01/29:
Craig Westerland (Univ. of Melbourne): Stabilization of Hurwitz spaces.
Abstact:
We will describe the Hurwitz space of branched covers of the disc, and
study its group completion under "pants multiplication." We will
show that this stabilization is a double loop space, and give some
evidence that its rational stable homology is very small. If time
permits, we will relate this to some number-theoretic conjectures
regarding the asymptotic growth of the number of branched covers of the
line in finite characteristic.
02/12: Jonathan Scott
(Cleveland State Univ.): Koszul
complexes of quadratic operads and morphisms up to strong homotopy.
Abstract:
Morphisms that commute with a certain algebraic structure up to strong
homotopy is normally described using "standard constructions" such as
the bar construction or the Chevalley-Eilenberg complex for associative
and Lie algebras, respectively. The operads for associative and
Lie
algebras are quadratic operads. Any quadratic operad has a Koszul
complex, that turns out to be a co-ring over the operad and hence
determines a comonad on the category of algebras. We show that
the
resulting Kleisli category is precisely the category of algebras and
morphisms up to strong homotopy, and extend the result to strong
homotopy algebras. This is joint work with Kathryn Hess (EPFL).
02/28: Leonid Chekhov
(Steklov Institute Moscow): Graph description of Teichmuller spaces of Riemann surfaces with
orbifold points
Abstract:
The fat graph (combinatorial) description of Teichmuller
spaces of Riemann
surfaces with holes (R.C.Penner and V.V.Fock) was recently advanced to the
case of Riemann surfaces with orbifold points (Fock+Goncharov, L.Ch.). The
Thurston theory of these surfaces is equivalent to the one of windowed
surfaces by Kaufmann and Penner, but probably the most interesting object
to investigate are algebras of geodesic functions (governed mainly by
Goldman bracket). There will be, first, a short excursion into the graph
description of Teichmuller spaces followed by introducing general algebras
of geodesic functions (classical and quantum). We then specify these
algebras to two important cases related to A_n and D_n algebras.
04/23: Ralph Kaufmann
(Purdue):
Stringy Singularities
Abstract: We discuss stringy constructions for singularities with symmetries.
05/07: Brian Munson
(Wellesley):
A stable range description of the space of link maps
Abstract:
For smooth manifolds P, Q, and N, let Link(P,Q;N) denote the
space of smooth maps of P in N and Q in N such that their images are disjoint. I will discuss the connectivity of a "generalized linking
number" from the homotopy fiber of the inclusion of Link(P,Q;N) into
Map(P,N)xMap(Q,N) to a certain cobordism space of manifolds over a space
which is a homotopy theoretic model for the intersections of P and Q. The
proof of the connectivity uses some easy statements about connectivities
in the world of smooth manifolds as a guide for obtaining similar
estimates in a setting where the tools of differential topolgy do not
apply. This is joint work with Tom Goodwillie.
