Second Meeting of the UK Metric Geometry and Analysis Network

Friday 30 Aug 2024, 09:00 → 15:30 Europe/London

MCS2068 (Department of Mathematical Sciences)

Fernando Galaz-García (Durham University), Martin Kerin (Durham University)

This is the second meeting of the UK Metric Geometry and Analysis Network, a network which brings together working groups from Cardiff University, Durham University, University of Oxford, University of Stirling, the University of Warwick and beyond.

The network focuses mainly on topics in differential geometry and geometric analysis which include:

•    Alexandrov spaces
•    Manifolds with lower curvature bounds
•    RCD spaces
•    Lorentzian geometry
•    Geometric PDEs

Speakers

• Asma Hassannezhad (Bristol)
• Ana Lucía García Pulido (Stirling)
• Mattia Magnabosco (Oxford)
• Ambrose Yim (Cardiff)

Programme

The event will take place in room MCS2068 of the Mathematics and Computer Science building.

• 09:00 - Welcome
• 09:15 - Ana Lucía García Pulido:  Bi-Lipschitz embeddings of measures with partial transportation distances 
  
• 10:15 - Coffee Break
• 10:30 - Ambrose Yim:   Reconstructing the topology of manifolds
  
• 11:30 - Community Updates
• 11:45 - Mattia Magnabosco: Failure of the curvature-dimension condition in sub-Finsler manifolds
  
• 12:45 - Lunch Break
• 14:15 - Asma Hassannezhad: Pleijel Nodal domain theorem for the Robin problem on Lipschitz domains
  
• 15:15 - End

Organisers

Fernando Galaz-García & Martin Kerin Durham University.

Arrival Information

The nearest train station is Durham (DHM), which is 10-15 minutes from the centre of the city.

The Mathematics and Computer Science building is about 20-25 minutes away from the city centre on foot.  

Here's a map showing the Mathematics and Computer Science building to the south of the city centre, and the train station to the north on the west of the river.

On-campus Accommodation

If you are interested in staying in an ensuite room on campus on the night of 29 August (cost £63pp), including breakfast on 30 August, please make a reservation here by Tuesday, 13 August.

Support

We gratefully acknowledge the support of the London Mathematical Society through the Joint Research Group scheme.

This network is supported by a project that has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant agreement No. 802689; PI: A Mondino).

The network is also supported by a UKRI Future Leaders Fellowship [grant number MR/W01176X/1; PI: J Harvey].

1 Welcome

2 Bi-Lipschitz embeddings of measures with partial transportation distances

The question of bi-Lipschitz embeddability of Wasserstein spaces into classical Banach spaces has attracted much attention. The importance of such embeddings can be seen, for instance, in the proof of Almgren's partial regularity theorem.

In optimal transport, Wasserstein distances are the prime examples transportation metrics to compare measures of the same total mass. This talk will consider the partial transportation metric defined by Figalli and Gigli, which removes this mass constraint. We will begin with the definition of these spaces and discuss new relationships with other transportation metrics. We will then present new results regarding their bi-Lipschitz embeddability into Hilbert space. This is joint work with D. Bate.

Speaker: Ana Lucía García Pulido (Stirling)

10:15 Coffee Break

3 Reconstructing the topology of manifolds

Given finitely many samples on a manifold, can we infer its topological invariants? We discuss some state of the art methods and theoretical bounds for inferring the homology of a manifold from a sufficiently dense point sample. In particular, we describe how these methods are motivated by dynamical systems and Morse theory. Conversely, we also show how these methods can help infer topological invariants associated to gradient dynamics from finite data.

Speaker: Ambrose Yim (Cardiff)

4 Community Updates

5 Failure of the curvature-dimension condition in sub-Finsler manifolds

The Lott–Sturm–Villani curvature-dimension condition CD(K,N) provides a synthetic notion for a metric measure space to have curvature bounded from below by K and dimension bounded from above by N. It has been recently proved that this condition does not hold in any sub-Riemannian manifold equipped with a positive smooth measure, for every choice of the parameters K and N. In this talk, we investigate the validity of the analogous result for sub-Finsler manifolds, providing two results in this direction. On the one hand, we show that the CD condition fails in sub-Finsler manifolds equipped with a smooth strongly convex norm and with a positive smooth measure. On the other hand, we prove that, for the sub-Finsler Heisenberg group, the same result holds for every reference norm. Additionally, we show that the validity of the measure contraction property MCP(K,N) on the sub-Finsler Heisenberg group depends on the regularity of the reference norm.

Speaker: Mattia Magnabosco (Oxford)

12:45 Lunch Break

6 Pleijel Nodal domain theorem for the Robin problem on Lipschitz domains

The classical Courant nodal domain theorem states that the number of nodal domains of the k-th Dirichlet eigenfunction is bounded above by k. Pleijel later showed that only a finite number of eigenfunctions realised this bound. There have been extensions and improvements of Pleijel's result to the Robin and Neumann boundary conditions requiring boundary regularity that is much stronger than being merely Lipschitz. De Ponti, Farinelli, and Violo recently proved Pleijel's nodal domain theorem for a class of metric measure spaces, which, in particular, applies the Pleijel theorem to the Neumann problem on a Lipschitz domain. We discuss how building upon their and previous results, we can obtain Pleijel's theorem for the Robin problem with any parameter on Lipschitz domains.
This is joint work with Katie Gittins, Corentin Léna, and David Sher.

Speaker: Asma Hassannezhad (Bristol)

15:15 End