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Index Theorems with Spectral Localizers

Waldemarson, Erik LU (2025) In Master's Theses in Mathematical Sciences FMAM05 20251
Mathematics (Faculty of Engineering)
Abstract
The Atiyah--Singer index theorem is a foundational result in mathematics, providing a deep connection between the analytical properties of an elliptic differential operator and the topological properties of its underlying manifold. This thesis focuses on a numerical approach to index theory for generalized Toeplitz operators. Building on the work of Loring and Schulz-Baldes, we utilize the spectral localizer method, which relates the analytical index of a Toeplitz operator to the signature of an associated finite-dimensional matrix, the spectral localizer. The contributions of this work are a simplified proof of the key formula, and new, less restrictive parameter bounds.
Popular Abstract
One of the most surprising discoveries in mathematics is that solving equations can reveal the hidden shape of the space they live in. At the heart of this connection lies the index, an integer that acts as a bridge between two worlds of mathematics. On one hand, we can define the index of an operator analytically using equations. Think of an operator as a machine that takes something in, does something to it, and produces an output. On the other hand, the index is also a topological invariant: unchanged by bending or stretching the underlying space.

This thesis develops a new way to understand these index numbers. Traditionally, computing an index requires heavy theoretical machinery. The spectral localizer method offers a more... (More)
One of the most surprising discoveries in mathematics is that solving equations can reveal the hidden shape of the space they live in. At the heart of this connection lies the index, an integer that acts as a bridge between two worlds of mathematics. On one hand, we can define the index of an operator analytically using equations. Think of an operator as a machine that takes something in, does something to it, and produces an output. On the other hand, the index is also a topological invariant: unchanged by bending or stretching the underlying space.

This thesis develops a new way to understand these index numbers. Traditionally, computing an index requires heavy theoretical machinery. The spectral localizer method offers a more concrete approach: it translates the problem into studying a large but finite square matrix -- a square grid of numbers, much like a spreadsheet, but rich in mathematical structure.

To analyze such a matrix, mathematicians look at its spectrum, a collection of special numbers called eigenvalues. Just as the spectrum of light splits into different colors, the spectrum of a square matrix reveals its hidden structure. Some eigenvalues come out positive, others negative. The signature of the matrix is simply the difference between how many positive and negative eigenvalues appear.

The spectral localizer method offers us this: The index of a special type of operators, called Toeplitz operators, is equal to half the signature of a finite square matrix.

The contributions of this work are twofold: a simpler proof of the central formula that makes the spectral localizer method work, and more flexible conditions under which it can be applied. Together, these results make index theory — once thought of as purely abstract — more accessible to explicit calculation and computer-based exploration. (Less)
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author
Waldemarson, Erik LU
supervisor
organization
course
FMAM05 20251
year
type
H2 - Master's Degree (Two Years)
subject
keywords
Index, Index theorem, Spectral Localizer, Toeplitz operator, Eigenvalues
publication/series
Master's Theses in Mathematical Sciences
report number
LUTFMA-3581-2025
ISSN
1404-6342
other publication id
2025:E32
language
English
id
9210825
date added to LUP
2025-11-24 10:07:38
date last changed
2025-11-24 10:07:38
@misc{9210825,
  abstract     = {{The Atiyah--Singer index theorem is a foundational result in mathematics, providing a deep connection between the analytical properties of an elliptic differential operator and the topological properties of its underlying manifold. This thesis focuses on a numerical approach to index theory for generalized Toeplitz operators. Building on the work of Loring and Schulz-Baldes, we utilize the spectral localizer method, which relates the analytical index of a Toeplitz operator to the signature of an associated finite-dimensional matrix, the spectral localizer. The contributions of this work are a simplified proof of the key formula, and new, less restrictive parameter bounds.}},
  author       = {{Waldemarson, Erik}},
  issn         = {{1404-6342}},
  language     = {{eng}},
  note         = {{Student Paper}},
  series       = {{Master's Theses in Mathematical Sciences}},
  title        = {{Index Theorems with Spectral Localizers}},
  year         = {{2025}},
}