In the first half of the project we have used physical ideas (Feynman diagrams) to construct (rational or real) models for configuration spaces of points on manifolds.
Our models exhibit all the necessary algebraic structures to be used in the Goodwillie-Weiss calculus on the topological side.
They are hence the main building blocks of the bridges between the quantum field theory on the physical side and the topology.
We have already used these bridges, as far as built, to solve a few open conjectures and problems in topology. These include:
i) We showed that for a simply connected manifold the real homotopy type of the configuration space only depends on the real homotopy type of the manifold.
Generally, we provided models for configuration space of points on manifolds capturing their real or rational homotopy type.
ii) We computed the rational homotopy type of higher dimensional knot and embedding spaces (in codimension at least 3)
iii) We determined the rational homotopy type of the framed little n-disks operad, and the real homotopy types of configuration spaces of framed points.
iv) We computed the automorphisms of the (rationalized) little n-disks operad, expressing the answer through a graph complex of Feynman diagrams.
v) We defined a higher genus version of the Grothendieck-Teichmüller Lie algebras.
vi) We studied the cohomology of and algebraic structures on several graph complexes, as well their interrelation.
The results have presented to the scientific community through many conference and workshop talks be project members during the duration of the project.
