# Seminar archive

The m:iv programme held a series of seminars that took place at the University of Leicester. Dr Leschke organised a regular seminar on Geometry and Visualisation for the spring semester 2018 on Mondays at 4.00pm-5.00pm in MA119.

## Graham Smith

- 30 August 2019, 3.00pm - 5.00pm
- MA117

## T Wani

**Minimal surfaces: A quaternionic approach**

In this talk, I will discuss the main tools of quaternionic calculus on a Riemann surface and use it to connect the theory of conformal minimal immersions from a Riemann surface into the Euclidean 3-space to the theory of integrable systems. This connection will helps us to use the tools and techniques of Integrable Systems in studying the minimal immersions. At the end, I will discuss the technique of dressing operation to generate new harmonic maps and new minimal immersions.

- 6 June 2019, 2.00pm - 3.00pm
- MA119

## L Ferrer

**Minimal surfaces in ℍ2 × ℝ**

One of the most interesting advances of the field of minimal surfaces in the recent decades has been the extension of this classical theory to simply connected homogeneous 3-dimensional spaces. The aim of this talk is to present some results on minimal surfaces in one of this spaces, ℍ2 × ℝ, obtained jointly with F. Martin, R. Mazzeo and M. Rodriguez. In order to do this we start with a review of the theory of minimal surfaces in ℝ3 and ℍ2 × ℝ. We recall definitions, examples and some of the most useful tools in this setting.

- 30 April 2019, 2.00pm - 3.00pm
- MA119

## B Nelli

**Hypersurfaces with constant higher mean curvature**

We give an overview of some old and new results about the shape of hypersurfaces whose one symmetric function of the principal curvatures is constant.

- 26 February 2019, 2.00pm - 3.00pm

## M Padilla

**JReality and Houdini**

- 23 January 2019, 2.00pm - 3.00pm
- MA119

## Wei Yeung Lam

**Holomorphic quadratic differentials on graphs**

- 21 January 2019, 2.00pm - 3.00pm
- MA119

## Joseph Cho

**Discrete omega surfaces - a rough introduction**

Omega-surfaces were discovered by Demoulin, and one of the motivations of studying Omega-surfaces arise from the fact that well-known classes of surfaces such as constant mean curvature surfaces, constant Gaussian curvature surfaces, and linear Weingarten surfaces are all examples of Omega-surfaces. Demoulin showed that Omega-surfaces are the class of surfaces that envelop a pair of isothermic sphere congruences, allowing one to apply the rich theory of isothermic surfaces to examine the properties of Omega-surfaces. Recent interest in integrable systems has led to renewed interest in Omega-surfaces, exemplified in works by Burstall, Cecil, Ferapontov, Hertrich-Jeromin, Musso, Nicolodi, etc.. In this talk, we look at discrete Omega-surfaces. First, we give different characterizations of discrete Omega-surfaces: via Konigs dual lifts, via Moutard lifts, and via gauge-theoretic description. Then, using the transformation theory developed for discrete isothermic surfaces, we discuss Darboux transformations of discrete Omega-surfaces. This talk is based on the joint work with Fran Burstall (University of Bath), Udo Hertrich-Jeromin (Technische Universität Wien), Mason Pember (Technische Universität Wien), and Wayne Rossman (Kobe University).

- 17 July 2018, 4.10pm - 5.00pm
- MA119
- Joseph's profile

## Joshua Cork

**Symmetric calorons and the rotation map**

For quite some time, there has been much interest in Yang-Mills instantons. These are manifested as finite-action anti-self-dual connections over some four manifold. They possess a very rich geometry, providing examples of hyperkahler metrics, and various applications to physics. A great deal of the progress in understanding the most basic examples of these objects has been through studying fixed points under isometries. In this talk, I will discuss calorons, which are instantons on S^1\times R^3. In particular, I shall present a classification result of cyclically symmetric calorons, where the cyclic groups considered are coupled to a non-spatial isometry known as the rotation map.

- 17 July 2018, 3.00pm - 3.50pm
- MA119

## Gerasim Kokarev

**Conformal volume, eigenvalue problems, and related topics**

I will give a short survey about the classical inequalities for the first Laplace eigenvalue on Riemannian manifolds, such as the Reilly inequality, the Li-Yau inequality, and tell about related history and questions. I will then discuss results concerning their versions for the higher Laplace eigenvalues and concerning estimates for the number of negative eigenvalues of Schrodinger operators.

