a.kovalev a.kovalev

Consortium Distinguished Lecture Series
Alexei Kovalev

Coassociative fibrations

Alexei Kovalev, University of Cambridge

Wednesday, April 12, 2023, 5:00pm

Ungar Building, Room 528B

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Abstract: The Euclidean space R7 possesses some amazing geometrical features as it is a unique higher-dimensional vector space admitting a structure akin to the vector product in R3. As a particular attribute, this so-called G2 structure allows us to define some intriguing submanifolds. One class of such submanifolds is known as co-associative 4-folds. Remarkably, it can be shown that these coassociative submanifolds are volume-minimizing among all nearby submanifolds if the respective G2 structure is torsion-free. In particular, this bears an analogy to special Lagrangian submanifolds of Calabi–Yau manifolds. I will explain the role of differential forms in setting up a G2 geometry in 7 dimensions and the deformation theory of coassociative submanifolds. Then I will give informal description of different constructions of examples where the deformation family "fills" the ambient 7-manifold M, thereby defining a fibration of M by coassociative submanifolds (with some singular fibres).

On nearly parallel G2-manifolds.

Thursday, April 13, 2023, 5:00pm

Ungar Building, Room 528B

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Abstract: A nearly parallel G2–structure on a 7-manifold can be given by a 3-form φ of special algebraic type satisfying a differential equation dφ = τ * φ for a non-zero constant τ. We consider nearly parallel G2–structures on the Aloff–Wallach spaces and on regular Sasaki–Einstein 7-manifolds. We give a construction and explicit examples of associative 3-folds (a particular type of minimal submanifolds) in these spaces. Joint work with M. Fernández, A. Fino and V. Muñoz.