Περίληψη: This is an elementary talk about the Mobius function and its role in elementary number theory and its relationship with the Riemann Zeta function. We show how, via the structure of Legendre’s prime counting function, the Mobius function is expressed recursively in terms of itself. This recursion shows clearly how the Mobius function is far from random. It is more like a magic coin that will not tolerate long runs of heads (+1) or tails (-1). Thus Mobius behaves more sedately than a simple random coin. This gives a heuristic argument in favor of the Riemann Hypothesis. We will discuss these matters with the help of the computer.

Τίτλος: “Generalized Colorings, Penrose Formulas and Multiple Virtual Knot Theory”

Περίληψη: This self-contained talk discusses a generalization of virtual knot theory. Multi-virtual knot theory uses a multiplicity of types of virtual crossings. As we will explain, this multiplicity is motivated by the way it arises first in a graph-theoretic setting. As a consequence, we begin with graph theory, and then proceed to the topology. We begin by discussion on the work of Roger Penrose in his paper “Applications of Negative Dimensional Tensors” which contains a remarkable method for counting the Tait colorings of a planar cubic graph. We generalize the Penrose formula to colorings of arbitrary cubic graphs and generalized colorings of cubic graphs with specific perfect matchings. These graph theoretic explorations are of interest in their own right and they involve a use of multiple virtual crossings that sows the seed for the multi-virtual knot theory.