An approach to the four colour theorem via Donaldson- Floer theory

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• Tomasz Mrowka (Massachusetts Institute of Technology)
• Wednesday 16 August 2017, 12:00-13:00
• Seminar Room 1, Newton Institute.

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SYGW05 - Symplectic geometry - celebrating the work of Simon Donaldson

This talk will outline an approach to the four colour theorem using a variant of Donaldson-Floer theory.

To each trivalent graph embedded in 3-space, we associate an instanton homology group, which is a finite-dimensional Z/2 vector space. Versions of this instanton homology can be constructed based on either SO(3) or SU(3) representations of the fundamental group of the graph complement.  For the SO(3) instanton homology there is a non-vanishing theorem, proved using techniques from 3-dimensional topology: if the graph is bridgeless, its instanton homology is non-zero. It is not unreasonable to conjecture that, if the graph lies in the plane, the Z/2 dimension of the SO(3) homology is also equal to the number of Tait colourings which would imply the four colour theorem.   This is joint work with Peter Kronheimer.

This talk is part of the Isaac Newton Institute Seminar Series series.

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