So we hit the pistes during monday morning, those of us who actually
already are here. Me, Bruno Vallette (Hi Stockholm!), Arne Weiner, Miles
Gould, Paul Eugene Parents and Jonathan Scott, Dev Sinha and Muriel
Livernet. Skiing was MARVELOUS. Me, Arne and Miles shot off on our own,
and damn did we have a good time.
As I'm writing this, they're still out there - I went back when the pain
in my legs caused tears in my eyes for just turning on the skis. The
techniques were solid as concrete. The muscles not so much. It took half
an hour in the sauna to get to the point where I actually was able to
Among the more amusing things that happened was that when I was stopping
to wait up for Arne and Miles in a narrow forest path, I ended up
standing too close to the edge, which subsequently gave up and dropped
me down into a few meters of powder just under a fern tree. Getting out
of there was awkward - to begin with my legs were in the wrong angle to
get out of the ski bindings; and once Miles helped me out, the only
reason I wasn't buried in snow to my shoulders was that I packed it as I
stood on it, and the carrying point ended up being roughly waistdeep.
Go on. You try it. Get up from waistdeep loose powder snow. Straight up,
a meter or so, onto that hardened shell that you once were skiing on.
It's an extremely amusing and rather hard exercise.
Now, for the mathematics. The (only) talk today was by Bruno Vallette,
Manin products and Koszul duality
For associative quadratic algebras, given as a quotient of the free
tensor algebra on a space [tex]V[/tex] by [tex]A(V,R)=T(V)/(R)[/tex],
Manin defines two different products, by [tex]A(V,R)\circ
A(W,S)=A(V\otimes W,\tau(R\otimes W^2+V^2\otimes S))[/tex] for
[tex]\tau a\otimes b\otimes c\otimes d=a\otimes c\otimes b\otimes
d[/tex] the standard twisting homomorphism; and [tex]A(V,R)\bullet
A(W,S)=A(V\otimes W,\tau(R\otimes S))[/teX]. The main and most
relevant result here is that the two products are Koszul dual to each
other, i.e. [tex](A\circ B)^!=A^!\bullet B^![/tex].
Using the concept of lax 2-monoidal categories, it is possible to
generalize this to basically any possible interesting category - with
the same particular construction viable for Algebras, Nonsymmetric and
symmetric operads, dioperads, coloured operads, properads, PROPs (and
probably ½PROPs as well).
Notes on this will be forthcoming, and I'll post here once I get a