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Autumn 2016

Leeds Geometry Seminar

Organiser: Gerasim Kokarev

Time and venue:
Wednesday 3pm, Roger Stevens LT12. Coffee/Tea served in the Common Room after 4pm.
Programme:

5 October 2016
Dima Panov (KCL) Real line arrangements with Hirzebruch property 4pm (unusual time)
Abstract. A line arrangement of 3n lines in the complex projective plane satisfies Hirzebruch property if each line intersects others in n+1 points. Hirzebruch asked if all such arrangements are related to finite complex reflection groups. We give a positive answer to this question in the case when the line arrangement is real, confirming that there exist exactly four such arrangements.

12 October 2016
Felix Schulze (UCL) Ricci flow from spaces with isolated conical singularities
Abstract. Let (M,g_0) be a compact n-dimensional Riemannian manifold with a finite number of singular points, where at each singular point the metric is asymptotic to a cone over a compact (n-1)-dimensional manifold with curvature operator greater or equal to one. We show that there exists a smooth Ricci flow starting from such a metric with curvature decaying like C/t. The initial metric is attained in Gromov-Hausdorff distance and smoothly away from the singular points. To construct this solution, we desingularize the initial metric by glueing in expanding solitons with positive curvature operator, each asymptotic to the cone at the singular point, at a small scale s. Localizing a recent stability result of Deruelle-Lamm for such expanding solutions, we show that there exists a solution from the desingularized initial metric for a uniform time T>0, independent of the glueing scale s. The solution is then obtained by letting s->0. We also show that the so obtained limiting solution has the corresponding expanding soliton as a forward tangent flow at each initial singular point. This is joint work with P. Gianniotis.

26 October 2016
Yorkshire and Durham Geometry Day (to be held at the University of York)

2 November 2016
Peter Topping (Warwick) A gradient flow for the harmonic map energy that finds minimal immersions
Abstract. In this talk I will explain a gradient flow of the classical harmonic map energy in which both a map from a surface and the metric on that surface are allowed to evolve. In principle, the flow wants to find minimal immersions. However, in general, there may not be a minimal immersion to converge to, and the flow must do something more exotic. We will explain what happens, and hopefully get to some forthcoming work that completes the foundational theory. This is joint with Melanie Rupflin.

16 November 2016
Katrin Leschke (Leicester) Integrable system methods for minimal surfaces
Abstract. Minimal surfaces, that is critical points of the area functional, are the best understood class of surfaces. This is due to the fact that minimal surfaces are given in terms of meromorphic functions, the Weierstrass data: the wealth of methods from Complex Analysis can be used to study minimal surfaces. On the other hand, in the recent study of properly embedded minimal planar domains by Meeks, Perez and Ros surprisingly algebra-geometric properties of the hierarchy of the Cortege-de Vries equation have been used in an essential way. This motivates to investigate how to further exploit integrable systems to approach open problems such as the Finite Topology Conjecture of Hoffman and Meeks. In my talk, I will explain three ways to see minimal surfaces as integrable systems, how they are connected and how they link to the Finite Topology Conjecture for spheres (which was proven by Lopez-Ros).

23 November 2016
Norbert Peyerimhoff (Durham) Eigenvalue estimates for the magnetic Laplacian on Riemannian manifolds
Abstract. In this talk I will introduce basic concepts in connection with the magnetic Laplacian on a manifold and will then discuss various eigenvalue estimates for this operator. These estimates are analogues of well known results for the classical Laplacian on functions: Cheeger and higher order Cheeger inequalities, Lichnerowicz type inequalities, as well as higher order Buser inequalities on manifolds with lower Ricci curvature bounds. This material is based on joint work with Michela Egidi, Carsten Lange, Shiping Liu, Florentin Muench, and Olaf Post.

30 November 2016
Hanming Zhou (Cambridge) Geometric inverse problems for connections
Abstract. Given a compact Riemannian manifold with strictly convex boundary, we consider the inverse problem of determining an arbitrary connection and the Higgs field from its parallel transport along geodesics. We show unique determination, modulo gauge transformations, for manifolds admitting strictly convex functions in dimension three and higher and for generic ones in arbitrary dimension. The proofs involve a reduction to the geodesic X-ray transform with a matrix weight. We also apply our methods to solve some inverse problems in quantum state tomography and polarization tomography. Part of the talk is based on a joint work with G. P. Paternain, M. Salo and G. Uhlmann.

7 December 2016
Joe Cook (Loughborough) Symmetry and Low Lying Eigenvalues
Abstract. I will give a brief background to isoperimetric problems, and present a Faber-Krahn style inequality for hyperbolic cylinders. The Bolza surface and Klein Quartic are the most symmetric compact Riemann surfaces of genus two and three respectively. It is possible to prove results about their Laplace spectra based on the representation theory of their automorphism groups. I will discuss this in detail for the Bolza surface, including an application of my isoperimetric result to prove that the first eigenspace is three dimensional. If time permits, I will then describe a few interesting properties of the Klein Quartic, including an initial result on the first eigenspace.

http://www1.maths.leeds.ac.uk/~pmtgk/seminar/autumn2016.html
Last modified: 1 Aug 2016