Report from Villars (4 in a series)

I haven't been able to get around to skiing since the last update - I may, or may not, go out in the slopes after this updates. The weather is growing warmer and wetter; and doesn't really invite to skiing as it previously did.

However, we have had more talks. First out, yesterday evening, was Pascal Lambrechts

Coformality of the little ball operad and rational homotopy types of spaces of long knots

The theme of interest for this talk was long knots; i.e. embeddings of [tex]\mathbb R[/tex] into [tex]\mathbb R^d[/tex] such that outside some finite region in the middle, the embedding agrees with the trivial embedding [tex]t\mapsto(t,0,0,\dots,0)[/tex]. The space of all such is denote [tex]\mathcal L[/tex], and the item of study is more precisely the rational homology and rational homotopy of the fiber of the inclusion of [tex]\mathcal L[/tex] into the space of all immersions of [tex]\mathbb R[/tex] into [tex]\mathbb R^d[/tex].

Vassiliev approached this by constructing a spectral sequence converging to the homology for high enough d, and where the first term has a combinatorial description in terms of chord diagrams. The idea, building on some preparatory work by Dev Sinha, is to replace the long knot with a sequence of very many distinct points on the knot. Thus you end up in the configuration space of many knots, which can be put in the context of being (a compactification of) configuration spaces; which in turn form a cosimplicial space. These spaces, in fact, are (more or less) the Fulton-MacPherson operad, and using this correspondence, Lambrechts is able to show that the spectral sequence by Vassiliev collapses.

Paolo Salvatore had a speech on mainly the same theme; however, I haven't really gotten any substantial notes out of that speech.

Today, however, started with Dev Sinha

The duality between Lie and Comm revisited

Introducing a function called a Configuration pairing; which takes a directed graph on [n] and a rooted, half-planar tree with leafs in [n] to an integer; by sending to 0 if the map that takes an edge to the lowest vertex of the path in the tree between the corresponding leafs is not bijective; and sending to (-1):sup:k if there are k edges such that the corresponding path runs right to left otherwise, we can get a function on the pairs that vanishes on anti-symmetry and Jacobian identity of the trees; and on anti-symmetry, loops and the Arnold identity - meaning [tex]a\to b\to c + b\to c\to a + c\to a\to b[/tex] vanishes - on the graph half; we get a perfect pairing between the slices Lie(n) generated by the trees and the slices Eil(n) generated by the graphs. Lie(n) has a basis consisting of tall trees (i.e. (((((((1,i1),i2),....,in)) and Eil(n) a basis of long graphs (i.e. paths beginning with 1). The pairing matches a tree and a graph if they have the same indices in the same order.

Using this, and involving the binomial operad, he then manages by introduction of co-Lie-brackets by removing an edge of the graph and taking antisymmetric differences to make the Eil operad describe Lie co-algebras; and thus gives a way to construct a linear duality between DG-Lie-algebras and DG-Lie-Coalgebras which does not go the way over bar and cobar constructions.

After Sinha, Martin Markl spoke on

Cohomology operations and the Deligne conjecture

He characterizes the renaissance of operad theory as being all about calculating Kontsevich graph (co)homology in various settings. More specifically, he looks on the operad of "natural operations" on multilinear functions - i.e. anything given by free compositions and pointwise multiplications. This turns out to be a rather big object, and many conjectures and questions were asked and discussed. In the ideal case, Markl wishes, the structure of the this operad will give information about the theory it's based in.

Finally, Jonathan Scott had a speech on

Co-rings of operads

In which he gives a way of viewing an operad as a monoid, whose left modules are the algebras and right modules are the co-algebras of the operad.

That is, basically, all there is to report at this point. For all things, just poke me if you're interested in more details.