Abstract:  Knot theory is the branch of topology that deals with projections of a simple closed loop in R³ that are planar isotopic to a simple closed polygonal curve with a finite vertex set.  Here we explore the equivalence of the three simplest knots, the unknot, the trefoil knot, and the figure-eight knot using fundamental combinatorial and pictorial methods of knot theory.  More specifically, the knot invariants tricolorability and the Jones Polynomial are proven to maintain their characteristics under Reidemeister moves and are used in proving that the most fundamental projections of the unknot, the trefoil knot, and the figure-eight knot are not topologically equivalent to each other.

Abstract:  This talk will discuss the genital HPV infection, caused by the human papillomavirus, responsible for 70 percent of cervical cancer in the United States.  Recently, a vaccine, Gardasil, was approved for use to defend against the HPV infection and consequently cervical cancer.  To date, models addressing the efficacy of the vaccine have assumed full immunity provided by the vaccine after one injection.  However, Gardasil is administered in three doses over a six-month period.  A Markov model used to determine the affects of the three dose structure on overall instances of HPV and cervical cancer in the United States will be discussed, and specifically how it affects the variance of HPV cases as transmission rates vary across injection states.