In 1961, Karamata studied the intersection patterns of diagonals in regular n-gons inscribed in the unit circle, leading to a fascinating limit formula involving Euler's dilogarithm function. For a fixed radius $r \in [0,1]$, let $I_n(r)$ denote the number of intersection points of diagonals (counted with multiplicity) lying within a circle of radius r, concentric with the unit circle. Karamata showed that $ \lim_{n \rightarrow \infty} I_n(r)/I_n(1) = 2/\pi^2 Dilog(r^2)$ where Dilog is Euler’s dilogarithm function.

In this talk, we revisit this classical result and present a proof of its continuous analogue, where the regular n-gon is replaced by the continuous circle. We also introduce and analyze a maximal version of the problem, where the number of intersections is maximized over all possible probability measures supported within radius r.

A meromorphic connection on $\mathbb C P^1$ is a collection of local meromorphic systems of ODEs, with suitable gluing conditions on the overlaps. Meromorphic connections come with monodromy data: indeed, the analytic continuation of a solution along a loop encircling a singularity generally yields a new solution, which may differ from the original one.

A natural question —historically arising from Hilbert's twenty-first problem— is whether one can deform a meromorphic connection slightly without changing its monodromy. In 1981, Jimbo, Miwa, and Ueno discovered that the solutions of the Painlevé equations parametrize isomonodromic deformations for certain classes of meromorphic connections.

In this talk, we will explain all the unfamiliar terms appearing in this abstract, with the goal of introducing the foliation induced by the Painlevé V equation and presenting some results concerning its compactification.

Amongst complex manifolds, Stein manifolds are notable for the property that the topological classification of their vector bundles coincides with the holomorphic one. That is, two vector bundles over a Stein manifold are holomorphically equivalent if and only if they are topologically equivalent. For this and other reasons, Stein manifolds are fundamental to understanding the theory of complex manifolds in general. This property may also be viewed as a manifestation of Gromov's $h$-principle, for those who like using such language. The purpose of this talk is to provide a short introduction to the theory of Stein manifolds, as presented by Grauert--Remmert, with a view to presenting the essential ideas behind the proof of this property.

This work investigates the statistical properties of solutions to shell models of turbulence, focusing on coupled Sabra and passive-scalar transport equations. It is known that these statistics are intermittent, i.e., break the scaling symmetries considered in classical theories by Kolmogorov-Obukhov and Obukhov-Corrsin. A particular consequence is the existence of  anomalous scaling laws, which govern the distributions of velocity and passive-scalar. To explain this phenomenon, we define and analyze the hidden scaling symmetry for solutions and statistical measures of the ideal system. This symmetry may be recovered in the statistical sense, even when all traditional scaling symmetries are broken. 

By Alexander's theorem, every link in the 3-sphere can be represented as the closure of a braid. Lorenz links and twisted torus links are two families that have been extensively studied and are well-described in terms of braids. In this talk, we will present a natural generalization of Lorenz links and twisted torus links that produces all links in the 3-sphere. This provides a simpler braid description for all links in the 3-sphere. 

Dada uma superfície elítica complexa, Ulmer e Urzúa definiram uma folheação (não analítica) tal que as curvas de torção são folhas dessa folheação e demonstraram uma fórmula para o número de tangencias a essa folheação de uma curva algébrica, não de torção. 

Em trabalho conjunto com Ulmer, definimos uma família a dois parâmetros de curvas analíticas na superfície elítica, contendo as folhas da folheação de Ulmer e Urzúa, e demonstramos uma fórmula para o número de tangencias de ordem superior a essa família de uma curva algébrica, não de torção.

Além disso, em característica positiva, obtemos um resultado similar sobre uma família de curvas que, de uma certa forma, é intermediaria às duas famílias complexas mencionadas acima.