Refined tautological relations on moduli spaces of curves
The key purpose of this project is to study tautological relations on the level of closed differential forms or even integral cochains (i.e. we don't take classes in cohomology).
The moduli space M_g of curves of genus g is a central object of study in modern geometry and topology. A key question is to describe the algebra of all universal cohomology classes for curves, that is, the cohomology algebra of M_g itself. As this question seems too hard at the moment, one focuses on the so-called tautological algebra, a subalgebra of cohomology generated by some "obvious'' classes obtained using diagonals. A central problem is to find (all) relations between these generators.
The key purpose of this project is to study tautological relations on the level of closed differential forms or even integral cochains (i.e. we don't take classes in cohomology). A central place is occupied by the so-called Green's functions associated to the diagonals. The refined tautological algebra that one obtains in this way turns out to be relevant for string theory, and in fact part of the project is motivated by an influential paper of August 2013 by string theory pioneers Eric d'Hoker and Michael Green focusing on an invariant recently discovered by Shouwu Zhang and Nariya Kawazumi and studied extensively by the applicant.
Part of this project is to put the work by d'Hoker and Green in a broader mathematical context, and study the ``higher'' Zhang-Kawazumi invariants suggested in that work. The new theory that lies waiting to be discovered is without doubt both extremely beautiful and combinatorially very rich."