- 17 July 2018, 2.00pm - 2.50pm
- MA119
- Gerasim's profile

## Yuta Ogata

**Constant mean curvature surfaces and positon-like solutions**

The classical Bianchi-Baecklund transformation for constant mean curvature surfaces in Euclidean 3-space has been studied by many researchers. In this talk, we introduce the method to construct positon-like solution of elliptic sinh-Gordon equation via successive Bianchi-Baecklund transformations with a single spectral parameter. We also show the recipe of the corresponding constant mean curvature surfaces of positon-like solutions.

- 17 July 2018, 12.00pm - 12.50pm
- MA119
- Yuta's profile

## Gudrun Szewieczek

**Combescure transformations – a helpful tool for the study of Guichard nets**

Two surfaces are Combescure transformations of each other if they have parallel tangent planes along corresponding curvature lines. The existence of particular Combescure transformations can be used to characterize various classes of integrable surfaces, e.g. the Christoﬀel transformation for isothermic surfaces. In this talk we investigate this concept for triply orthogonal systems and discuss the subclass of Guichard nets, which arise as special coordinate systems of 3-dimensional conformally ﬂat hypersurfaces.

- 17 July 2018, 11.10am - 12.00pm
- MA119
- Gudrun's profile

## Jenya Ferapontov

**Dispersionless integrable systems in 3D and Einstein-weyl Geometry** (based on joint work with Boris Kruglikov)

For several classes of second-order dispersionless PDEs, we show that their characteristic varieties define conformal structures which must be Einstein-Weyl in 3D (or self-dual in 4D) if and only if the PDE is integrable by the method of hydrodynamic reductions. This demonstrates that the integrability of dispersionless PDEs can be seen from the geometry of their characteristic varieties.

- 17 July 2018, 10.00am - 10.50am
- MA119
- Jenya's profile

## Professor Joeri Van Der Veken

**Lagrangian submanifolds of S3 x S3**

- 13 March 2018, 2.00pm - 3.00pm
- MA119

## Professor Fran Burstall

**TBA**

- 20 February 2018, 2.00pm - 3.00pm

## Professor Fran Burstall

**Linear Weingarten surfaces in Lie sphere geometry**

I shall give an elementary take on some joint work with Hertrich-Jeromin and Rossman which describes how linear Weingarten surfaces in 3-space are related to pairs of isothermic surfaces in the space of 2-spheres. Both smooth and discrete cases will be treated. No prior knowledge of the area will be assumed.

- 19 February 2018, 4.00pm - 5.00pm

## Keomkyo Seo

**Necessary conditions for submanifolds to be connected in a Riemannian manifold**

It is well-known that any simple closed curve in ℝ3 bounds at least one minimal disk, which was independently proved by Douglas and Radó. However, for any given two disjoint simple closed curves, we cannot guarantee existence of a compact connected minimal surface spanning such boundary curves in general. From this point of view, it is interesting to give a quantitative description for necessary conditions on the boundary of compact connected minimal surfaces. We derive density estimates for submanifolds with variable mean curvature in a Riemannian manifold with sectional curvature bounded above by a constant. This leads to distance estimates for the boundaries of compact connected submanifolds. As applications, we give several necessary conditions and nonexistence results for compact connected minimal submanifolds, Bryant surfaces, and surfaces with small L2 norm of the mean curvature vector in a Riemannian manifold.

- 16 January 2018, 2.00pm - 3.00pm

## K Grosse-Brauckmann

- 24 November 2017, 2.00pm - 3.00pm
- MA119

## Josef F. Dorfmeister

**Willmore surfaces in spheres and harmonic maps**

- 10 November 2017, 2.00pm - 3.00pm
- MA119

## Dr Ross Ogilvie

**Moduli of harmonic tori in S^3**

The primary focus of this talk is to discuss harmonic maps from a torus to the 3-sphere, which may be characterised by means of their 'spectral curves'. We will give a description of the moduli space of harmonic maps whose spectral curve is genus zero or one, and compare this to the similar case of constant mean curvature tori in S^3.

