Andrew Stacey

Andrew Stacey
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By: Andrew Stacey
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Andrew Stacey

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Mon, 3rd Aug 2009 (Seminars)

Comparative Smootheology: Workshop on Loops, Strings and Moduli Spaces

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.


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.

This is based partly on Preprints/smthcat

This talk is a beamer presentation and uses TikZ for the drawings. I now use lots of my own macros for beamer presentations so it's not practical to make the source available. However, I'm still learning how best to do the pictures (particularly 3-dimensional ones) so I'm including the source code of some of the pictures in case they are of interest to others. I don't claim any kind of expertise, but I make it available just in case it's useful to anyone.

Please do not print out the beamer or trans versions of this talk. They are intended solely for on-screen viewing (the trans version has just one overlay per frame so is a little simpler to step through). If you really want to print something out (why?) then print out the handout version. Better still, take a look at the original article: Preprints/smthcat.

  • Presentation version in PDF. This is what is actually shown on the screen.
  • Slides version in PDF. This was the backup version for an OHP in case of computer failure. For the most part, it is a one slide per frame version of the presentation.
  • Handout version in PDF. Note: this uses pgfpages to format it as 4-up. With double-sided printing, this only uses 5 pieces of paper. Neat, huh? This version skips a few "redundant" slides to keep the number of pages down. No actual information is lost, though.
  • Pictures: In the presentation, some of these contain overlay commands (demonstrating the fantastic integration of TikZ with beamer; almost as if they were made for each other) which I've left out of the samples.
    • Cobordism: PDF SRC.
    • Cobordism of cobordisms: PDF SRC.
    • Smooth map of manifolds: PDF SRC SVG. Although this picture appears in the original version of this talk, the method used to produce the torus is different. This picture compiles much faster as the torus is built up from coloured rectangles rather than rotated circles.

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Last modified on:
Mon, 2nd Nov 2009