Spiders are mostly blind. Yet they manage to `see’ the world by processing the vibration data they collect from their webs. In a sense, the spider’s web is an extension of the spider’s body. Would a dense enough spider’s web approximate continuous space, thereby allowing the spider to apply standard techniques from calculus? While it seems that spiders have lasted long enough without working this out, I will discuss a continuous Laplace-type operator that allows us to study diffusive-wave-quantum behaviour on high-density networks.

Recently I have developed a great interest in the stochastic interpretation of deterministic PDEs. In this informal talk, I will present some basic aspects of this theory, and hopefully illuminate some essential themes in modern PDE analysis with which I believe every mathematician should be acquainted. (Thus “random” in the title is interpreted both in the colloquial and mathematically rigorous senses.)

Our protagonists in this story will be the heat equation and Brownian motion. We will encounter a cast of colourful characters including stochastic integrals, It^{o} diffusions, and operator semigroups and their generators.

In this talk I will provide a brief introduction to the research of mathematics education, focusing on the tertiary level as it is most applicable to members of our School. A selection of educational theories and frameworks will be introduced in general and then applied to the context of mathematics. Finally, we will discuss what motivates and emanates from academic research in tertiary mathematics education, and why it may be beneficial (or even essential) to be on our individual agendas and the School as a whole (with some interesting case studies presented).

Mathematical models constructed to model real-world systems are often complex, nonlinear and of high dimensions, which could make them hard to tackle, either analytically or numerically. It is often desirable if we could find a simpler equation with fewer dimensions which can still capture the dynamics of the original system well.

In the setting of differential equations with multiple time-scales, in which dominant variables evolve on a time-scale much slower than that of the other variables, there are results which show that, under certain conditions, such reduced models can be obtained via techniques known as averaging and homogenization. It turns out that the effective dynamics of the slow variables is often in the form of stochastic differential equations, with coefficients determined from the dynamics of the fast variables. In this talk I will present some derivation of reduced dynamics via perturbation expansion of the corresponding backward Kolmogorov equations, together with some numerical demonstrations.

Multistability is a common phenomenon which naturally occurs in complex networks. If coexisting attractors are numerous and their basins of attraction are complexly interwoven, the long-term response to a perturbation can be highly uncertain. I will discuss a paper I read which examines the uncertainty in the outcome of perturbations to the synchronous state in a Kuramoto-like representation of the British power grid, and how this phenomenon can be related to transient chaos through chaotic invariant sets in the basin landscape.

Consider an ant eating its way through a Krispy Kreme. For whatever reason, you want to track how many times the ant winds around the hole in the centre. The duality between winding numbers and intersection numbers means you only need to look at how the ant travels through one appropriately chosen cross-section of the donut to do this. In a sense, this means we can glean global information about the ant’s path from local info only.

The Maslov index is a winding number for a path of Lagrangian subspaces of a symplectic Hilbert space. It can be used to generalise Sturm-Liouville theory to Hamiltonian systems. In this talk, I will discuss the basic definitions and topological properties of the Maslov index, ants eating their way through donuts, and an application to the spectral (in)stability problem of a nonlinear wave.