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
However, we have had more talks. First out, yesterday evening, was
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
That is, basically, all there is to report at this point. For all
things, just poke me if you're interested in more details.