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Johannes Nordström: Lectures

January 9, 2023 (jointly with Matt Turner)
TITLE: Examples of asymptotically conical G2-instanstons

ABSTRACT: We present examples of G2-instantons with dilation-invariant asymptotics on the “C7” asymptotically conical G2-metric on the anticanonical bundle of CP1 x CP1. The examples have cohomogeneity one which reduces the problem to solving an ordinary differential equation. We find solutions to these equations using a dynamical systems approach. This is joint work with Karsten Matthies.

Slides of Lecture (second part)

September 8, 2022
TITLE: Invariants of twisted connected sum G_2-manifolds

ABSTRACT: When trying to apply the h-cobordism theorem to prove that two manifolds are diffeomorphic, obstructions can often be captured in terms of invariants that measure the failure of a manifold with boundary to satisfy relations that hold for manifolds without boundary. I will discuss invariants of this kind for 7-manifolds with G_2-structures or stable complex structures, and recent progress with Crowley and Goette on applying these to holonomy G_2-manifolds obtained by variations of the twisted connected sum construction.

Slides of Lecture

June 10, 2022
TITLE: Rational homotopy theory of closed 7- and 8-manifolds

ABSTRACT: Deligne, Griffiths, Morgan and Sullivan proved that any closed Kähler manifold is formal in the sense of rational homotopy theory; thus their the rational homotopy type is completely determined by the cohomology algebra. However, it remains an open problem whether closed 7-manifolds with holonomy G_2 and closed 8-manifolds with holonomy Spin(7) must be formal. Looking for counter-examples is hard because the known constructions of G_2 and Spin(7)-manifolds are complicated, so even evaluating Massey products (which must vanish on a formal space) is involved.

I will describe new rational homotopy invariants, certain 4 and 5-tensors on the cohomology defined in a style similar to Massey products, but with less dependence on choices. On closed manifolds, they determine all Massey triple and fourfold products, but also capture some information that the Massey products do not. For closed simply-connected 7- and 8-manifolds their vanishing is not only necessary for formality but also sufficient, providing a more convenient test for formality than was previously available. This is joint work with Diarmuid Crowley and Csaba Nagy.

June 9, 2022
TITLE: Topology of extra-twisted connected sum G_2-manifolds

ABSTRACT: We have a good understanding of what invariants are needed to classify closed 2-connected 7-manifolds (possibly equipped with a homotopy class of G_2-structures) up to diffeomorphism. This involves obvious primary invariants like the cohomology, but also more subtle secondary invariants defined in terms of spin coboundaries.

The twisted and extra-twisted connected sum constructions produce a variety of examples of closed G_2-manifolds, starting from complex algebraic geometric data like Fano 3-folds. One of the G_2 invariants, nu, can be computed via an intrinsic analytic reformulation. For twisted connected sums, all the other invariants are known too, in part thanks to a computation by Alge Wallis that uses almost complex coboundaries instead of spin ones. I will give an overview of these results, and report on work in progress on computing the full invariants of a class extra-twisted connected sums. This is joint work with DIarmuid Crowley and Sebastian Goette.

Slides of Lecture

September 14, 2020 (Jointly with Mark Haskins)
TITLE: Complete cohomogeneity one solitons for Bryant’s closed G2-Laplacian flow

ABSTRACT: We will report briefly on some of our recent results, numerical investigations (and resulting conjectures) and work still-in-progress on complete noncompact cohomogeneity one solitons in Bryant’s closed G2-Laplacian flow (G2-solitons for short). Our results include the construction of new complete noncompact (asymptotically conical) shrinking and expanding (gradient) G2-solitons on the total space of the bundle of anti-self dual 2-forms on S^4 and CP^2; we conjecture the existence of a 1-parameter family of asymptotically conical steady (gradient) G2-solitons on the total space of the bundle of anti-self dual 2-forms of CP^2 and describe some analytic and numerical evidence supporting this conjecture. In other more widely studied geometric flows (e.g. Ricci flow, codimension 1 mean curvature flow and Lagrangian mean curvature flow) solitons are key to understanding singularity formation and for attempting to continue the flow after the first singular time. We discuss very briefly how our results reveal both some similarities and differences with the behaviour of solitons currently known in Ricci flow and Kahler-Ricci flow.

Slides of Lecture

September 12, 2019
TITLE: Topology in Special Holonomy

ABSTRACT: Nordstrom will discuss what scant information has been available about obstructions to the existence of G2-holonomy metrics on closed 7-manifolds and then go on to explain some recent progress on the topology of known and recently constructed examples. Many of these 7-manifolds belong to classes where diffeomorphism classification results have been proved or are within reach, making it possible to map out a portion of the landscape of closed G2-manifolds. In particular, we can exhibit phenomena such as homeomorphic, but not diffeomorphic, examples and the non-connectedness of the moduli space of G2-metrics on specific 7-manifolds.
Slides of lecture

June 7, 2019
TITLE: Gluing constructions for associatives

ABSTRACT: I will discuss two related gluing constructions of associative submanifolds that I have been promising to complete for a long time. One is to construct associative 3-spheres in many twisted connected sum G2-manifolds as analogously twisted connected sums of complex lines. The other is to resolve self-intersection singularities of immersed associatives by gluing in Lawlor necks. If the immersed associative is unobstructed, then it will deform uniquely under small deformations of the ambient G2-structure. If the self-intersection is transverse in a 1-parameter family of G2-structures (so in particular, the immersed associative fails to be embedded at a single parameter t0) this yields a second family of associatives that is created/destroyed at t0, as envisaged by e.g. Donaldson and Joyce.

January 8, 2019
TITLE: Building blocks for twisted connected sums

ABSTRACT: The twisted connected sum method of Kovalev and Corti-Haskins-Nordström-Pacini allows the construction of large numbers of closed manifolds with holonomy G_2. The “building blocks” of the construction are a certain kind of closed Kähler 3-folds. The aim of the talk is to review the basics of the construction (and the “extra-twisted” version of Crowley-Goette-Nordström), and to discuss some algebraic questions concerning building blocks, whose answers would further expand the supply and variety of examples of G_2-manifolds.

January 9, 2019
TITLE: The matching problem for twisted connected sums

ABSTRACT: Applying the twisted connected sum construction of G_2-manifolds requires finding special diffeomorphisms between anticanonical K3 surfaces in building blocks. The aim of this talk is to explain how this problem can be solved by establishing sufficient conditions for a generic element of a family of K3s to appear as an anticanonical divisor in some element of a set of blocks. We will also discuss the matching problems involved in constructing associatives in or instantons on twisted connected sums by gluing, in joint work with Menet and Sa Earp.

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.