Scientific

The basement membrane (BM) is a nanoporous extracellular matrix that surrounds most tissues and blocks the passage of cells, primarily composed of covalently cross-linked collagen IV fibres and laminins. While BM breaching has traditionally been thought to be mediated by protease-mediated degradation, the failure of protease-targeting clinical trials to reduce metastasis suggests the existence of alternative protease-independent mechanisms. Recent studies indicate that invasive cells extend filopodia capable of remodelling plastic extracellular matrices. However, the covalent cross-links in collagen IV fibres are very strong, prompting the question of how filopodia might facilitate BM invasion. Collagen IV fibres undergo turnover---a dynamic process of protein renewal that may create transient weak spots in the BM. We hypothesise that filopodia exploit these weak spots during turnover to initiate and expand pores, enabling protease-independent invasion. We propose a mathematical biophysical model to test the plausibility of this mechanism using biologically relevant protruding and turnover conditions obtained from experimental observations in the literature. Invasive cells are represented as energetic biomembranes using geometric-surface partial differential equations, allowing the formation of filopodial protrusions, while the BM is modelled as a barrier with collagen IV cross-links that stochastically transition between active and inactive states. The results of the model contrasted with experimental observations identify two subpopulations of filopodia in invasive cells: thin, short-lived filopodia that contribute to global BM degradation, and long-lived, widening filopodia that locally stabilise and enlarge pores where invasion can eventually occur. Under suitable conditions, the model predicts that random turnover and filopodia can synchronise, leading to progressive pore enlargement. Further, pore enlargement arise from the collaboration of several filopodia entering and leaving the same region of the BM at different times. Although our results cannot demonstrate that this mechanism occurs in vivo, they place turnover as a plausible contributor to protease-independent invasion.

Host–parasite interactions often resemble an evolutionary arms race, where each side must continually adapt just to keep up. These dynamics can produce oscillations in allele frequencies—often called Red Queen dynamics—that are a hallmark of host–parasite coevolution. Despite their prominence, we still have an incomplete understanding of how they influence broader evolutionary outcomes. Much of the existing theory has focused on their role in the evolution of sex and recombination, leaving their consequences for other life-history traits largely unexplored. In this talk, I will explore the mechanisms that generate coevolutionary cycles and the role of eco-evolutionary feedbacks in shaping them. I will then discuss how these cycles influence the evolution of parasite virulence.

Vertex operator algebras (VOAs) are algebraic objects that arose in the study of infinite dimensional lie algebras, mathematical physics, and in the classification of finite simple groups. These days they are understood to give rise to vector bundles on moduli spaces of algebraic curves that are useful in a variety of areas of mathematics and physics. In number theory one frequently encounters them via their incarnation on modular curves. In this talk we will recall background on VOAs and modular forms, and we will give a concrete description of the corresponding VOA bundles in terms of modular forms. We will also describe their connection with quasi-modular forms, which arises naturally from the VOA structure.

Although we often introduce group theory to students using groups of symmetries, we tend to move quickly away from these intuitive representations into the realm of axioms and deductions. There are plenty of good reasons for this, not least of which is that the objects or pictures we study the symmetries of, often fail to give us effective intuition about basic properties of the symmetry group.

I will discuss research on representing groups as groups of symmetries of some basic combinatorial structures (graphs, directed graphs, graphs with special properties, posets, etc.) and how this approach can provide improved intuitive understanding of the group.

I will also discuss research on asymptotic results about symmetries of combinatorial objects. The overall lesson of this aspect of the research is that not only is symmetry rare, but even when some amount of symmetry is required or imposed, it is rare for additional symmetry to arise spontaneously.

In this talk, I will share my experiences working in data science related roles in Hong Kong and Canada. I will describe how I explored different career paths and possibilities during my studies, despite starting with a weak academic background in applied math. I will also introduce the challenges I faced in finding internship opportunities and adapting to industry work, along with the strategies I used to overcome them. Finally, I will share some insights into how mathematical thinking and data-driven approaches in math and stats have influenced my career trajectory.