Abstracts for MPI-Oberseminar

Following the ideas of Quillen and by means of model category structures, Hinich, Goerss and Hopkins have developped a cohomology theory for (simplicial) algebras over a (simplicial) operad. Thank to Koszul duality theory of operads, we describe the cotangent complex to make these theories explicit in the differential graded setting. We recover the known theories as Hochschild cohomology theory for associative algebras and Chevalley-Eilenberg cohomology theory for Lie algebras and we define the new case of homotopy algebras.

One more interaction between theoretical physics and mathematics will be discussed. There is a famous combinatorial problem to enumerate the so-called alternating-sign matrices, which are a generalization of the permutation matrices. This problem was solved by Doron Zeilberger in 1995. Much simpler solution was given by Greg Kuperberg in 1996. It was based on the one-to-one correspondence between the alternating-sign matrices and the states of the statistical six-vertex model.

Representations up to homotopy of Lie algebras have attracted recently much attention. On the other hand J. Baez has introduced a way to build a homotopy Lie algebra out of a Lie algebra and an n-cocycle. We show in this work a common framework enabling to generalize both notions (replacing Lie algebras by homotopy Lie algebras) and extend them for other types of algebras (commutative and associative). The main tool is the language of homological vector fields on products of formal manifolds. This is a joint work with John Baez.

In this talk we will discuss the following problem: Given a triple of integers (r,s,t), which finite simple groups are quotients of the triangle group T(r,s,t)? This problem has many applications, especially concerning Riemann surfaces and Beauville surfaces. In the talk we will focus on the group theoretical aspects of this problem.