This is a joint work with G.Garkusha. In the talk I will discuss the construction of fibrant replacements for spectra consisting of Thom spaces (suspension spectra of varieties and algebraic cobordism MGL being the motivating examples) that uses the theory of framed correspondences. As a consequence we get a description of the infinite loop space of MGL in terms of Hilbert schemes.

Let $G$ be a compact connected Lie group and $K$ a closed connected subgroup. In 1950 Cartan determined the real cohomology ring of the homogeneous space $G/K$. The formula can be written as a torsion product involving the cohomologies of the classifying spaces BG and BK, which are polynomial algebras. Around 1970 this result was extended to other coefficients through work of Baum, Gugenheim-May, Husemoller-Moore-Stasheff, Munkholm and Wolf. However, the isomorphism was only shown to be additive in this case. I will present a recent result saying that the isomorphism is in fact multiplicative and natural in the pair $(G,K)$, provided that 2 is invertible in the coefficient ring. I will also discuss the main new ideas of the proof, which involve homotopy Gerstenhaber algebras.

This is to just get started. We will assign topics and speakers. We will also recall some results from last year.