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From Hodge theory to combinatorics and geometry

Objective

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 to
commutative 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.

Coordinator

THE HEBREW UNIVERSITY OF JERUSALEM
Net EU contribution
€ 1 456 197,00
Address
Edmond J Safra Campus Givat Ram
91904 Jerusalem
Israel

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Activity type
Higher or Secondary Education Establishments
Other funding
€ 1 456 197,00

Participants (1)

KOBENHAVNS UNIVERSITET
Denmark
Net EU contribution
€ 491 928,00
Address
Norregade 10
1165 Kobenhavn

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Region
Danmark Hovedstaden Byen København
Activity type
Higher or Secondary Education Establishments
Other funding
€ 491 928,00