# name
The Riemann–Hilbert Correspondence for Flat Connections on Principal Bundles

# metadata.dc.description[].value
Treballs Finals de Grau de Matemàtiques, Facultat de Matemàtiques, Universitat de Barcelona, Any: 2025, Director: Ignasi Mundet

# metadata.dc.description.abstract[].value
This thesis aims to prove the Riemann--Hilbert correspondence between representations of the fundamental group of a smooth manifold $M$ and flat connections on principal bundles over $M$. We begin by introducing the necessary background on Lie groups and Lie algebras, and then develop the theory of connections on principal bundles, leading to the notions of parallel transport, holonomy, and curvature. The core of this work focuses on flat connections—those with vanishing curvature—whose holonomy gives a well-defined group homomorphism from the fundamental group of $M$ to the structure group of the bundle. As a concrete example, we study flat $S^{1}$-principal bundles over compact, connected, and orientable $2$-manifolds without boundary, and prove that the set of such bundles modulo isomorphism is naturally identified with $(S^{1})^{2g}$, providing a concrete example of the Riemann--Hilbert correspondence.

Notation: In this work, $M$ will denote a smooth manifold unless stated otherwise, and if $p$ is a point in $M$, the tangent space to $M$ at $p$ will be denoted by $T_pM$. Also, if $F : M \to N$ is a smooth map between smooth manifolds, its differential at $p \in M$ will be written as $d_pF : T_pM \to T_{F(p)}N$. If $X$ is a vector field in $M$, its value at $p \in M$ will be denoted by $X_p$. The set of smooth vector fields in $M$ will be written as $\mathfrak{X}(M)$.

# metadata.dc.rights[].value
cc-by-nc-nd (c) Marçal Herraiz Bayó, 2025

# metadata.dc.source[].value
Treballs Finals de Grau (TFG) - Matemàtiques

# metadata.dc.title[].value
The Riemann–Hilbert Correspondence for Flat Connections on Principal Bundles