booklet - CUMC - Canadian Mathematical Society

GROUP-STRUCTURES ON SPHERES, AND THE HOPF FIBRATIONSAIFUDDIN SYEDn-Spheres, S n are extremely important topological spaces. As they arise naturally inmany subfields of math and physics, the importance of understanding their structureand properties cannot be overstated. In particular, we will investigate which spherescan be viewed as groups. Spheres such as S 0 and S 1 have naturally imposed groupsassociated with them: Z and U(1), respectively. We shall see that there is a correspondencebetween the class of spheres with this property and real normed divisionalgebras R, C, H, and O. We will then introduce the concept of a fibration and give examples.Finally, we will investigate the relation between these spheres and the projectivespaces of R, C, H, and O, and use it to construct what are called the Hopf fibrations.Time permitting we will discuss an application the Hofp fibration to the quibit systemin quantum physics.Required Background: Basic point set topologySUM OF CUBES IS SQUARE OF SUMSAMER SERAJInspired by intriguing mathematical fact is that for every natural n,1 3 + 2 3 + · · · + n 3 = (1 + 2 + · · · + n) 2 ,we explore, for each n, the Diophantine equation representing all non-trivial sets of nintegers with this propertya 3 1 + a3 2 + · · · + a3 n = (a 1 + a 2 + · · · + a n ) 2 .We find definite answers to the standard question of infinitude of the solutions aswell as several other surprising results. The material is from a paper by Dr. EdwardBarbeau and Samer Seraj, to be published in NNTDM, Vol. 19, No. 1.Required Background: Elementary number theoryGAME THEORY AND S5 MODAL LOGIC: ASYMPTOTICS AND APPLICATIONSSAMUEL REIDWe present a temporal-theoretic formalism for game theory with a motivating exampleof the game of Nim where a winning strategy is presented in the formalism. Weprove that the number of distinct games on a set W with cardinality n is the numberof partial orders on a set of n elements. By generalizing this theorem from temporalmodal frames to S5 modal frames, it is proved that the number of isomorphism classesof S5 modal frames F = < W, R > with |W| = n is equal to the partition function p(n).We use these results to prove that an arbitrary modal frame is an S5 modal frame withprobability zero.Required Background: First-Order Logic44