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Johannes Nordström: Lectures

January 12, 2018
TITLE: G2-instantons on twisted connected sums

ABSTRACT: The twisted connected sum construction of G_2-manifolds glues two pieces along a common boundary that is a product of a K3 surfaces and a 2-torus. I will discuss joint work with Gregoire Menet and Henrique Sa Earp on constructing G_2-instantons on such G_2-manifolds. In the example I will describe, both halves of in the construction admit positive-dimensional moduli spaces of Hermitian-Yang-Mills instantons. Ensuring that their images in the moduli space of ASD instantons on the boundary K3 intersect transversely allows the application of a gluing theorem for G_2-instantons due to Sa Earp and Walpuski.

September 11, 2017
TITLE: New asymptotically conical G2-manifolds

ABSTRACT: Until now, the only examples of asymptotically conical G_2-manifolds that have been proved to exist are the three cohomogenity one examples of Bryant and Salamon. I will discuss cohomogeneity one examples on the total spaces of complex line bundles over S2 x S3, whose asymptotic links are quotients of the standard nearly Kähler S3 x S3.

June 5, 2017
TITLE: Twisted connected sum G2-manifolds

June 6, 2017
TITLE: Topological invariants of G2-manifolds

September 6, 2016
TITLE: Complete and conically singular G2-manifolds of cohomogeneity one

ABSTRACT: Bryant and Salmon’s cohomogeneity 1 examples of complete, asymptotically conical G_2-manifolds provide a model for desingularising compact G_2-manifolds with conical singularities; however no examples of the latter are yet known, and there are also no further known examples of asymptotically conical G_2-manifolds. Theoretical physicists such as Cvetic-Gibbons-Lu-Pope and Brandhuber-Gomis-Gubser-Gukov have considered complete cohomogeneity 1 G_2-manifolds that are “asymptotically locally conical”–the model at infinity is a circle bundle over a cone–and which in a 1-parameter family converge to an asymptotically conical manifold. However, only some of these families have been studied rigorously (Bazaikin-Bogoyavlenskaya).

I will discuss joint work in progress with Foscolo and Haskins on these families, and some of their limits, which include a new asymptotically conical G_2-manifold and a conically singular G_2-manifold with locally conical asymptotics. The latter may provide an avenue to construction of compact G_2-manifolds with conical singularities.