A dedicated search of the CMB sky, driven by implications of conformal
cyclic cosmology (CCC), has revealed a remarkably strong signal, previously
unobserved, of numerous small regions in the CMB sky that would appear to be
individual points on CCC's crossover 3-surface from the previous aeon, most
readily interpreted as the conformally compressed Hawking radiation from
supermassive black holes in the previous aeon, but difficult to explain in
terms of the conventional inflationary picture.

The usual finite dimensional Grassmannians are well known to be classifying spaces for vector bundles. It is maybe a less known fact that one has certain natural connections on the Stiefel bundles over them, which also have a universality property. I will show how these connections are constructed and explain how this viewpoint can be used to rediscover Chern-Weil theory. Finally, we will see how a certain stabilized version of this, called the restricted Grassmannian, admits a similar construction, which can be used to show that it is a smooth classifying space for differential K-theory.

Morse theory explores the topology of a smooth manifold $M$ by looking at the local behaviour of a fixed smooth function $f : M \to \mathbb{R}$. In this talk, I will explain how we can construct ordinary homology by looking at the flow of $\nabla f$ on the manifold. The talk should serve as an introduction to Morse theory for those new to the subject. At the end, I will state a new(ish) proof of the functoriality of Morse homology.

We will discuss some recent results with Martin Bridson about 
Sidki's construction X(G). In particular, if G is a finitely presented
group then X(G) is a finitely presented group. We will discuss as well the
result that if G has polynomial isoperimetric function and the maximal
metabelian quotient of G is virtually nilpotent then X(G) has polynomial
isoperimetric function. Part of the arguments we will use have homological
nature.

Mechanical percolation is a phenomenon in materials processing wherein ‘filler’ rod-like particles are incorporated into polymeric materials to enhance the composite’s mechanical properties. Experiments have well-characterized a nonlinear phase transition from floppy to rigid behavior at a threshold filler concentration, but the underlying mechanism is not well understood. We develop and utilize an iterative graph compression algorithm to demonstrate that this experimental phenomenon coincides with the formation of a spatially extending set of mutually rigid rods (‘rigidity percolation’). First, we verify the efficacy of this method in two-dimensional fiber systems (intersecting line segments), then moving to the more interesting and mechanically representative problem of three-dimensional fiber systems (cylinders). We show that, when the fibers are uniformly distributed both spatially and orientationally, the onset of rigidity percolation appears to co-occur with a mean field prediction that is applicable across a wide range of aspect ratios.

Disease transmission and rumour spreading are ubiquitous in social and technological networks. In this talk, we will present our last results on the modelling of rumour and disease spreading in multilayer networks.  We will derive analytical expressions for the epidemic threshold of the susceptible-infected-susceptible (SIS) and susceptible-infected-recovered dynamics, as well as upper and lower bounds for the disease prevalence in the steady state for the SIS scenario. Using the quasistationary state method, we numerically show the existence of disease localization and the emergence of two or more susceptibility peaks in a multiplex network. Moreover, we will introduce a model of epidemic spreading with awareness, where the disease and information are propagated in different layers with different time scales. We will show that the time scale determines whether the information awareness is beneficial or not to the disease spreading.