This section contains notes produced for various seminars that I have given. This is not a full list of the talks that I have given; for that list consult my CV in the Professional section of this website.
If a link to a PDF or PostScript version is not immediately visible, click on the See more ... link after the abstract to see a list of versions available.
The latter talks were designed using Beamer and I have included the LaTeX source in case they are useful as examples of how to - and how not to - use Beamer.

How to make a pearl:

1. Find something irritating.
2. Cover it in something most think vaguely disgusting.
3. Leave it to one side until someone comes along and says "That's beautiful".

If you are an oyster, this involves grit and mucus (generally agreed upon as being a bit disgusting by all except small children) and the result is a pearl. For a mathematician, irritating things include: being told something can't be done, being told that two things are similar but for there to be no connection between the two, and being told that something is true but with no reason behind it. Category theory grew out of getting rid of these irritations by covering them with structures and axioms (which most find faintly displeasing but a few find fascinating). But, over time, it has produced its fair share of gems.

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Being a mathematician requires a careful balance of two important character traits: tenacity and laziness. A mathematician will not let go of a problem until an answer has been found, but equally a mathematician would rather find an easy answer than a difficult one, even if finding the easy answer takes longer than the more direct one.

Laziness also means that mathematicians are extremely reluctant to give up a good theory. If a theorem was useful in one area, it is reasoned, it ought to be useful in others as well - even if there is no apparent connection between the two.

One of the most successful, certainly the most practical, areas of mathematics is calculus. And so mathematicians have spent hundreds of years pushing calculus into areas that it should never have been taken to so that now calculus appears in just about every area of mathematics, from number theory to discrete dynamics.

Differential topology is one of the milder outposts of calculus, yet even there one can encounter spaces that would make Isaac Newton's head spin. In some spaces, the back of your head is the front, and the middle of the top of the right hand side of the front is the back. Even Einstein would get lost here (though due to relativity, it would not be him that was lost but the universe around him would be lost).

In these spaces, it's wise to have a look first from a safe distance before venturing further in.

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Talk given at the Workshop on Loops, Strings and Moduli Spaces held at the Chern Institute, Nankai University, 3rd - 7th August 2009.

Similar in content to the talk give at Ottawa in May 2009, but aimed at a different audience so emphasises different aspects of the theory.

Abstract:

Smooth manifolds are extremely nice spaces. The fact that they have charts means that a vast amount of the theory of Euclidean spaces can be easily transferred to manifolds. This makes for a very useful subject.

However, the charts also make manifolds very fragile: it is easy to do something to a manifold that makes it no longer a manifold. Taking a quotient by a group action is one such, looking at mapping spaces is another. Often, specific operations can be fixed - orbifolds fix the quotienting, infinite dimensional manifolds fix the mapping spaces - but systematic case-by-case fixing is a little unsatisfying. Over the years there have been several attempts to build a suitable category of "smooth objects" generalising smooth manifolds. The general method is to take some property that all manifolds have, which can be defined in a more robust way than charts.

In this talk I shall review some of these attempts, focussing particularly on the similarities between them. I shall try to motivate my own favourite: Frölicher spaces. In addition, it is worth mentioning that the majority of these categories come under the heading of "sets with structure". There have also been attempts to do away with the "sets with" part of this and I shall talk about why one might wish to do this.

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Invited talk at the Workshop on Smooth Structures in Logic, Category Theory, and Physics held at Ottawa University, 1st - 3rd May 2009.

Updated to the version actually given (essentially just added more pictures)

Abstract:

Smooth manifolds are extremely nice spaces. The fact that they have charts means that a vast amount of the theory of Euclidean spaces can be easily transferred to manifolds. This makes for a very useful subject.

However, the charts also make manifolds very fragile: it is easy to do something to a manifold that makes it no longer a manifold. Taking a quotient by a group action is one such, looking at mapping spaces is another. Often, specific operations can be fixed - orbifolds fix the quotienting, infinite dimensional manifolds fix the mapping spaces - but systematic case-by-case fixing is a little unsatisfying. Over the years there have been several attempts to build a suitable category of "smooth objects" generalising smooth manifolds. The general method is to take some property that all manifolds have, which can be defined in a more robust way than charts.

In this talk I shall review some of these attempts, focussing particularly on the similarities between them. I shall try to motivate my own favourite: Frölicher spaces. In addition, it is worth mentioning that the majority of these categories come under the heading of "sets with structure". There have also been attempts to do away with the "sets with" part of this and I shall talk about why one might wish to do this.

