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The lectures will hold in MALL1, Level 8, School of Mathematics, The University of Leeds. Coffee and tea will be served on Level 9. Programme: 11.30 Coffee and Reception 12.00 Felix Schulze (Warwick) Singularities along the Lagrangian mean curvature flow of surfaces Abstract. It is an open question to determine which Hamiltonian isotopy classes of Lagrangians in a Calabi-Yau manifold have a special Lagrangian representative. One approach is to follow the steepest descent of area, i.e. the mean curvature flow, which preserves the Lagrangian condition. But in general such a flow will develop singularities in finite time, and it has been open how to continue the flow past singularities. We will give an introduction to the problem and explain recent advances where we show that in the simplest possible situation, i.e. the Lagrangian mean curvature flow of surfaces, when the singularity is the special Lagrangian union of two transverse planes, then the flow forms a “neck pinch”, and can be continued past the singularity. This is joint work with Jason Lotay and Gábor Székelyhidi. 13.00 Lunch 14.00 Marie-Amelie Lawn (Imperial) G-invariant spin structures on homogeneous spaces and their special spinors Abstract. We investigate the relationship between the existence of spinors which are parallel or special for particular connections (usually with torsion) on homogeneous spaces and the related geometric structure. In particular we study the large class of homogeneous spin manifolds G/H with G-equivariant spin structures: in this case, finding invariant spinors can be explicitly done by computing the invariant subspace for the composition H → Spin(n) → GL(V) where V is the Spin representation; this translates the problem into representation theory, and more specifically invariant theory. The elements of this subspace will be exactly the invariant sections or ”special spinors” we are looking for. Conversely the geometric properties of the homogeneous space will be characterized by the existence of such spinors. As an example, we will give a complete classification of all possible invariant spinors in each of the nine different Riemannian homogeneous decompositions of the sphere. 15.00 Ilyas Khan (Oxford) Uniqueness of Asymptotically Conical Gradient Shrinking Solitons in G_2-Laplacian Flow Abstract. (joint with M. Haskins and A. Payne) Riemannian 7-manifolds with holonomy equal to the exceptional Lie group G_2 are intensely studied objects in diverse domains of mathematics and physics. One approach to understanding such manifolds is through natural flows of 3-forms called G_2-structures, the most prominent of which is Bryant’s Laplacian flow. In general, Laplacian flow is expected to encounter finite-time singularities and, as in the case of other flows, self-similar solutions should play a major role in the analysis of these singularities. In this talk, we will discuss recent joint work with M. Haskins and A. Payne in which we prove the uniqueness of asymptotically conical gradient shrinking solitons of the Laplacian flow of closed G_2 structures. We will particularly emphasize the unique difficulties that arise in the setting of Laplacian flow (in contrast to the Ricci flow, where an analogous result due to Kotschwar and Wang is well-known) and how to overcome these difficulties. 16.00 Tea Break 16.30 Jakob Stein (UCL) Gauge theory on asymptotically local conical G2 metrics Abstract. Asymptotically locally conical (ALC) G2 metrics can be viewed as higher-dimensional analogues of the Ricci-flat Taub-NUT metric, and large classes of examples have recently been constructed by Foscolo-Haskins-Nordstrom. In the co-homogeneity one setting, the complete examples can be realised by gluing an asymptotically conical G2 metric into a conically singular G2 metric with an asymptotically locally conical end: this suggests a similar gluing approach to describing their gauge theory, currently being investigated in an ongoing joint work with Matt Turner (Bath). In this talk, I will report on the current progress on this work, and suggest some further ideas to be investigated in future. 18.00 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: 10 May 2023 |