Cotti, GiordanoFlorentino, Carlos Armindo ArangoInácio, Tomás de Oliveira Santos2025-02-132025-02-1320252024http://hdl.handle.net/10400.5/98423Tese de mestrado, Matemática, 2025, Universidade de Lisboa, Faculdade de CiênciasIn this thesis, we explore the classical and modern theory of differential equations in the complex domain. We begin with a concise exposition of the main classical results, discussing concepts such as monodromy, the Stokes phenomenon, and isomonodromic deformations. These are presented in a predominantly analytical manner, so we also provide a brief geometric approach. Next, we highlight two situations where isomonodromic deformations play a central role: Frobenius structures and Joyce structures. All concepts related to these, as well as the relationships between them, are clarified. In the second half of the thesis, we examine both the analytical and geometric aspects of these structures, starting with the introduction of the associated Riemann-Hilbert-Birkhoff problem. While this problem is highly non-trivial in the Joyce case, we showcase an example where a solution is found. We also explore a new method of solving this problem, which involves inverting a non-commutative power series. By involving sums over rooted trees and integer partitions, we uncover a strong connection between the Riemann-Hilbert-Birkhoff problem and Number Theory. We examine this model in detail, particularly focusing on the two-dimensional case. Finally, we adopt a more geometric perspective. The Poisson geometry inherent in Joyce structures gives rise to a symplectic geometry, which in turn induces a hyperkähler structure on the tangent bundle of the deformation parameter space. In this framework, the flatness condition of the connection is equivalent to Plebanski’s second heavenly equation. This equivalence naturally motivates the construction and ´ study of the associated twistor space, which is simplified by the presence of the hyperkähler structure. The twistor space serves as a powerful tool, offering various geometric interpretations of the concepts presented in this thesis. We explore these geometries in some detail, though we expect the reader to be familiar with the underlying basic theory.engDeformações isomonodrómicasproblemas de Riemann-Hilbert-Birkhoffsérie de potências não comutativageometria hyperkählerespaço twistorTeses de mestrado - 2025master thesis