We prove that the positive-dimensional part of the torsion locus of the Ceresa normal function in $\mathcal {M}_g$ is not Zariski dense when $g\geq 3$. Moreover, it has only finitely many components with generic Mumford-Tate group equal to $\mathrm {GSp}_{2g}$; these components are defined over $\overline {\mathbb Q}$, and their union is closed under the action of $\mathrm {Gal}(\overline {\mathbb Q}/\mathbb Q)$. More generally, we study the distribution of the torsion locus of arbitrary admissible normal functions.

We find upper and lower bounds on the number of rational points with bounded denominators that are contained in a rectangular neighbourhood of some $n$-dimensional $p$-adic integer. To find the upper bound, we use lattice point counting techniques on $p$-adic approximation lattices, and for the corresponding lower-bound statement, a classical pigeonhole principle-style argument is used. We apply this counting result to prove a statement in the setting of weighted simultaneous $p$-adic Diophantine approximation on coordinate hyperplanes. For the lower-bound Hausdorff dimension result, we construct a local ubiquitous system of rectangles and then apply the recent Mass Transference Principle result of Wang and Wu (Math. Ann., 2021).

A known condition for the integrability of the one-dimensional Fourier transform, in which the derivative is assumed to belong to the real Hardy space, is extended to the $n$-dimensional case, in terms of the $n$-th Riesz derivative. We obtain results for the Riesz derivatives of other orders. The function spaces involved in this study are compared and analyzed.

Two-sort species yield differential equations for functional digraphs of Cayley permutations. From these, we obtain an explicit formula for fixed-point-free Cayley permutations and prove that their proportion tends to $1/e$, as for permutations and endofunctions. Our approach also yields counting formulas when the functional digraph is a tree, forest, or connected.

We give a new proof of a result by Fathi, which states that, to any homeomorphism of a closed surface which is isotopic to a pseudo-Anosov homeomorphism, we can associate a stable and an unstable invariant partition of the surface with properties which are similar to the unstable and the stable foliation of a pseudo-Anosov homeomorphism.

Malaria remains a significant global health challenge, with sub-Saharan Africa bearing the majority of the burden. While vector control measures such as pyrethroid-based insecticidal nets and indoor residual spraying have significantly reduced malaria incidence, the emergence of insecticide resistance in Anopheles mosquito populations threatens these gains. Resistance develops through genetic mutations under prolonged selection pressure, complicating control efforts and necessitating a deeper understanding of its evolutionary dynamics. This study introduces a novel mathematical framework to investigate the emergence and spread of insecticide resistance in mosquito populations. By modelling insecticide resistance as a continuous (quantitative) trait influenced by multiple genes, we capture its variability and evolutionary transient dynamics. We propose an age-structured mosquito population model using integro-differential equations, where the resistance trait influences life-history parameters such as mortality and reproduction. Our approach provides new insights into how resistance emerges and spreads within mosquito populations over time. We analyse the model’s properties, including the existence of a unique maximal bounded semiflow, and derive conditions for the existence and stability of steady states. Through parameterization and simulations, we explore the transient and long-term dynamics of resistance evolution under different scenarios. The results offer valuable insights into the evolutionary mechanisms driving insecticide resistance and inform the design of sustainable vector control strategies.