Mathematics

‘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.

A vanishing sum of roots of unity (VSRU) is a finite list $z_1,\ldots,z_K$ of $N$-th complex roots of unity whose sum is zero. While there are many simple examples—including the famous "beautiful equation" of Euler, $e^{i \pi} + 1 = 0$—such sums become extremely complex as the parameter $N$ attains more complex prime power divisors (and we will see several classical examples illustrating this idea, as well as new examples from my work).

One fruitful line of inquiry is to seek a quantitative relationship between the prime divisors of $N$, their associated exponents, and the cardinality parameter $K$. A theorem of T.Y. Lam and K.H. Leung from the early '90's states: $K$ must always be (at least) as large as the smallest prime dividing $N$. This generalizes the well known observation that that sum of all $p$-th roots of unity (where $p$ is any prime number) must vanish; and, one notices that Euler's equation is one example of this fact.

In this talk, we will discuss two significant strengthenings of this result (one due to myself and I. Łaba, another due to myself, G. Kiss, I. Łaba and G. Somlai), which are derived from complexity measurements for polynomials with integer coefficients which have many cyclotomic polynomial divisors. As applications, we give connections in two other areas of mathematics. The first is in the study of integer tilings: additive decompositions of the integers $Z = A+B$ as a sum set, where each integer is represented uniquely. The second application is to the Favard length problem in fractal geometry, which asks for bounds upon the average length of the projections of certain dynamically-defined fractals onto lines.

This talk is based upon my individual work, as well as my joint work with I. Łaba, as well as my joint work with G. Kiss, I. Łaba and G. Somlai. All are welcome, and the first 15-20 minutes will include introductory ideas and examples for all results discussed in the latter portion of the talk.

Lagrangian submanifolds are a central object of study in symplectic topology. Their rigidity properties have been uncovered via Floer theory since the early ’90’s. The talk will briefly review the subject, in particular how triangulated category structures naturally arise in this context through work of Donaldson, Kontsevich, Fukaya, and others. Further, will be discussed the more recent, natural role of persistence theory, in the sense common in data science. Finally, we will outline how complexity measurements based on persistence methods reflect topological and dynamical invariants, such as topological entropy.