Mathematics

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.

In this talk, we first introduce the Brumer-Rosen-Zantema exact sequence (BRZ), a four-term sequence related to strongly ambiguous ideal classes in finite Galois extensions of number fields. Then, using BRZ, we obtain some known cohomological results in the literature concerning Hilbert's Theorem 94, the capitulation map, and the Principal Ideal Theorem. This is a joint work with Ali Rajaei (Tarbiat Modares University) and Ehsan Shahoseini (Institute for Research in Fundamental Sciences).

The rainbow and the brain have in common that frequencies are produced. In both cases there is a function of frequency, f, called the power spectral density (PSD). In both cases invasive investigation spoils the investigated object. This talk will describe using noninvasive electroencephalography (EEG) to evaluate the PSD of the brain, via stochastic modelling of associated brain structure. We explore the popular question: does the human brain manifest the mysterious property called "1/f"? Is the PSD of the brain proportional to the function "f to the power -a", for some a > 0, and hence scale-free? What would that mean about the brain? Independent of these fascinating questions, the exponent, a, has many successful applications as a diagnostic of brain disorders and treatments.