The actual abstracts are here: math.virginia.edu/2019/09/mid-at…

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Then the homotopy groups of H(M) tell you about obstructions to h-cobordisms being trivial.

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(A trivial h-cobordism over M just looks like MxI, which is diffeomorphic to M, so I *think* the idea here is that if you can find an h-cobordism H between M and N, then M~N, and if H is trivial, this forces M≅N.)

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Given a group G, you can consider representations of G into vector spaces and get something called the "representation ring". If you replace vector spaces with sets, you get the "Burnside ring", which is supposed to capture how G can act on sets.

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Then Segal's conjecture (theorem!) essentially says that you can recover the Burnside ring of G from the stable homotopy groups of the classifying space BG (modulo many many details)

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The idea is to produce a type of localization that "inverts" these operations.

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In operads, we can do a similar construction: given an operad O and a submonoid W, we can "localize at W" to get L_W(O).

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So there is some natural sequence W -> O -> L_W(O), and we can consider the pairs of categories arising from W and O, say C(W) and C(O).

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An open question here is whether or not there are any equivalences here, i.e. does the operad localization commute with the categorical one?

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