From Hodge theory to combinatorics and geometry

We propose an approach to the Dodziuk-Singer conjecture, a central conjecture in geometric topology, specifically concerning the great challenge that understanding aspherical manifolds can pose, based on newly developed tools from combinatorial commutative algebra and combinatorial Hodge Theory, and discuss several intermediate problems along the way.

The main idea is based on a connection tocommutative algebra via the partition complex, an interpretation of local cohomology that allows for a translation between data contained in the L2 cohomology of a manifold and Lefschetz properties of toric varieties associated to them.Additionally, we outline connections to other approaches to the Dodziuk-Singer conjecture as well as special cases, such as the Hopf and Charney-Davis conjectures, and propose ideas to connect the different aspects of the viewpoints into one.Finally, we discuss problems related to the methods proposed, in particular focussing on unrealised and unexploited relations between combinatorics, Hodge theory and geometry.

We discuss in particular deformations of polyhedra and metrics, as well as expansion and connectivity.