I will talk about link and three-manifold invariants defined in terms of a non-semisimple finite ribbon category C together with a choice of tensor ideal and a trace on it. If the ideal is all of C, these invariants agree with those defined by Lyubashenko in the 90’s, and as we show, they only depend on the Grothendieck class of the objects labelling the link. These invariants are therefore not able to determine non-split extensions, or they are algebraically weak. However, we observed an interesting phenomenon: if one chooses an intermediate proper ideal between C and the minimal ideal of projective objects, the invariants become algebraically much stronger because they do distinguish non-trivial extensions. This is demonstrated in the case of C being the super-modular category of an exterior algebra. That is why these invariants deserve to be called “non-semisimple”. This is a joint work with J. Berger and I. Runkel.

In this talk, I will describe some of the ways in which mathematical modelling contributed to the Covid-19 pandemic response in New Zealand. New Zealand adopted an elimination strategy at the beginning of the pandemic and used a combination of public health measures and border restrictions to keep incidence of Covid-19 low until high vaccination rates were achieved. The low or zero prevalence for first 18 months of the pandemic called for a different set of modelling tools compared to high-prevalence settings. It also generated some unique data that can give valuable insights into epidemiological characteristics and dynamics. As well as describing some of the modelling approaches used, I will reflect on the value modelling can add to decision making and some of the challenges and opportunities in working with stakeholders in government and public health.        

Gotzmann's regularity and persistence theorems provide tools which allow us to find explicit equations for the Hilbert scheme Hilb_P(P^n). A natural next step is to generalise these results to the multigraded Hilbert scheme Hilb_P(X) of a smooth projective toric variety X. In 2003 Maclagan and Smith generalise Gotzmann's regularity theorem to this case. We present new persistence type results for the product of two projective spaces, and time permitting discuss how these may be applied to a more general smooth projective toric variety.