Pure PGR Seminar

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• Spring term

Organised by:

• Paul Leask
• Simon May

Spring Term

17 Feb

Tathagata Ghosh — Deformation Theory of Asymptotically Conical Spin(7)-Instantons

Abstract: In this talk we discuss the deformation theory of instantons on asymptotically conical $\mathrm{Spin}(7)$-manifolds where the instanton is asymptotic to a fixed nearly $G_2$-instanton at infinity. By relating the deformation complex with Dirac operators and spinors, we apply spinorial methods to identify the space of infinitesimal deformations with the kernel of the twisted Dirac operator on the asymptotically conical $\mathrm{Spin}(7)$-manifold.

24 Feb

Joel Costa da Rocha — Parametrizing positroid cells using bicolored tilings

Abstract: We define bicolored tilings as a disk with a collection of smooth curves with a coloring map on the tiles that these curves delimit. Postnikov diagrams can be viewed as the image of reduced tilings under the Scott map. Using bicolored tilings, we can parametrise positroid cells in the Grassmannian. The postroid cell associated to a tiling is invariant under flip equivalence and reduction of tilings.

10 Mar

Thomas Galvin — The Penrose Transform

Abstract: We will discuss the Penrose Transform which is an aspect of twistor theory. The Penrose Transform relates solutions to PDEs in $\mathbb{R}^n$ to holomorphic objects in an associated complex manifold called the twistor space. We will mostly focus on the example of the Laplace equation in $\mathbb{R}^3$ and finish with a generalization developed by Murray (1985) for the case of any homogeneous polynomial differential operators in $\mathbb{R}^n$.