In 1961, Jovan Karamata proved a remarkable identity involving the Dilogarithm function and intersection of diagonals of regular polygons. We reframe the problem and give different proofs for the result. We also investigate what happens when we consider different approaches to it.

Fourier Restriction Theory, i.e., the study of the interaction between the Fourier transform and the curvature of surfaces, is a central topic in harmonic analysis. In this area, there are some particular cases in which an optimal inequality can be achieved.

In this talk, we will discuss some of these cases and explore some discrete analogues. This is based on a joint work with Diogo Oliveira e Silva.

Since their introduction, generalized Toeplitz structures (in the sense of L. Boutet de Monvel and V. Guillemin) over contact manifolds have found numerous applications in fields such as CR geometry, analysis, and quantization. More recently, S. Zelditch introduced the concept of dynamical Toeplitz operators to study the dynamics of quantized contact transformations. These operators, closely tied to the geometry of the underlying manifold, have demonstrated significant applications in both geometry and analysis.

The aim of this talk is to provide a simple introduction to the geometric and analytical context of these operators on Grauert tube boundaries and to present, as an application, their connection to the complexification of Laplacian eigenfunctions.

A Ferrers diagram is a graphical way of representing an integer partition. A q-series is a series in which the ratio of the nth term to the next is a rational function of qⁿ. With reference to the origins of the subject in the work of Sylvester, I will present a short introduction to the use of Ferrers diagrams in giving combinatorial interpretations of q-series identities. I will then move on to more recent developments involving a sort of generalized partition, called an overpartition. Finally, I will describe some further generalizations and related open problems.

In this talk, I'll explore how tools from conformal geometry help us study global features of spacetime using local differential geometry techniques. Specifically, I'll focus on a particular conformal representation of Minkowski spacetime, which provides insights into the behaviour of massless fields near spatial infinity. We'll take a look at how this representation can be applied to analyze the asymptotics of the wave equation and use that to calculate asymptotic conserved quantities for the field.