The talk will be about two theorems. One proved in joint work with Tilman Bauer, Nitu Kitchloo, and Dietrich Notbohm says:
Theorem: Let B be a CW complex and X be the loop space on B. Then X is homotopy euivalent to a closed, smooth, parallellizable manifold if and only if the homology of X is finitely generated.
A Lie group is homotopy equivalent to the loops on its classifying space, so the theorem indicates that finite loop spaces are a lot like Lie groups. There is however the following theorem, which is a strenghtening of a famous result due to Hilton and Roitberg. This is joint work with Kasper Andersen, Timan Bauer and Jesper Grodal.
Theorem. There exists a finite loop space X whose rational cohomology is different from the rational cohomology of any Lie Group. The rank is 66 and this is the minimal rank for such an example (dimension is 1254).