Yorkshire and Durham Geometry Days

2 May 2018, Leeds

The lectures will hold in MALL2, Level 8, School of Mathematics, The University of Leeds. Coffee and tea will be served on Level 9.

11.00 Coffee and Reception

11.30 Alexandr Buryak (Leeds)
Intersection theory on the moduli space of Riemann surfaces with boundary and matrix integrals.
Abstract. The information about the integrals of basic cohomology classes over the moduli space of Riemann surfaces, the so-called intersection numbers, can be effectively described using a certain special solution of the Korteweg-de Vries equation. This is the subject of E. Witten's conjecture, which was motivated by theories of two-dimensional quantum gravity. The conjecture was proved by M. Kontsevich, who also presented a matrix integral describing the intersection numbers. I will talk about analogous results and conjectures for the moduli space of Riemann surfaces with boundary, which we obtained in joint works with A. Alexandrov and R. Tessler.

12.30 Lunch

13.30 Lorenzo Foscolo (Heriot-Watt)
Infinitely many new families of complete cohomogeneity one G2-manifolds.
Abstract. G2-manifolds are the 7-dimensional Riemannian manifolds with holonomy the exceptional compact Lie group G2. G2-manifolds are Ricci-flat and conversely most known constructions of Ricci-flat metrics involve some form of holonomy reduction. I will present joint work with Mark Haskins and Johannes Nordstroem on SU(2)xSU(2)-invariant G2-holonomy metrics. Because they are Ricci-flat, G2-manifolds can only admit continuous symmetries when they are non-compact or incomplete. We obtain many complete and incomplete G2-metrics with interesting prescribed asymptotic geometry and/or singular behaviour. More precisely, we obtain infinitely many 1-parameter families of complete G2-metrics with non-maximal volume growth and so-called ALC asymptotics, infinitely many asymptotically conical G2-metrics, and the first known example of a G2-metric that has an isolated conical singularity in the interior, but is otherwise complete. Our infinitely many asymptotically conical examples are particularly noteworthy since only the three classical Bryant-Salamon examples from 1989 were previously known.

14.30 Joe Cook (Loughborough)
Properties of Eigenvalues on Riemann Surfaces with Large Symmetry Groups
Abstract. We will discuss the problem of bounding the Laplace eigenvalues of highly symmetric compact Riemann surfaces, such as the Bolza surface and the Klein quartic. Surfaces of constant negative curvature can be obtained as the quotient of the hyperbolic plane by the action of a Fuchsian group, or by gluing together hyperbolic pairs of pants, and are classified topologically by their genus. The Laplace spectra of such surfaces cannot be computed explicitly, however, we can use the symmetry of the surfaces to decompose the closed eigenvalue problem into Dirichlet, Neumann, and mixed boundary problems on the fundamental domain of the symmetry group. These problems can then be analysed using variational methods and isoperimetric inequalities to produce upper and lower bounds. As well as discussing this approach in the case of the Bolza surface, we will compare some numerical computations of eigenvalues for interesting surfaces of genus 3.

15.00 Yuguo Qin (Durham)
Non-isometric Riemannian manifolds with equal equivariant spectrum.
Abstract. In this talk, we will examine the two methods that people used to systematically construct isospectral non-isometric Riemannian manifolds, the Sunada method and the torus action method, and show that both methods can be used to produce equivariantly isospectral non-isometric Riemannian manifolds.

15.30 Tea Break

16.00 Martin Guest (Waseda University)
The tt*-Toda equations
Abstract. As an example of the topological-antitopological fusion equations, Cecotti and Vafa made a number of conjectures regarding a special case of the well-known 2-dimensional Toda equations, in a series of papers in the 1990's. By combining several techniques (p.d.e., isomonodromy theory, loop groups) in joint work with Alexander Its and Chang-Shou Lin, we have been able to confirm the predictions of Cecotti and Vafa for these "tt*-Toda equations". The important information is contained in the monodromy data (Stokes matrices and connection matrices) of an associated meromorphic connection. In general this kind of data is hard to compute, but for the tt*-Toda equations the computation is facilitated by the high degree of symmetry.

17.30 Dinner in the city


Leeds is easily accessible by train and has direct inter-city links with major destinations in the UK. In particular, if you are travelling from London, there is a direct high-speed train from King's Cross railway station with average journey time of 140 minutes.

From the railway station, the University campus is within walking distance of approximately 15-20 minutes.

The Google map of the university campus can be found here; on the campus map from the university web-pages the School of Mathematics is located in the building number 84.

History and organizers:

Yorkshire and Durham Geometry Days are jointly organised by the Universities of Durham, Leeds and York, and occur at a frequency of three meetings per academic year. Financial support is provided by the London Mathematical Society through a Scheme 3 grant, currently administered by the University of York.

The current local organizers are:

John Bolton & Wilhelm Klingenberg, University of Durham

Derek Harland & Gerasim Kokarev, University of Leeds

Ian McIntosh & Chris Wood, University of York

Previous organizers: John Wood (Leeds, 2000-2015), Jurgen Berndt (Hull, 2000-2004), Martin Speight (Leeds, 2003-2016).

Archive of previous meetings can be found here.

Last modified: 28 April 2018