This talk is based on two papers by Philip Griffiths and Joseph Harris from the 1970s: “A Poncelet Theorem in Space” and “On Cayley’s explicit solution to Poncelet Porism”. It demonstrates the application of the theory of genus one Riemann surfaces (elliptic curves) in the classical geometry problem known as the Poncelet Theorem.

Poncelet Theorem: Let C and D be two smooth conics generally situated in the projective plane. Then, there exists a closed polygon inscribed in C and circumscribed about D if and only if there are infinitely many such, one with a vertex at any given point in C.

This talk will focus on the techniques used to solve differential and discrete equations, as well as how to study interesting phenomena which solutions exhibit, for example; boundary layers, shifting timescales and asymptotics. The talk will be quite introductory in nature and will require little background. We will start briefly with differential equations and then move onto additive and multiplicative type discrete equations trying to draw analogies between the three.

Knot theory is a field that abounds with simple yet intractably difficult problems, for example, telling if a drawing of a knot can be untangled into a simple circle or decomposing a knot into all of its prime factors. The difficulty of these problems grows greater yet when considering different types of generalised knots, such as virtual knots (knots drawn on orientable surfaces of genus g) or welded knots (knotted rings flying in four dimensions).

This talk will be a brief tour of generalised knot theory. We will look at some of the tools topologists use to tackle these questions, and, time permitting, I will show you a nice counterexample from my honours thesis, proving that some of the tools that work for regular knots no longer work in the generalised setting.