Mathematics

I will discuss several recent results on the Turán density of long cycle-like hypergraphs. These results (due to Kamčev–Letzter–Pokrovskiy, Balogh–Luo, and myself) all follow a similar framework, and I will outline a general strategy to prove Turán-type results for tight cycles in larger uniformities or for other "cycle-like" hypergraphs.

One key ingredient in this framework, which I hope to prove in full, is a hypergraph analogue of the statement that a graph has no odd closed walks if and only if it is bipartite. More precisely, for various classes C of "cycle-like" r-uniform hypergraphs — including, for any k, the family of tight cycles of length k modulo r — we equiivalently characterize C-hom-free hypergraphs as those admitting a certain type of coloring of (r-1)-tuples of vertices. This provides a common generalization of results due to Kamčev–Letzter–Pokrovskiy and Balogh–Luo.

In this talk we will introduce the modular method, the approach followed by Wiles to prove Fermat’s Last Theorem. We will explain the role of elliptic curves, modular forms, and Galois representations in this framework, and discuss how the method has evolved in recent years.

‘Reef halos’ are rings of sand, barren of vegetation, encircling reefs. However, the extent to which various biotic (e.g., herbivory) and abiotic (e.g., temperature, nutrients) factors drive changes in halo prevalence and size remains unclear. The objective of this study was to explore the effects of herbivore biomass, primary productivity, temperature, and nutrients on reef halo presence and width. First, we conducted a field study using artificial reef structures and their surrounding halos, finding that halos were more likely to be observed with high herbivorous fish biomass, and halos were larger under high temperatures. There was a distinct interaction between herbivorous fish biomass and temperature, where at high fish biomass, halos were more likely to be observed under low temperatures. Second, we incorporated environmental drivers into a consumer-resource model of halo dynamics. Certain formulations of temperature-dependent vegetation growth caused halo width and fish density to change from a fixed to an oscillating system, supporting the idea that environmental drivers can cause temporal fluctuations in halo width. Our unique combination of field-based and mechanistic modeling approaches has enhanced our understanding of the role of environmental drivers in grazing patterns, which will be particularly important as climate change causes shifts in marine systems worldwide.

Fix a positive integer $n$ and a graph $F$. A graph $G$ with $n$ vertices is called $F$-saturated if $G$ contains no subgraph isomorphic to $F$ but each graph obtained from $G$ by joining a pair of nonadjacent vertices contains at least one copy of $F$ as a subgraph. The saturation function of $F$, denoted $\mathrm{sat}(n, F)$, is the minimum number of edges in an $F$-saturated graph on $n$ vertices. This parameter along with its counterpart, i.e. Turan number, have been investigated for quite a long time.

We review known results on $\mathrm{sat}(n, F)$ for various graphs $F$. We also present new results when $F$ is a complete multipartite graph or a cycle graph. The problem of saturation in the Erdos-Renyi random graph $G(n, p)$ was introduced by Korandi and Sudakov in 2017. We survey the results for random case and present our latest results on saturation numbers of bipartite graphs in random graphs.

Mathematical biology offers powerful tools to tackle pressing problems at the interface of health and public policy. In this talk, I will share two vignettes demonstrating how mathematical and simulation modelling can be applied to tobacco regulatory science. The first uses a Markov state transition framework to capture population-level dynamics of two tobacco products, each with a flavour option. This structure highlights the challenges of modelling high-dimensional systems, parameter inference from sparse data, and representing policy interventions as modifications to initiation, cessation, and product switching rates. The second vignette focuses on social network modelling, where adolescent tobacco use is primarily shaped by peer influence and network structure. In this setting, stochastic processes and graph-based models describe how behaviours propagate and stabilise within adolescent populations. Together, these examples illustrate how applied mathematics can bridge data and policy in public health.