337 words on Uni
Theoretically I had two seminar talks to give on Monday. But due to my flu, I wasn't able to prepare one of them and thus had to postpone it. Quite embarrassing. Luckily I had prepared enough of the one for the other seminar, so it only needed finishing touches which I managed despite having headache and all.
I spoke on the 1976 paper The Homology Theory of the Closed Geodesic Problem by Micheline Vigué-Poirrier and Dennis Sullivan. While the article gives statements on geodesics in conjunction with another paper, it doesn't deal itself with the geodesics but rather uses the theory of minimal DGAs to associate a statement on the Betti numbers of the free loop space of a manifold to a statement on the cohomology ring of the manifold itself. While it looked quite hard a first, it turned out to me mostly linear algebra.
When trying to work out the proofs of the various propositions in the paper, I actually stumbled about a few things. Most notably, I couldn't actually prove them. It turned out that some of the prerequisites for the propositions are stated more general than they should be: for (almost) arbitrary DGAs rather than the special DGAs you get as minimal models for a compact simply-connected manifold. Before finding this, Jan-Philipp and me spent quite a bit of time trying to prove this missing statements, until Thomas eventually came up with a counter-example, thus showing we have to make additional assumptions. Looking at a couple of other arguments more closely, revealed the same dependence on those particular kind of DGA, which has the correct properties by construction.
Did I mention that notation in the paper sucks, sort-of?
Note that the symbol
Λ is used in three distinct ways.
After the giving talk I was quite exhausted and had to take some additional aspirins. Probably I should have stayed in bed altogether - but that way I'd have an ugly mountain of talks-to-be-given waiting after recovery.
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