In this talk, we discuss the rational dilation problem on domains in $\mathbb{C}^n$. Among the various approaches to this problem, we focus on techniques from operator space theory, highlighting the connection between rational dilation and complete contractivity. En route, we discuss the case when the domain is a unit ball. We determine the cotangent operator space structure at the origin of the tetrablock and use it to construct a commuting triple of operators for which the tetrablock is a spectral set but not a complete spectral set, thereby establishing the failure of rational dilation on the tetrablock.

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.

TBA