Abstract One of the fundamental pillars of partial differential equations is the a priori estimates, which provide quantitative information about a solution, regardless of its existence. One of the main results in this field is the Maximum Principle, which plays a crucial role throughout the theory of PDEs, acting as an essential tool in various proofs and having important direct applications, such as the uniqueness of solutions. In this context, throughout this lecture, my goal will be to contextualize some versions of the Maximum Principle, exploring generalizations over time as well as its utilities and the role of this concept in my field of research.

Abstract The classical theory of functions show that a holomorphic function f  on a domain U of C (complex plane) does not always have a maximal domain of definition _" inside" C, we're talking about multivalued function (Think for instance about log (z)=int ( from 1 to z) du/u). Now, for a holomorphic vector field X the multivaluedness is a "local" obstruction for X to be complete. The aim of this talk is to shed some light on this concept and to show the importance of the class of vector fields that all its integral curves have maximal domain of definition on C. We will also  discuss its relation with foliation theory!

Abstract TBA

Abstract  An arrangement of pseudolines is a colection of unbounded x-monotone simple curves in the euclidean plane where each pair of curves intersects exactly once. Levi introduced them in the late 1920s as a natural generalization of line arrangements, relaxing the requirement of straightness while maintaining their topological properties. These structures are all over combinatorics, yet finding a closed formula to enumerate the set of non-isomorphic pseudoline arrangements remains an open problem. In this talk, we present a recursive formula to count a specific subset of pseudoline arrangements, known as shellable arrangements, and provide some bounds for their enumeration.

Abstract In this talk we introduce the notion of compatible algebras: given two products of the same type (associative, Lie, Leibniz, etc.) on a vector space, they are said to be compatible if their sum is also of the same type.

We focus on the class of compatible Lie algebras, by starting with some basic definitions and results which are analogous to those of Lie algebras. Then, we explore the more complex notions such as nilpotency, solvability, semisimplicity and representation theory of compatible Lie algebras, highlighting the results that are analogous to the non-compatible case and those that differ significantly.