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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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    Also, radical is, like, totally egregious. Bill S. Preston said so.2012-02-08

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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.