booklet - CUMC - Canadian Mathematical Society

theory have been extended to virtual knots. This talk will include an introduction tovirtual braid theory and the virtual braid group, highlighting some recent results andconnections to virtual knot theory.ROTH’S THEOREM IN THE PRIMESERIC NASLUNDIn 1939, Van Der Corput proved that the set of primes P contains infinitely manynon trivial three term arithmetic progressions. In 2003, Green proved an anologue ofRoth’s theorem, and showed that any subset A ⊂ P with positive relative density mustcontain infinitely many three term arithmetic progressions. Suppose that A ⊂ {1, . . . , N}is a set of primes containing, and let α = |A|/π(N) be the relative density of A in{1, . . . , N}, where π(N) denotes the number of primes in the set {1, . . . , N}. Helfgottand Roton improved Green’s quantitative result, proving A must contain a three termarithmetic progression whenlog log log Nα ≫(log log N) 1/3 ,We improve their density bound, showing that if there exists ɛ > 0 such thatα ≫ ɛ1(log log N) 1−ɛ ,then A contains a three term arithmetic progression.Required Background: AnalysisTHE FIVE COLOUR THEOREMERIN MEGERIf we want to colour a map so that countries with a common border have differentcolours, how many colours do we need? In 1976, it was shown, with the use of acomputer, that we can always do this with at most four colours. This is known as the4-colour theorem, which has yet to be reproduced by a human. With only a few resultsfrom graph theory, it is easily shown that we require at most 5 colours (the 5-colourtheorem). I will be presenting Heawood’s 1890 proof of this theorem.27