algebraic topology, partial differential equations, and medicinal biology. Louis currently works at Draper Laboratory in Cambridge, MA and hopes to pursue a PhD. in mathematics next year.