My PhD thesis for the layperson

🚨🚨🚨 this post is under construction 🚨🚨🚨

I recently completed my PhD thesis, Stability conditions and canonical metrics (available here). In this post I will make some attempt to explain the key ideas of my work and its surrounding context, aimed at a non-specialist.

The main body of work of my thesis (i.e. Part I) combines two different threads of academic investigation together: the search for canonical geometric structures, and the use of physics as a guide towards finding interesting mathematics. Part II of my thesis primarily follows just that first thread (although like most aspects of modern geometry, it is secretly influenced by the second).

Since my thesis is a thesis in geometry, it is necessary to take a fairly considerable detour into the history and context of what geometry is in modern mathematics. This is mostly necessary because, since geometry and the geometry we teach in school is so old, the gap between what the average person understands to mean geometry (i.e. geometry of shapes in the plane, and perhaps in 3-space) and what geometry constitutes now is more like a vast chasm.

What is geometry?

Geometry (from geo-metron: measurement of the Earth) is (more-or-less) the oldest mathematical subject. Serious investigation began at least as far back as 3000 BC in ancient Babylonia, and the first textbook in geometry, Euclid’s Elements, was written circa 300 BC. A continuous thread of development has proceeded from those ancient times, radically changing what geometers have been interested in, what is most important to understand, the applications of the subject, and even its entire foundations multiple times.

In those ancient days, geometry was primarily concerned with the study of Euclidean spaces (the flat plane, and normal 3-dimensional space). Polygonal objects in these spaces (polygons in the plane, or polyhedra in space) were essentially completely understood in the Elements.¹

Platonic solids

Already at this time, some of the most basic questions in geometry were being asked (and answered!). The crowning achievement of the Elements, contained in the final Book XIII, is the construction and classification of all Platonic solids.² Propositions 13 to 17 give constructions of the tetrahedron, octahedron, cube, icosahedron, and dodecahedron, and Proposition 18 proves that only these 5 Platonic solids exist; there are no more. This is a fundamental kind of problem in mathematics, a classification problem: Choose an interesting type of mathematical object (in this case Platonic solids), and completely classify them (in this case there are exactly 5).

Of all the things Euclid could have tried to classify, why did he choose the Platonic solids? Ignoring the considerable historical and philosophical context around them, one can still distill several convincing reasons:

1. Being regular polyhedra, the Platonic solids are nice, uniform shapes.
2. Being convex, the Platonic solids are bounded in size, and so easy to imagine, think about, and understand.
3. The Platonic solids appear quite naturally all around us.

These properties (and others) manifest as a certain aesthetic quality to the Platonic solids, which we take to justify our focused interest in them (of course, such aesthetic qualities are precisely why Plato and others were so enamoured by them before Euclid completed their mathematical study). The universality of this aesthatic quality is worth remarking on: the layperson as much as the expert would agree that a tetrahedron or an icosahedron are, in some sense, much “nicer” looking shapes than your average irregular polyhedron.

The nature of space

Condensing the next two-thousand-odd years of mathematical development into the gap between paragraphs, we now turn to a foundational question in geometry: what is a shape and what is a space.

To the layperson, geometry is the study of shapes. That is, things with some size, structure, position, volume, or area, which live in the plane or in normal space, and which you could imagine holding in your hand. Things like a ball.

This natural intuition about geometry is of course highly applicable to the normal world, and not to be discounted lightly, but it does betray a kind of unconcious bias. Namely, it is at all times taken for granted that shapes, such as a ball, live somewhere. Thus our normal conception of a ball is actually the conception of a ball, the infinite 3-space it lives in, and perhaps even an observer hanging around somewhere in that ambient space to look at the ball.³ How confident can we be that our understanding of the ball is really due to its fundamental nature, and not a consequence of the ambient space in which it lives? Is this distinction even important? After all, if we live in normal 3-space, who cares about balls which might not?

Mathematics is all about the removal of extraneous details to get to the heart of the matter, and mathematicians doubted the fundamental importance of the normal 3-space of Euclidean geometry for millenia since the Elements appeared. The so-called “parallel postulate problem” seeked to justify the primacy of normal 3-space unsuccessfully from antiquity all the way until the 1800s, until a radical transformation of the foundations of geometry occurred: the transition from shapes to spaces.

In the early 1800s, Carl Friedrich Gauss developed the theory of surfaces. Of his many contributions to the subject of geometry, the most philosophically important is his Theorma Egregium (remarkable theorem), which more-or-less states the following:

The basic geometric properties of a surface do not depend on the particular way it happens to be embedded in an ambient 3-space, and therefore the ambient 3-space is seemingly superfluous.

