Yorkshire-Durham Geometry Day (18 April 2023)

The Geometry Day will take place at the University of York on 18 April 2023.
All talks will be held in the Topos and online participation will be available via Zoom. The Zoom information will be made available through the YDGD email list.

Information on getting to the Maths Department

Schedule

Time	Speaker	Title
11:30-12:30	Viveka Erlandsson (Bristol)	Counting geodesics of given commutator length
12:30-2:00	LUNCH	LUNCH
2:00-3:00	Aleksander Doan (UCL)	Holomorphic Floer theory and the Fueter equation
3:10-4:10	Jaime Santos-Rodriguez (Durham)	On the fundamental groups of RCD spaces
4:104:40	AFTERNOON TEA	AFTERNOON TEA
4:40-5:40	Cordelia Webb (York)	Superminimal equivariant minimal surfaces in complex hyperbolic space

Previous Yorkshire-Durham Geometry Days

7 December 2022 (hosted by Durham)
13 May 2020 (virtual event hosted by Leeds)
11 December 2019 (hosted by Durham)
The University of Leeds maintains a list of previous Yorkshire-Durham Geometry Days (up to and including 2019/20).

Abstracts

11:30 Viveka Erlandsson (Bristol)
Title. Counting geodesics of given commutator length
Abstract. It’s a classical result by Huber that the number of closed geodesics of length bounded by L on a closed hyperbolic surface is asymptotic to exp(L)/L as L grows. This result has been generalized in many directions, for example by counting certain subsets of closed geodesics. One such result is the asymptotic growth of those that are homologically trivial, proved by Katsura-Sunanda and Phillips-Sarnak. A homologically trivial curve can be written as a product of commutators, and in this talk I will describe the asymptotic growth of those that can be written as a product of g commutators (in a sense, those that bound a genus g subsurface). As a motivating example of our methods, we will give a geometric proof of Huber’s original theorem.
This is joint work with Juan Souto.

2:00 Aleksander Doan (UCL)
Title. Holomorphic Floer theory and the Fueter equation
Abstract. Lagrangian Floer homology is a powerful invariant associated with a pair of Lagrangian submanifolds in a symplectic manifold. I will discuss a conjectural refinement of this invariant for a pair of complex Lagrangian submanifolds in a complex symplectic manifold. The refined invariant should no longer be a homology group but a category, mimicking the well-known Fukaya-Seidel category, an invariant associated with a holomorphic function on a complex manifold. This proposal leads to many interesting problems in geometric analysis which so far remain largely unexplored. I will talk about some of these problems and discuss the special case of cotangent bundles.
This talk is based on joint work with Semon Rezchikov.

3:10 Jaime Santos-Rodriguez (Durham)
Title. On the fundamental groups of RCD spaces
Abstract. The class of

R C D (K, N)

$RCD(K,N)$ -spaces for given

K \in R

$K \in \mathbb{R}$ ,

N \in [1, \infty)

$N \in [1, ∞)$ consists of proper metric measure spaces that satisfy a synthetic notion of having Ricci curvature bounded below by

K

$K$ and Hausdorff dimension bounded above by

N

$N$ . Most notably this class is known to be stable under measured Gromov-Hausdorff convergence and examples of such spaces include Riemannian manifolds and Alexandrov spaces.
A very classical problem in geometry is to study the relation between curvature and topology. So, one can then for instance consider studying properties of the fundamental group of these spaces. However, since

R C D (K, N)

$RCD(K,N)$ -spaces are not necessarily topological manifolds some work was needed before one could talk about their Universal covers, this was done by Mondino and Wei. More recently, Wang proved that the Universal cover is indeed simply connected.
In this talk we will first focus on properties and examples of RCD-spaces before presenting some of the results obtained regarding their fundamental groups.
This is based on joint work with Sergio Zamora-Barrera.

4:40 Cordelia Webb (York)
Title. Superminimal equivariant minimal surfaces in complex hyperbolic space
Abstract. The moduli space of equivariant minimal surfaces in complex hyperbolic space over a compact surface X, acquires an analytic structure via an embedding into the direct product of the Teichmuller space of X and the character variety of the fundamental group of X. This latter space is homeomorphic to the moduli space of

P U (n, 1)

$PU(n, 1)$ -Higgs bundles via the Non-abelian Hodge correspondence. Such minimal surfaces also admit a harmonic sequence of maps. We explore this relation between harmonic sequences and Higgs bundles to, in particular, better understand the critical points of the moduli space under the

C^{*}

$\mathbb{C}^*$ action.

Last updated 17 April, 2023