Yorkshire Durham Geometry Day 2022

Wednesday 7 Dec 2022, 09:00 → 17:50 Europe/London

Department of Mathematical Sciences at Durham University

 

Yorkshire Durham Geometry Days

The Yorkshire and Durham Geometry Days (YDGD) are jointly organised by the Universities of Durham, Leeds, and York and occur at a frequency of three meetings per year. Financial support is currently provided by LMS Research Grant Number 32225.

This iteration will take place at the Department of Mathematical Sciences of Durham University on 7 December 2022.

Speakers

• Mauricio Che (Durham)
• Rhiannon Dougall (Durham)
• Martin Kerin (Durham)
• Kohei Suzuki (Bielefend and Durham)

The talks will be in room L 50 (Psychology), from 1 pm - 3 pm, and in room MCS 2068 (Mathematics & Computer Science), from 4:30 pm - 6 pm.

We will meet at 12:30 pm in the Mathematics and Computer Science building lobby and then go to the Psychology building (a short walk from the MCS building). 

At 3 pm the mathematics departmental colloquium will take place in the Scott Logic Theatre in the MCS building. The speaker will be Michael Magee.

Organisers

John Bolton, Fernando Galaz-García & Wilhelm Klingenberg, Durham University.
Derek Harland & Gerasim Kokarev, University of Leeds.
Ian McIntosh & Chris Wood, University of York.

 

Recordings of past Yorkshire Durham Geometry Days are available at Durham's Department of Mathematical Sciences YouTube Playlist.

 

1 Double disk bundles

When searching for examples satisfying certain geometric properties, it is often convenient to examine manifolds constructed by gluing simple pieces together. One common example of such a construction involves gluing disk bundles together along their common boundary. On the other hand, many geometric phenomena impose strong topological conditions on the underlying manifold, such as the existence of a decomposition into a union of disk bundles (glued along a common boundary).

Given that they arise frequently from these two different viewpoints, it thus makes sense to study manifolds which decompose as a union of disk bundles in their own right. In this talk, I will report on joint work with J. DeVito and F. Galaz-García in this direction.

Speaker: Martin Kerin (Durham University)

2 On the Geometry of Configuration Spaces and Particle Systems

The configuration space U(X) over a base space X is the space
of all locally finite point measures on X. The space U(X) being equipped
with the vague topology, the L^2-transportation distance and a point
process, it is a Polish extended metric measure space. In this talk, we
show that U(X), equipped with the Poisson point process, satisfies
synthetic lower Ricci curvature bounds if and only if so does X. As a
byproduct, we obtain the Sobolev-to-Lipschitz property on U(X), which
confirms the conjecture by Röckner-Schied (J. Funct. Anal. '99). We
discuss several applications to the corresponding infinite-particle
systems such as the integral Varadhan short-time asymptotic of the heat
flow on U(X) and a new characterisation of ergodicity of particle
systems in terms of the L^2-transportation distance. If time allows, we
also explain the case beyond the Poisson point process. This talk is
based on the joint work with Lorenzo Dello Schiavo (Institute of Science
and Technology Austria).

Speaker: Kohei Suzuki (Durham University)

3 Departmenttal Research Colloquim: Spectra and dynamics of hyperbolic surfaces

A hyperbolic surface is a surface, in the intuitive sense, with a geometry that is negatively curved at every point with the same curvature (-1) everywhere. These are not easy to visualize, but there are many of them.

Two interesting things to study on a hyperbolic surface are the dynamics of the geodesic flow (classical mechanics) and the Laplacian differential operator (quantum mechanics). The geodesic flow is chaotic and so the Laplacian there belongs to a field of study known as quantum chaos. Although these systems are far from being solvable in any sense, they are often the first place that we can see anticipated physical phenomenon rigorously. This is because for a hyperbolic surface, there is further structure (representation theory) that bridges the classical and quantum mechanics.

I will explain all this in simple terms, covering a range of paradigms that hyperbolic surfaces provide us to study.

If I have time, I'll then highlight some recent results in the field.

Speaker: Michael Magee (Durham University)

16:00 Break

4 Metric geometry of spaces of persistence diagrams

Persistence diagrams are fundamental objects in topological data analysis. They are pictorial representations of persistence homology modules, which in turn describe topological features of a data set when viewed at different scales or levels, i.e. along a filtration. However, given a data set, it is possible to obtain different persistence diagrams depending on the filtration, so it is natural to study the space of all persistence diagrams. Such space has several interesting geometric and topological properties.

In collaboration with Fernando Galaz-García (Durham University), Luis Guijarro (Universidad Autónoma de Madrid) and Ingrid Membrillo-Solis (University of Southampton, London Metropolitan University), we study a family of functors that assign to each metric pair $(X, A)$ a metric space of persistence diagrams $\mathcal{D}_p(X, A)$ with points in $X$ and finite $p$-persistence with respect to $A$. This construction in the case $(X, A) = (\mathbb{R}^2, \Delta)$ give as a result the usual spaces of persistence diagrams. We will present a continuity result with respect to the Gromov-Hausdorff convergence in the setting of metric pairs, as well as some other properties already known for the usual spaces of persistence diagrams which hold in this generality.

Speaker: Mauricio Che (Durham University)

5 Amenability and Growth of Closed Geodesics for Regular Covers

For compact negatively curved Riemannian manifolds, a classical object of study is the exponential growth rate of closed geodesics, which is the same as the exponential growth rate of volume in the universal cover, and also equal to the topological entropy of the geodesic flow. In this talk, we discuss the question of the growth rate of closed geodesics in noncompact regular covering manifolds and how this relates to the group of deck transformations. We give some historical context to this problem and discuss the perspectives in the dynamics community.

Speaker: Rhiannon Dougall (Durham University)