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An Introduction to Hitchin Systems

Neumayer, Manuel LU (2023) In Bachelor’s Theses in Mathematical Sciences MATK11 20231
Mathematics (Faculty of Engineering)
Mathematics (Faculty of Sciences)
Centre for Mathematical Sciences
Abstract
This thesis aims to introduce the tools that are needed to understand — and historically led to the inception of — Hitchin systems. We lay out a basic theory of Hamiltonian systems on symplectic manifolds, and prove the Liouville-Arnold theorem, which states that integrable Hamiltonian systems admit coordinates in which their solution is basically linear. We then introduce a theory of holomorphic structures on complex vector bundles, following a 1983 paper by Atiyah and Bott on the Yang-Mills equations in proving a one-to-one correspondence between such structures and the Dolbeault operator. Finally, the thesis gives an overview of Nigel Hitchin’s 1987 paper Stable Bundles and Integrable Systems.
Popular Abstract
Have you ever wondered how scientists can send a rover into space and know that it will rendezvous with Mars in exactly 6 years, on a specific date, even down to the minute? Ever since Newton, a branch of physics known as classical mechanics has given us "superpowers" when it comes to predicting the evolution of intricate dynamical systems, such as our solar system.

Mathematicians have always felt inspired by the rich mathematics that governs nature, and many exciting interactions between pure mathematics and physics have emerged. In 1987, mathematician Nigel Hitchin unraveled a new one, somewhere that no one expected. He discovered that a fascinating mathematical object known as a moduli space had a natural connection to classical... (More)
Have you ever wondered how scientists can send a rover into space and know that it will rendezvous with Mars in exactly 6 years, on a specific date, even down to the minute? Ever since Newton, a branch of physics known as classical mechanics has given us "superpowers" when it comes to predicting the evolution of intricate dynamical systems, such as our solar system.

Mathematicians have always felt inspired by the rich mathematics that governs nature, and many exciting interactions between pure mathematics and physics have emerged. In 1987, mathematician Nigel Hitchin unraveled a new one, somewhere that no one expected. He discovered that a fascinating mathematical object known as a moduli space had a natural connection to classical mechanics, allowing the application of a theory physicists and mathematicians had developed over centuries to a removed, abstract area of mathematics that was barely three decades old.

Hitchin's discovery had a profound impact. In 2010, it contributed to a research program known as the Langlands Program, specifically to the proof of the Fundamental Lemma, a result that has been awarded the Fields Medal, the highest honor in mathematics.

This thesis gives a concise overview of the theory of classical mechanics that Hitchin applied to the moduli space and proves important results surrounding the construction of the moduli space. It concludes with an outline of Hitchin's 1987 paper, hopefully equipping the reader with everything they need to embark on this thrilling area of mathematics. (Less)
Please use this url to cite or link to this publication:
author
Neumayer, Manuel LU
supervisor
organization
course
MATK11 20231
year
type
M2 - Bachelor Degree
subject
publication/series
Bachelor’s Theses in Mathematical Sciences
report number
LUNFMA-4151-2023
ISSN
1654-6229
other publication id
2023:K25
language
English
id
9137030
date added to LUP
2023-09-13 15:29:48
date last changed
2023-09-13 15:29:48
@misc{9137030,
  abstract     = {{This thesis aims to introduce the tools that are needed to understand — and historically led to the inception of — Hitchin systems. We lay out a basic theory of Hamiltonian systems on symplectic manifolds, and prove the Liouville-Arnold theorem, which states that integrable Hamiltonian systems admit coordinates in which their solution is basically linear. We then introduce a theory of holomorphic structures on complex vector bundles, following a 1983 paper by Atiyah and Bott on the Yang-Mills equations in proving a one-to-one correspondence between such structures and the Dolbeault operator. Finally, the thesis gives an overview of Nigel Hitchin’s 1987 paper Stable Bundles and Integrable Systems.}},
  author       = {{Neumayer, Manuel}},
  issn         = {{1654-6229}},
  language     = {{eng}},
  note         = {{Student Paper}},
  series       = {{Bachelor’s Theses in Mathematical Sciences}},
  title        = {{An Introduction to Hitchin Systems}},
  year         = {{2023}},
}