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Autumn 2017

Leeds Geometry Seminar

Organiser: Gerasim Kokarev

Time and venue:
Wednesday 3pm, Roger Stevens LT09. Coffee/Tea served in the Common Room after 4pm.
Programme:

4 October 2017
Josh Cork (Leeds) Symmetric calorons and the rotation map
Abstract. There has long been interest in finding symmetric solutions to variational problems, and integrable systems. We shall look at calorons - periodic anti-self-dual connections - and their symmetries. In particular, unlike other similar objects, calorons reveal a somewhat special discrete symmetry called the rotation map, which acts like a gauge transformation but is not periodic itself. We shall state results regarding the fixed points under cyclic symmetry groups incorporating the rotation map, give an overview of the different techniques involved in constructing such solutions.

18 October 2017
Stuart Hall (Newcastle) Bounding the invariant spectrum of the Laplacian
Abstract. I will report on an ongoing project with Tommy Murphy to understand spectrum of the Laplacian of a Riemannian metric invariant under the action of a compact Lie Group G. We are interested in questions such as can you bound the kth-invariant eigenvalue independent of the metric? Are such bounds sharp and are there smooth metrics that achieve them? The only large family of manifolds that have a substantial set of answers to such questions are toric Kahler manifolds. I will try to detail what is known here and how techniques might be generalised to other settings.

25 October 2017
Fritz Hiesmayr (Cambridge) Index and spectrum of minimal hypersurfaces arising from the Allen-Cahn construction
Abstract. The Allen-Cahn construction is a method for constructing minimal surfaces of codimension 1 in closed manifolds. In this approach, minimal hypersurfaces arise as the weak limits of level sets of critical points of the Allen-Cahn energy functional. In my talk I will give a brief overview of this construction, and then present my work relating the variational properties of the hypersurfaces arising in the limit to those of the Allen-Cahn energy functional. For instance, bounds for the Morse indices of the critical points lead to a bound for the Morse index of the limit minimal surface; time permitting I will sketch a proof of this.

1 November 2017
Giuseppe Tinaglia (KCL) Collapsing ancient solution of mean curvature flow
Abstract. Understanding the geometry of ancient solutions for mean curvature flow is key to study singularities of mean curvature flow. In this talk, I will describe the construction of the unique compact convex rotationally symmetric ancient solution of mean curvature flow contained in a slab. This is joint work with Bourni and Langford.

8 November 2017
Asma Hassannezhad (Bristol) Higher order Cheeger inequalities for the Steklov eigenvalues
Abstract. In 1970 Cheeger obtained a beautiful geometric lower bound for the first nonzero eigenvalue of the Laplacian in term of an isoperimetric constant. Inspired by the Cheeger inequality, Cheeger type inequalities for the first nonzero Steklov eigenvalue have been studied by Escobar, and recently by Jammes. The generalization of the Cheeger inequality to higher order eigenvalues of the Laplacian in discrete and manifold settings has been studied in recent years. In this talk, we study the higher-order Cheeger type inequalities for the Steklov eigenvalues. It gives an interesting geometric lower bound for the k-th Steklov eigenvalue. It can be viewed as a counterpart of the higher order Cheeger inequality for the Laplace eigenvalues, and also as an extension of Escobar's and Jammes' results to the higher order Steklov eigenvalues. This is joint work with Laurent Miclo.

15 November 2017
Kirill Krasnov (Nottingham) The Bryant-Salamon construction of metrics of holonomy G2 and its generalisation.
Abstract. As Calabi-Yau manifolds in (real) dimension 2n are manifolds of special holonomy SU(n) and admit covariantly constant spinors, 7D metrics that admit a covariantly constant spinor are special holonomy with holonomy contained in the exceptional group G2. As in the Calabi-Yau case, such manifolds are automatically Ricci flat. G2 manifolds are of particular importance for string theory, because 11D supergravity can be compactified on them to four dimensions, while preserving supersymmetry.

First complete (but non-compact) examples of G2 holonomy manifolds were constructed by Bryant and Salamon in 1989. In one variant of the construction, it provides a G2 holonomy metric in the total space of an R3 bundle over an anti-self-dual Einstein manifold. There are two complete G2 metrics that can be constructed this way, because there are only two positively curved gravitational instantons - S4 and CP2. The corresponding Bryant-Salamon metrics are asymptotically cones over the twistor spaces of S4 and CP2.

After reviewing the above, I will describe a generalisation of the Bryant-Salamon construction in which one obtains a G2 holonomy metric in the total space of an R3 bundle over a four-dimensional base, but the base is no longer required to be a gravitational instanton. Instead, a certain system of four-dimensional PDE's on an SU(2) connection needs to be solved. Every solution can then be lifted to a 7D metric of G2 holonomy. This provides many new local examples of G2 holonomy manifolds.


22 November 2017
Martins Bruveris (Brunel University London) Riemannian geometry of the diffeomorphism group
Abstract. This talk will survey the relationship between right-invariant Riemannian metrics on the diffeomorphism group and PDEs. I will concentrate in particular on Euler's equation for incompressible fluids, which can be interpreted as the geodesic equation on the volume-preserving diffeomorphism group. Afterwards I will discuss some mathematical results about the Riemannian geometry of diffeomorphism groups equipped with Sobolev metrics, with emphasis on completeness results and the theorem of Hopf-Rinow.

29 November 2017
Gerasim Kokarev (Leeds) Conformal volume, eigenvalue problems, and related topics
Abstract. I will give a short survey on the classical inequalities for the first Laplace eigenvalue on Riemannian manifolds (such as the inequality in terms of the L^2-norm of the mean curvature due to Reilly in 1977, the inequality in terms of the conformal volume due to Li and Yai in 1982, and due to El Soufi and Ilias in 1986), tell about related history and questions. I will then discuss results concerning their versions for the higher Laplace eigenvalues as well as estimates for the number of bound states of Schrodinger operators.

6 December 2017
Alan Beardon (Cambridge) Conformal mappings and the convergence of domains
Abstract. Caratheodory proved what is now known as the `Kernel theorem' which says that, in a certain sense, a sequence of simple connected domains converges to a simply connected domain if and only if the corresponding conformal mappings converge. The hypotheses of his theorem are quite restrictive, and in this talk we shall explore the idea of releasing some of these restrictions by rewriting the result in terms of the hyperbolic metrics of the domains instead of the conformal maps.

Previous Geometry Seminars:

Academic Year 2016/17: Autumn 2016, Spring 2017
http://www1.maths.leeds.ac.uk/~pmtgk/seminar/autumn2017.html
Last modified: 9 Nov 2017