Date | Speaker | Title |

5 October | ||

12 October | ||

19 October | ||

29 October at 2pm in the Dusa McDuff Room Note the date, location and time! | Derek Harland (Leeds) | Cyclic monopole chains and parabolic Higgs bundles |

2 November | ||

9 November | Corina Keller (Montpellier) | Generalized Character Varieties and Quantization via Factorization Homology |

16 November | Sergey Cherkis (Arizona) | Bow Construction of Yang-Mills Instantons on Taub-NUT space |

23 November | Lorenzo Foscolo (UCL) | Anti-self-dual instantons and codimension-1 collapse |

30 November | ||

7 December |

Date | Speaker | Title |

18 January | ||

1 February | ||

15 February | ||

1 March | ||

15 March |

Date | Speaker | Title |

19 April | Markus Upmeier (Aberdeen) | Cobordism categories with applications to enumerative invariants |

26 April | Stuart Hall (Newcastle) | Kähler Quantisation and the geometry of molecular surfaces |

3 May | John Parker (Durham) | Fenchel-Nielsen coordinates for SL(3,C) representations of surface groups |

10 May | Lashi Bandara (Brunel) | Index theory and boundary value problems for general first-order elliptic differential operators |

2019-20

This is joint work with Calum Ross.

I will discuss a new approach to these index-theoretic questions based on cobordism categories, of which I shall construct a new categorical group representation. The main application produces canonical orientations for Donaldson-Thomas invariants for Calabi-Yau 4-folds and sheaves with \(c_2=0\). Finally, I will comment on how the general case can be solved using flag structures, a new concept that arises naturally from the point of view cobordism categories.

One such description comes from Kähler quantisation - a technique developed around 20 years ago to provide numerical approximations to Calabi-Yau and other Kähler-Einstein metrics. I'll explain the technique and how it can be applied to the shapes in question as well as give some idea how effective it is. If I have time I might touch on other related descriptors one could consider based on ideas from Riemannian geometry.

This is joint work with Daniel Cole, Thomas Murphy and Rachael Pirie.

This material is joint work with my student Rodrigo Davila.

APS showed that local boundary conditions are topologically obstructed for index theory. Therefore, a central theme emerging from the work of APS is the significance of non-local boundary conditions for first-order elliptic differential operators. An important contribution from APS was to demonstrate how their crucial non-local boundary condition for the index theorem could be obtained by a spectral projection associated to a so-called adapted boundary operator. In their application, this was a self-adjoint first-order elliptic differential operator.

The work of APS generated tremendous amount of activity in the topic from the mid-70s onwards, culminating with the Bär-Ballmann framework in 2010. This is a comprehensive machine useful to study elliptic boundary value problems for first-order elliptic operators on measured manifolds with compact and smooth boundary. It also featured an alternative and conceptual reformulation of the famous relative index theorem from the point of view of boundary value problems. However, as with other generalisations, a fundamental assumption in their work was that an adapted boundary operator can always be chosen self-adjoint.

Many operators, including all Dirac-type operators, satisfy this requirement. In particular, this includes the Hodge-Dirac operator as well as the Atiyah-Singer Dirac operator. Recently, there has been a desire to study more general first-order elliptic operators, with the quintessential example being the Rarita-Schwinger operator on 3/2-spinors. This operator has physical significance, arising in the study of the delta baryon, analogous to the way in which the Atiyah-Singer Dirac operator arises in the study of the electron. However, not only does the Rarita-Schwinger operator fail to be of Dirac-type, it can be shown that outside of trivial geometric situations, this operator can never admit a self-adjoint adapted boundary operator.

In this talk, I will present work with Bär where we extend the theory for first-order elliptic differential operators to full generality. That is, we make no assumptions on the spectral theory of the adapted boundary operator. The ellipticity of the original operator allows us to show that, modulo a lower order additive perturbation, the adapted boundary operator is in fact bi-sectorial. Identifying the spectral theory makes the problem tractable, although departure from self-adjointness significantly complicates the analysis. Therefore, we employ a mixture of methods coming from pseudo-differential operator theory, bounded holomorphic functional calculus, semi-group theory, and maximal regularity to extend the Bär-Ballman framework to the fully general situation.