From 43031668c3aa8a70953dd298f3a5d08f8fc1ecc4 Mon Sep 17 00:00:00 2001
From: Luke Naylor <l.naylor@sms.ed.ac.uk>
Date: Wed, 14 Aug 2024 15:11:18 +0100
Subject: [PATCH] Add a lay summary
---
tex/lay-summary.tex | 23 +++++++++++++++++++++++
tex/main.tex | 2 ++
2 files changed, 25 insertions(+)
create mode 100644 tex/lay-summary.tex
diff --git a/tex/lay-summary.tex b/tex/lay-summary.tex
new file mode 100644
index 0000000..d70c8e0
--- /dev/null
+++ b/tex/lay-summary.tex
@@ -0,0 +1,23 @@
+\chapter*{Lay Summary}
+
+In mathematics, geometric spaces can be studied via the vector bundles over them.
+Possibly the most famous example of this is the
+`\href{https://en.wikipedia.org/wiki/Hairy_ball_theorem}{hairy ball Theorem}'
+with the image a hairy ball with `tufts' at each pole.
+In mathematical terms, this is a statement about the existence of a non-vanishing
+section of the tangent bundle of the sphere.
+This can also expressed via a numeric invariant called the `degree', which encodes
+how many points a section must vanish at, or in the case of the hairy ball, how
+many `tufts' you inevitably get when trying to comb it.
+Compare this with a hairy doughnut, which you could comb without any tufts.
+This gives mathematicians a way to distinguish between different spaces, in particular
+by comparing a numerical invariant between their corresponding tangent spaces.
+
+In algebraic geometry, invariants such as these do not completely determine a vector bundle.
+This has led to the construction of `moduli spaces' which parametrise vector bundles with the
+same invariants. However these typically only parametrise a subset of `stable' bundles.
+Somewhat analoguously to introducing complex numbers which can sometimes be used to solve
+problems about real numbers, mathematicians have been moving towards studying a generalisation
+of vector bundles called the `derived category of coherent sheaves'.
+This thesis studies generalisations of the notion of `stable' in this more abstract setting,
+and in particular when different notions of `stable' coincide or differ.
diff --git a/tex/main.tex b/tex/main.tex
index 2b80abf..111f215 100644
--- a/tex/main.tex
+++ b/tex/main.tex
@@ -43,6 +43,8 @@ $\nu$-Walls}
\maketitle
+ \import{./}{lay-summary}
+ \addcontentsline{toc}{chapter}{Lay Summary}
\import{./}{abstract}
\declaration
--
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