Many terrestrial arthropods carry electric charge, as do the flowers, predators and prey they encounter. Over the past decade, it has become clear that bees, spiders, caterpillars and treehoppers can detect and respond to these aerial electric fields, through the same mechanosensory hairs they use to sense airflow and sound. But what can such a sense actually tell an animal? I will describe an approach to this question. We model hairs as charged, coupled oscillators and build a minimal “Mathematical Bee” to ask what the physics permits. In the model, a few hairs suffice to locate charges, navigate by fixed charges, and reconstruct a flower’s shape from a distance. The models make testable predictions. For example, wingbeats should produce a modulated-DC field, and hairs tuned to wingbeat frequencies should detect predators and prey from farther away. Experiments on bumblebees and spiders are consistent with both. I will close with treehoppers, whose extraordinary shapes may act as electrostatic lenses, concentrating electric fields on their sensory hairs, and with open questions about how electrical information is stored, processed and used. This is joint work with Daniel Robert, Ryan Palmer and other collaborators.

Distance set problems form a central theme in geometric measure theory and harmonic analysis, investigating the relationship between the geometric structure of a set and the distances determined by its points. A fundamental question in this area is the Falconer distance conjecture, which asks whether a set in Euclidean space with sufficiently large Hausdorff dimension must determine a set of distances of positive Lebesgue measure. In this talk, I will discuss some of the central questions and developments in distance set problems, with an emphasis on the role of harmonic analysis and geometric measure theory. Beginning with the Falconer distance problem, I will explore related questions concerning prescribed distance configurations, including finite chains and trees, and how the geometric structure and sparsity of sets influence the existence of such configurations. If time permits, I will also discuss some extensions of these problems to the Heisenberg group, where the underlying geometry leads to interesting variations of classical distance problems.

Scanning-tunnelling potentiometry measurements by the Brar experimental group reveal magnetic-field-dependent voltage patterns around a circular graphene p–n junction. Motivated by this collaboration, I will discuss a related theoretical question: what determines the effective transport size of a smooth density perturbation? Our theoretical work shows that local Ohm–Hall transport can produce an effective obstacle radius R_no ~ a α^(−1/β) ≫ a, where a characterizes the perturbation, β > 2 describes its density tail, and α = σ_xx/|σ_xy| ≪ 1. This amplification arises from competition between Hall circulation and dissipative leakage, without invoking viscosity. The resulting far-field dipole also poses an inverse problem: its magnitude alone cannot separate the perturbation size, the density profile and transport coefficients, while its orientation supplies complementary information. Returning to the graphene experiment, I will discuss annular features whose separation approximately follows the cyclotron diameter, as predicted by a kinetic calculation with a specified boundary. A different question about what experiments reveal arises in correlated electron systems: can electron transfer miss the lowest charged state? Motivated by the one-third-filled charge crystal proposed for twisted MoTe₂, I studied a twelve-site, four-carrier extended Hubbard model by exact diagonalization. In this model, the lowest vacancy state retains a reversed magnetic moment up to a field about fifteen times larger than that needed to align the background. Because carrier number in each valley is conserved, removing one carrier from the aligned background cannot create this state, so it is absent from the direct removal spectrum. This raises three questions: does the reversed moment remain bound to the vacancy in larger crystals, can the charge crystal modify this binding beyond the known triangular-lattice magnetic-polaron mechanism, and can weak valley mixing make the lower state visible? A related question is whether magnetic rearrangement changes the ordering and electron-transfer weight of fractionally charged excitations in MoTe₂.