Specifically what the remarkable theorem states is that the way a surface is curved can be determined entirely from the perspective of a being living on the surface, without the need to observe the curved shape of the surface externally! Thus for example one could determine that we live on the curved surface of the Earth, as opposed to a flat plane, purely by measurements performed on the Earths surface!⁴ This immediately dispenses with the idea that the ambient 3-space which a ball lives in is important. It is perfectly reasonable then to imagine maybe a whole universe happening on the surface of a ball, a non-Euclidean geometry, with no ambient 3-space such as our own containing it (see the book Flatland).

This change of perspective, viewing the shape as its own kind of space, shattered the foundations of geometry. Gauss’ student Bernhard Riemann put this on sound footing, when he defined mathematically what such a “shape/space” should be (now known as a Riemannian manifold in his honour). So powerful was the idea that all geometric objects should be thought of not as shapes in some ambient space, but spaces with an intrinsic nature in their own right, that the term shape more or less fell out of favour. Geometers now exclusively refer to the basic objects of geometry as “spaces.”⁵

An aside on kinds of spaces

So one definition of geometry, in the boardest sense, is “the mathematical study of spaces” (or to the layperson, “the mathematical study of shapes, remembering that what is most important is the intrinsic nature of the shape itself”). Like all attempts to try and clearly define mathematical disciplines, this definition would not satisfy a mathematician. Why? The notion of space is far too ambiguous.

The same problem was being encountered across all of mathematics at the turn of the 20^th century. Mathematics went through a foundational revolution at this time, arising out of Hilbert’s program, which attempted to formalize, starting from the simplest elements of basic logic, all of the existing mathematics of the time.

One of the many side effects of this program was to change the nature of how mathematical objects are thought about. A mathematical object is any idea, pattern, form, or structure which can be defined and reasoned about mathematically. Such an idea is so broad as to be essentially useless. The output of Hilbert’s program was to shift the focus of how mathematicians thought about their objects to the following:

1. First begin with the simplest (or most abstract) mathematical object you can think of.
2. Progressively add to this simplest object to build up more and more complex objects, until you can describe from the simplest elements what you were really interested in in the first place.

What is meant by “the simplest mathematical object” is largely a choice, but mathematicians of the early 20^th century settled on the notion of a set (for example you could also try and choose “number” or “type” or “category”). That is, a collection of elements (with absolutely no more information, even the nature of what those elements are!). The second point largely manifests as a set being described with an increasing number of adjectives which either add to or restrict its nature until you arrive at the thing you were actually interested in.

Why might this be a useful way of thinking? Despite it coming with an enormous intellectual overhead, it gives two advantages: Firstly, it helped formalize very complicated mathematics as part of Hilbert’s program, and secondly, it vastly and effectively expands our conception of what can be by starting at the very bottom and building up (rather than starting at the top about trying to go down).

As an example, a question like “does there exist a notion of space other than our familiar Euclidean 3-space which we live in” (the “parallel postulate question” from earlier) is obviously the result of a severely limited perspective on space! If you start from the bottom at a set and build up, along the way you will encounter many types of spaces which are in some ways very similar to our 3-space, and in other ways different: topological spaces, vector spaces, manifolds, etc.

So which from this menagerie are the spaces we really care about. What abstract mathematical notion most accurately captures our human intuition of “shape” or “space” whilst most accurately excluding those things which do not satisfy this intuition?

Manifolds

The answer is “manifolds.” A manifold is a space which may be constructed through the process of taking flat disks and gluing them together along their boundary in a nice smooth way. This construction process is general enough to produce almost everything of real geometric interest (with some small exceptions, leading to other notions of space like “variety” or “orbifold” and so on).

In light of the abstraction revolution, it should be noted that “manifold” is not at the peak of the tower of mathematical constructions. In fact the mathematical notion of a manifold is so abstract that it fails to describe notions like: length, area, volume, angles, relative position of points, etc. But as a demonstration of point 2. in the process of abstraction, one can add these properties to a manifold with a suitable adjective:

• length -> Finsler manifold
• angles -> Conformal manifold
• all of the above -> Riemannian manifold

A case study in modern geometry: the classification of closed orientable surfaces

To conclude our exploration of what geometry is, we will consider the crowning achievement of modern (i.e. post-Gauss) geometry, and contrast it with the study of Platonic solids.

Physics and geometry

Part I of my thesis: String theory

Part I of my thesis: Geometry

Part II of my thesis

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