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.

Subharmonic functions provide a more flexible way of studying holomorphic phenomena in one complex variable; for several complex variables, the natural analogue is plurisubharmonic functions. I will give an overview of this theory and its geometric applications, as well as explaining some background knowledge about complex geometry and what geometric analysis is, to try to keep the talk more self-contained.

The main motivation is that this allows us to study weak solutions to the complex Monge–Ampère equation (an important PDE in complex geometry). In particular, this generalises Yau’s famous proof of the Calabi conjecture to less regular data. I will also motivate trying to extend these techniques to a degenerate case, which I am studying for my PhD thesis.

We consider the Ricci flow in the setting of Kahler manifolds. The Ricci flow is a tool for deforming smooth manifolds with respect to their Ricci curvature in hopes of extracting a nice metric, whereas Kahler manifolds are smooth manifolds equipped with compatible Riemannian, complex, and symplectic structures. These properties enable the study of the Kahler-Ricci flow to be reduced to that of a complex Monge-Ampere type PDE. In the case that the existence time is necessarily finite, we show that under the existence of a map into complex projective space and a cohomology condition, we can find an L^4-like estimate on the Ricci curvature, which enables us to better understand the flow as it approaches its time of singularity.

There are two most notable canonical bases of complex irreducible representations of the symmetric group; the Kazhdan-Lusztig (KL) Basis and the classical Gelfand-Tsetlin (GT) basis. In this talk we recall the constructions of each basis, and introduce a family of generalised GT bases. Then we will explore the relationships between the KL-basis and the classical and generalised GT bases.

The finite-dimensional representations of a complex semisimple Lie algebra \mathcal{g} are well understood. They are completely reducible. This is no longer true for infinite-dimensional representations, where modules can be non-semisimple.

We look at category \mathcal{O}, a certain class of infinite-dimensional representations of a Lie algebra \mathfrak{g}. The Jantzen filtration is a filtration on distinguished objects in category \mathcal{O} which goes some way towards understanding their submodule structure. The filtration can be explicitly computed and has deep ties to Kazhdan-Lusztig theory. In this talk, I will introduce category \mathcal{O} and the Jantzen filtration through basic examples, and explain how the filtration depends on a deformation direction.

The content of this talk is based on Jaydeep Chipalkatti’s paper: “On the Poncelet triangle condition over finite fields”. Let \boldsymbol{P}^2(\mathbb{F}_q) be the projective plane over a finite field of order q with characteristic not equal to 2. A pair of non-singular conics (\mathscr{A},\mathscr{B}) in \boldsymbol{P}^2(\mathbb{F}_q) is said to satisfy the Poncelet triangle condition (PTC) if there exists a triangle inscribed in \mathscr{A} and circumscribed about \mathscr{B}. In this presentation, we will illustrate how to find a pair of non-singular conics satisfying PTC using Cayley’s condition and provide an example of Poncelet triangle construction. Moreover, we will discuss the main ideas leading to Chipalkatti’s main theorem which states that the asymptotic probability of obtaining a pair of non-singular conic satisfying PTC is \frac{1}{q} if char \mathbb{F}_q \neq 3 and \frac{2}{q} if char \mathbb{F}_q = 3.