Homology of labelled configuration spaces via spectral Lie algebra methods

The goal of this project is to produce new computations of the mod p homology of labelled configuration spaces of a general manifold with labels in an arbitray spectrum. These computations have remained outstanding challenges for decades except in special cases.

Recently developed techniques in stable homotopy theory, including spectral Lie algebras and synthetic homotopy theory, have proven to be useful in understanding the algebraic structure on the homology groups of labelled configuration spaces in universal cases, but are less attuned to the topology of the input manifolds.

In this project, I plan to attack the problem by incorporating the homotopy-theoretic methods with approaches that take into account the geometry of labelled configuration spaces.

New computations of these objects will have applications in a range of fields including stable homotopy theory, string topology, embedding calculus of manifolds, mathematical physics, and motion planning. The supervisor will be Prof. Nathalie Wahl at the department of Mathematics of the University of Copenhagen.