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A graph complex valued field theory

Project description

A universal topological field theory with values in graph complexes

Topology, historically a topic of mathematics, has gained critical practical value and significantly renewed interest with the discovery of topological materials and advances in accessing topological states of matter. The ERC-funded GRAPHCPX aims to create a universal topological field theory with values in graph complexes uniting areas of mathematical physics, topology, homological algebra, and algebraic geometry. It could provide: a precise topological interpretation of a class of well-studied topological field theories; new tools to study objects like configuration and embedding spaces and potentially diffeomorphism groups; and a wealth of new algebraic structures on graph complexes, some of the most important objects in the field.

Objective

The goal of the proposed project is to create a universal (AKSZ type) topological field theory with values in graph complexes, capturing the rational homotopy types of manifolds, configuration and embedding spaces.
If successful, such a theory will unite certain areas of mathematical physics, topology, homological algebra and algebraic geometry. More concretely, from the physical viewpoint it would give a precise topological interpretation of a class of well studied topological field theories, as opposed to the current state of the art, in which these theories are defined by giving formulae without guarantees on the non-triviality of the produced invariants.

From the topological viewpoint such a theory will provide new tools to study much sought after objects like configuration and embedding spaces, and tentatively also diffeomorphism groups, through small combinatorial models given by Feynman diagrams. In particular, this will unite and extend existing graphical models of configuration and embedding spaces due to Kontsevich, Lambrechts, Volic, Arone, Turchin and others.

From the homological algebra viewpoint a field theory as above provides a wealth of additional algebraic structures on the graph complexes, which are some of the most central and most mysterious objects in the field.
Such algebraic structures are expected to yield constraints on the graph cohomology, as well as ways to construct series of previously unknown classes.

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Topic(s)

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Funding Scheme

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ERC-STG - Starting Grant

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Call for proposal

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(opens in new window) ERC-2015-STG

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Host institution

EIDGENOESSISCHE TECHNISCHE HOCHSCHULE ZUERICH
Net EU contribution

Net EU financial contribution. The sum of money that the participant receives, deducted by the EU contribution to its linked third party. It considers the distribution of the EU financial contribution between direct beneficiaries of the project and other types of participants, like third-party participants.

€ 1 162 500,00
Address
Raemistrasse 101
8092 Zuerich
Switzerland

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Region
Schweiz/Suisse/Svizzera Zürich Zürich
Activity type
Higher or Secondary Education Establishments
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Total cost

The total costs incurred by this organisation to participate in the project, including direct and indirect costs. This amount is a subset of the overall project budget.

€ 1 162 500,00

Beneficiaries (1)

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