Time and place: Thursdays 3:30-4:30 in REC 116

Organizer: Ralph Kaufmann. Contact organizer.

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.