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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 Ben Lambert (Leeds) Alexandrov immersed mean curvature flow Abstract. In this talk I will introduce Alexandrov immersed mean curvature flow and extend Andrew's non-collapsing estimate to include Alexandrov immersed surfaces. This estimate implies an all-important gradient estimate for the flow and allows mean curvature flow with surgery to be extended beyond flows of embedded surfaces to the Alexandrov immersed case. This is joint work with Elena Mäder-Baumdicker. 13.00 Lunch 14.00 Lucas Ambrozio (IMPA) The width of embedded circles Abstract. Width is a classical geometric invariant of plane curves. It measures how narrow they are. Its definition, however, is based on Euclidean geometry and not easily generalisable beyond other geometries with constant curvature. We will discuss how the variational theory of the Riemannian distance function can be used to define a meaningful notion of width for curves embedded in any complete Riemannian manifold, of any dimension. In particular, this approach to the width naturally suggests the generalisation of another classical notion - curves of constant width - and helps to show the existence of geodesics that meet two points of the curve in certain geometrically special configurations - for instance, orthogonally. The talk will be based on joint work with Rafael Montezuma (UFC - Fortaleza) and Roney Santos (USP - São Paulo). 15.00 Ana Rita Pires (Edinburgh) Infinite staircases in the symplectic ball packing problem Abstract. The symplectic version of the problem of packing K balls into a ball in the densest way possible (in 4 dimensions) can be extended to that of symplectically embedding an ellipsoid into a ball as small as possible. A classic result due to McDuff and Schlenk asserts that the function that encodes this problem has a remarkable structure: its graph has infinitely many corners, determined by Fibonacci numbers, that fit together to form an infinite staircase. This ellipsoid embedding function can be equally defined for other targets, and in this talk I discuss for which other targets the function also has an infinite staircase. In the case when these targets can be represented by lattice (moment) polygons, the targets seem to be exactly those whose polygon is reflexive (i.e., has one interior lattice point). In a specific family of irrational polygons, the answer involves self-similar behavior akin to the Cantor set. This talk is based on various projects, joint with Dan Cristofaro-Gardiner, Tara Holm, Alessia Mandini, Maria Bertozzi, Tara Holm, Emily Maw, Dusa McDuff, Grace Mwakyoma, Morgan Weiler, and Nicki Magill. 16.00 Tea Break 16:30 Nicolò Cavalleri (UCL) Complete non-compact Spin(7)-manifolds from T2-bundles over asymptotically conical Calabi Yau manifolds Abstract. We develop a new construction of complete non-compact 8-manifolds with holonomy equal to Spin(7). As a consequence of the holonomy reduction, these manifolds are Ricci-flat. These metrics are built on the total spaces of principal T2-bundles over asymptotically conical Calabi Yau manifolds. The resulting metrics have a new geometry at infinity that we call asymptotically T2-fibred conical (AT2C) and which generalizes to higher dimensions the ALG metrics of 4-dimensional hyperkähler geometry. We use the construction to produce infinite diffeomorphism types of AT2C Spin(7)-manifolds and to produce the first known example of complete toric Spin(7)-manifold. 17:00 Tathagata Ghosh Instantons on Asymptotically Conical Spin(7)-Manifolds Abstract. In this talk we discuss instantons on asymptotically conical Spin(7)-manifolds where the instanton is asymptotic to a fixed nearly G2-instanton at infinity. We mainly discuss the deformation theory of AC Spin(7)-instantons by relating the deformation complex with Dirac operators and spinors and applying spinorial methods to identify the space of infinitesimal deformations with the kernel of the twisted negative Dirac operator on the asymptotically conical Spin(7)-manifold. As examples, we consider two important Spin(7) manifolds: R8, where R8 is considered to be an asymptotically conical manifold asymptotic to the cone over the round sphere S7, and Bryant--Salamon manifold asymptotic to the cone over the squashed sphere Sp(2) × Sp(1)/Sp(1) × Sp(1). We apply the deformation theory to describe deformations of Fairlie-Nuyts-Fubini-Nicolai (FNFN) Spin(7)-instantons on R8, and Clarke-Oliviera's instanton on the Bryant-Salamon manifold. We also calculate the virtual dimensions of the moduli spaces using Atiyah-Patodi-Singer index theorem and the spectrum of the twisted Dirac operators. If time permits, we discuss some results on AC instantons on R8 with gauge groups U(1) and SU(2) respectively. 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 |