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Oslo Topology Seminar, 17th April 2007

Partly repeated at Analysis and Topology in Interation in Cortona, 17th June 2008

I shall describe how to construct the Dirac operator on suitable loop spaces (i.e., loop spaces of string manifolds). There are two important aspects to the construction: firstly, that we use the space of smooth loops; secondly, that we first construct a cometric on the loop space. I shall also indicate how this method might or might not be adapted to define other operators on loop spaces, with particular interest in a semi-infinite de Rham operator.

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Algebraic Topology Special Session, BMC 2007

Infinite dimensional Riemannian manifolds have traditionally been divided into two types: strong and weak. One can generalise many of the standard constructions of Riemannian geometry to strong Riemannian manifolds but not to weak ones. In this talk I shall examine some of the basic results of Riemannian geometry in order to refine this classification. The purpose is to show that some constructions can be generalised to certain weak Riemannian manifolds. I shall conclude by describing which level of structure is available for the free loop space of a finite dimensional manifold and by explaining how this can be used in applications.

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Generalised cohomology theories are an important tool in algebraic topology. Part of the study of the theories themselves involves looking at their spaces of "operations" and "co-operations". In a paper in the Algebraic Handbook of Topology, Boardman, Johnson, and Wilson described the former as a comonad in a suitable category and the latter as an "enriched Hopf ring". The former of these is very elegant but not all that intuitive, the latter is less elegant and similarly not intuitive - at least, the "enriched" part. The missing part of these descriptions is a non-linear "tensor product", which was actually introduced by Tall and Wraith in 1970. I shall explain the various descriptions and show how both the unstable operations and co-operations have concise and straightforward descriptions as algebraic objects.

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Based on: Stable and Unstable Operations in mod p Cohomology Theories.

One of the pieces of baggage that comes with a graded cohomology theory is the family of operations. These are self-maps of the cohomology groups obeying certain obvious naturality conditions. There are two main types of operation: stable and unstable. An unstable operation acts only on the cohomology groups of a particular degree whilst a stable operation acts on the cohomology groups of any degree compatibly with the suspension isomorphism. It is clear, therefore, that a stable operation defines a family of unstable ones. However, even if one knows that an unstable operation came from a stable one it may not be easy to reconstruct that stable operation. What is remarkable about the Morava K-theories is that there is a straightforward way to do this.

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This is an introduction to the subject of the differential topology of the space of smooth loops in a finite dimensional manifold. It began as background notes to a series of seminars given at NTNU and subsequently at Sheffield. The topics covered are: the smooth structure of the space of smooth loops; constructions involving vector bundles; submanifolds and tubular neighbourhoods; and a short introduction to the geometry and semi-infinite structure of loop spaces. It is meant to be readable by anyone with a good grounding in finite dimensional differential topology.

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In this talk I describe a method by which one can construct over a suitable loop space an operator which is the analogue of the Dirac operator on a finite dimensional manifold.

The key step is to adapt an idea due to Jack Morava to construct an inner product on the cotangent bundle of the loop space. There is then a Hilbert bundle which is the fibrewise completion of the cotangent bundle. This bundle is used to construct the spin bundle so that the Clifford multiplication map extends to the domain of a connection allowing one to define the Dirac operator.

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I shall start by explaining just what "semi-infinite" means and how it gives rise to interesting structures in geometry. These structures are studied as part of String Theory and are closely related to Floer Theory. Manifolds which carry a "semi-infinite" structure include loop spaces and the theory seems to have particular simplicities when the original manifold is symplectic.

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In this seminar, we shall introduce semi-infinite manifolds and show how one may define cohomology theories dependent on this semi-infinite structure. In particular, we shall define a de Rham theory. With certain alterations, this is calculable for Wiener manifolds. The goal, however, is to calculate the theory for loop spaces and this is, as yet, an unsolved problem.

These were part of a short series of seminars given by myself and Paul Cooper, another graduate student at Warwick at the time, aiming to explain the connections between Mechanics and Geometry.

The original series was three lectures, of which I gave the first and third. Later, I gave two more seminars on similar themes.

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This seminar was given as a pre-seminar. John Jones was due to give a series of seminars on Elliptic Cohomology in the Geometry Seminar Series at Warwick and so I gave this seminar in the Junior Geometry Series as an introduction aimed at a lower level than John's talks were going to be aimed at.

The plain for this seminar was to go through an example of Elliptic Cohomology in action. The example chosen is called the A-hat-genus. It is actually a degenerate case and so is not strictly an elliptic genus. However, it is still a good introductory example.