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The lectures will hold in MALL2, Level 8, School of Mathematics, The University of Leeds. Coffee and tea will be served on Level 9. Programme: 11.00 Coffee and Reception 11.30 Florian Hanisch (Leeds) Index theory and supergeometry Abstract. During the 1980s, physicists proposed an approach to prove index theorems which is based on ideas from supersymmetry. It relates the index of certain operators to path integrals over super loopspace and the index theorems can then obtained by evaluating these functional integrals. We will explain the basic (super)geometric setup underlying these ideas and show that concepts from supergeometry lead us to a reformulation in terms of (equivariantly closed) differential forms on ordinary loopspace. We then discuss how such forms can be integrated and evaluated using an infinite-dimensional generalization of Berezin's integral. This yields a (rigorously defined) path integral on super loops and a rigorous version of the aforementioned proof. 12.30 Lunch 13.30 Thomas Madsen (Buckingham) Toric geometry of G_2-manifolds Abstract. G_2-manifolds are 7-dimensional and come with a Ricci-flat metric. When looking for examples, it is natural to include symmetry assumptions. This talk will focus on the case of torus symmetry and explain that one specific rank of the torus is of particular interest. I will then discuss what we currently know about these toric G_2-manifolds.If time permits, I shall also discuss how similar ideas apply to the setting of Spin(7)-manifolds. The talk is based on joint work with Andrew Swann, available in the preprints arXiv:1803.06646 and arXiv:1810.12962. 14.30 Adam Paxton (Oxford) Embeddedness of timelike maximal surfaces in (1+2) Minkowski space Abstract. Timelike maximal surfaces in Minkowski space R^{1+2} are the hyperbolic analogue of minimal surfaces in Euclidean space R^3. They have been suggested as models for some physical theories, and provide interesting problems in geometric analysis and wave equations. In this talk I will first give a brief introduction to the topic of timelike maximal surfaces in R^{1+2}. In particular I'll show that timelike maximal surfaces naturally give rise to a Cauchy problem, and I'll explain how timelike maximal surfaces may be constructed by conformal methods (analogously to the situation for minimal surfaces in R^3). In the main part of the talk I'll then present some recent results on the global geometry of timelike maximal surfaces in R^{1+2} and on singularity formation for the Cauchy evolution. 15.30 Tea Break 16.00 David Calderbank (Bath) Integrable GL2 geometry and quaternionic manifolds with symmetry Abstract. Moduli spaces of submanifolds of a complex manifold typically carry special differential geometric structures. In the case that the submanifolds are projective lines, the corresponding geometric structures are integrable GL2 geometries. Prototypical examples include Einstein--Weyl geometry and quaternionic geometry. This talk will present a general setting for integrable G-geometry, and explore how integrable GL2 geometries in different dimensions are related by symmetry reduction and generalizations thereof. In particular, we see that all (complex) integrable GL2 geometries arise as symmetry reductions of (complexified) quaternionic geometry. 17.30 Dinner in the city Travel: 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: 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.
http://www1.maths.leeds.ac.uk/~pmtgk/ydgd/ydgd2019.html
Last modified: 2 April 2019 |