- 22 June 2017, 11.00am - 12.00pm
- MA119

## Simon Cox

**The mathematics of soap bubbles**

A soap film spanning a wire frame is a minimal surface, i.e. it is a surface of zero mean curvature. A soap film enclosing a certain volume of gas, i.e. a bubble, is a surface of constant mean curvature. Hence an aqueous foam at equilibrium is a collection of cmc surfaces subject to volume constraints. The way in which soap films meet, known as Plateau's laws, are a consequence of the fact that soap films minimize their surface area. I will introduce the basics of foam structure and describe the Kelvin problem, which is the search for the least area partition of space into bubbles of equal volume, and some of the, often surprising, consequences of area-minimization in driving foam dynamics in different applications.

- 23 March 2017, 11.00am - 12.00pm
- MA119

## Francesca Tripoli

**On the topology of surfaces with the generalised simple lift property**

Motivated by the work of Colding and Minicozzi and Hoffman and White on minimal laminations obtained as limits of sequences of properly embedded minimal disks, Bernstein and Tinaglia introduce the concept of the simple lift property. Interest in these surfaces arises because leaves of a minimal lamination obtained as a limit of a sequence of properly embedded minimal disks satisfy the simple lift property. Bernstein and Tinaglia prove that an embedded minimal surface $\Sigma\subset\Omega$ with the simple lift property must have genus zero, if $\Omega$ is an orientable three-manifold satisfying certain geometric conditions. In particular, one key condition is that $\Omega$ cannot contain closed minimal surfaces. In my thesis, I generalise this result by taking an arbitrary orientable three-manifold $\Omega$ and introducing the concept of the generalised simple lift property, which extends the simple lift property. Indeed, it will be proved that leaves of a minimal lamination obtained as a limit of a sequence of properly embedded minimal disks satisfy the generalized simple lift property and one is then able to restrict the topology of an arbitrary surface $\Sigma\subset\Omega$ with the generalised simple lift property. Among other things, I prove that the only possible compact surfaces with the generalised simple lift property are the sphere and the torus in the orientable case, and the connected sum of up to four projective planes in the non-orientable case. In the particular case where $\Sigma\subset\Omega$ is a leaf of a minimal lamination obtained as a limit of a sequence of properly embedded minimal disks, we are able to sharpen the previous result, so that the only possible compact leaves are the torus and the Klein bottle.

- 2 March 2017, 11.00am - 12.00pm
- MA119

## Liam Collard

**The Weierstrass representation for minimal surfaces**

The Weierstrass representation allows any minimal surface to be represented by a pair of a meromorphic function and a holomorphic 1-form, the so-called Weierstrass data. In particular, new families of minimal surfaces can be created by transforming the Weierstrass data. In this talk I will introduce the Weierstrass representation, show that minimal surfaces are isothermic surfaces and discuss possible transformations. Throughout the talk examples of minimal surfaces will be shown.

- 23 February 2017, 11.00am - 12.00pm
- MA119

## Fran Burstall

**Introduction to discrete isothermic surfaces**

Isothermic surfaces, of which minimal surfaces are an example, are perhaps the simplest geometric integrable system and have smooth, discrete and semi-discrete incarnations. In this talk, after a brief account of the smooth theory to set the scene, I shall discuss various approaches to the discrete theory including a gauge-theoretic approach that applies in all three settings.

- 2 February 2017, 11.00am - 12.00pm
- MA119

## José M. Manzano

**Compact embedded minimal surfaces in S2xS1**

We will begin by surveying some topological obstructions for compact surfaces to be embedded minimally in different ambient three-manifolds. In the case of the Riemannian product S^2xS^1, we will show that a compact surface can be embedded minimally in S^2xS^1 if and only if it has odd Euler characteristic.To illustrate this result, we will construct compact embedded minimal surfaces in S^2xS^1 with a high number of symmetries, first by means of solutions to the Plateau problem with respect to suitable contours, and second by the so-called conjugate Plateau construction. Time permitting, we will explain how this conjugate technique can be extended to obtain constant mean curvature surfaces in H^2xR and S^2xR with prescribed symmetries. This is based on a joint work with Julia Plenhert and Francisco Torralbo.

- 28 November 2016, 4.00pm - 6.00pm
- MA119

## K Leschke and F Neumann

**jReality, 3D-XplorMath and SURFER**

An introduction on how the 3D projector in the VisLab can be used for 3D viewing of surfaces.

- 14 November 2016, 4.00pm - 6.00pm
- MA119