Time and venue:
Wednesday 3pm, Roger Stevens LT12. Coffee/Tea served in the Common Room after 4pm.
29 January 2020
Nuno Romão (University of Augsburg)
Pochhammer states on Riemann surfaces
Abstract. A basic fact about compact Riemann surfaces of genus g>1, equipped with a fixed riemannian metric, is that their L^2-cohomology (of universal or maximal abelian covers, say) concentrates in middle degree 1, with L^2-Betti number 2g-2. As a consequence, on such surfaces, nonzero harmonic p-forms valued in families of complex flat line bundles, and of finite L^2-norm, only exist for p=1. In fact, there is an infinite-dimensional Hilbert space H of such 1-forms, with finite von Neumann dimension 2g-2. In my talk, I shall construct a complete basis for this space depending on the choice of a particular kind of pair-of-pants decomposition of the surface. We have baptised the elements of such a basis `Pochhammer states', since they localise (in a precise sense) to 1-cycles lifting Pochhammer curves immersed on each pair of pants, and moreover they span all the ground states (waveforms) in supersymmetric quantum mechanics incorporating Aharonov-Bohm phases. Our construction provides a geometric picture for the fermionic space H strikingly analogous to the structure of graphite. Joint work with Marcel Bökstedt.
5 February 2020
Igor Wigman (KCL)
The defect of toral Laplace eigenfunctions and Arithmetic Random Waves
Abstract. We study the defect (or "signed area") distribution of toral Laplace eigenfunctions restricted to shrinking balls or radius above the Planck scale, in either random Gaussian scenario ("Arithmetic Random Waves"), or deterministic eigenfunctions averaged w.r.t. the spacial variable. In either scenario we exploit the associated symmetry of the eigenfunctions to show that the expectation (Gaussian or spacial) vanishes. Our principal results concern the high energy limit behaviour of the defect variance. This talk is based on a joint with with P. Kurlberg and N. Yesha.
12 February 2020
Lectures by Nicholas Young MALL2
19 February 2020 Joint Analysis/Geometry Seminar
Lucas Ambrozio (Warwick)
Volume of closed Riemannian manifolds and geometric invariants related to their area-minimising hypersurfaces
Abstract. Several variational methods used to find minimal submanifolds in a closed Riemannian manifold define critical levels of the area functional, and it is interesting to understand how much information these numbers contain about the ambient geometry itself. This can be done by comparison to other geometric invariants, for example curvature and volume. In this talk, we will focus on the space of unit volume metrics, mainly when the underlying manifolds are real projective spaces and the minimal submanifolds corresponding to the critical levels are of minimising type. This is joint work with Rafael Montezuma.
18 March 2020
Mauricio Bustamante (Cambridge)
Symmetries of exotic negatively curved manifolds
Abstract. Let N be a smooth manifold that is homeomorphic but not diffeomorphic to a closed hyperbolic manifold M. To what extent does N admit as much symmetry as M?. In this talk, I wil show that it's possible to find N with maximal symmetry, i.e. Isom(M) acts on N by isometries with respect to some negatively curved metric on N. For these examples, Isom(M) can be made arbitrarily large. On the other hand, one can also find N with little symmetry, i.e. no subgroup of Isom(M) of `small' index acts by diffeomorphisms of N. This is joint work with Bena Tshishiku.
Postponed to a later date
20 March 2020
Yorkshire and Durham Geometry Day in York. Postponed to a later date
25 March 2020
Albert Wood (UCL)
29 April 2020
TBC Unusual place: Roger Stevens LT10
6 May 2020
Florian Litzinger (QMUL) Unusual place: Roger Stevens LT10
13 May 2020
Yorkshire and Durham Geometry Day in Leeds. TBC