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Analytical and Geometric Aspects of the Isomonodromy Problem for Linear and Non-linear Connections
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Orientador(es)
Resumo(s)
In 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.
Descrição
Tese de mestrado, Matemática, 2025, Universidade de Lisboa, Faculdade de Ciências
Palavras-chave
Deformações isomonodrómicas problemas de Riemann-Hilbert-Birkhoff série de potências não comutativa geometria hyperkähler espaço twistor Teses de mestrado - 2025