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Are there any radical applications of algebraic topology? For example, in probability theory we look at sample spaces. Suppose the sample space is a torus (for example). Would computing homology groups and homotopy groups of the torus illuminate anything about the probability of certain events? If we have a random variable $X$ with some pdf $f_{X}(x)$ then we can transform it (e.g. $Y = X^2$) and find the pdf $f_{Y}(y)$ using the Jacobian. Is there any usefulness of computing various topological invariants of spaces in terms of probability theory?

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What do you mean by "radical"? – Qiaochu Yuan Feb 8 '12 at 18:53
I've heard of an application of homotopy theory to functional analysis (or maybe PDEs, not sure), but I don't know any more about it than that. – Charles Staats Feb 8 '12 at 19:04
Cohomology theories apparently have some use in engineering. – Jesse Madnick Feb 8 '12 at 19:26
To add to what was mentioned by Jesse Madnick, they also have applications in computational complexity theory. – André Nicolas Feb 8 '12 at 22:28
In most cases, the important thing with sample spaces is the sigma-algebra on them, which doesn't notice the kind of topological structure you mention. Using a torus for the sample space is no different from using a square or a 7-dimensional sphere. Your change of variable example is to do with structure on the space in which your random variable takes values, not with structure on the sample space. However, you can consider objects with interesting structure (locally compact groups, for instance) and study spaces of measures on them, but that seems slightly different from what you describe. – user16299 Feb 8 '12 at 22:39

If you're using "radical" in the 1980s sense, then there are lots of such things (in my mind). Something that might be cool is how things in algebraic geometry (or homological algebra) can be cast in the light of algebraic topology. A good read is Peter May's paper Derived Categories in Algebra and Topology.

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