# Research

My research interests lie in the field of Differential Geometry; the study of surfaces into 3-space using quaternionic holomorphic maps has immediate connections to classical complex-analytic geometry, visualization and conformal field theory, and applications to machine learning, physics, engineering and chemistry.

The focus in my recent research is in surface theory, where I study conformal maps into the 4 sphere. Surface theory models surfaces appearing in nature, such as soap films, soap bubbles, and provides methods to study and solve the underlying differential equations which appear for example in physics, chemistry and biology.

In papers with Burstall, Ferus, Pedit and Pinkall [FLPP], [BFLPP], we generalize methods and statements known in complex algebraic geometry (such as the Riemann-Roch theorem, Plücker formula) to complex quaternionic line bundles, and derive this way new results in surface theory. The main idea of our theory - which has come to be known as "Quaternionic Holomorphic Geometry" - is to consider the 4 sphere as quaternionic projective space and use the conformal maps into the 4 sphere as the holomorphic functions of the theory.

Particular interest lies on the transformation theory of surfaces: various surface classes are given by solutions to associated partial differential equations. Given one (simple) solution to the PDE, one hopes to construct new and more interesting examples by using transformations which preserve the surface class and are obtained by solving (simpler) ODEs, e.g. [L2].

Examples are Bäcklund and Darboux transformations which have been extensively studied in the past. In recent papers, we discussed such transformations for Willmore surfaces, for example, we classified Willmore tori with non-trivial normal bundle [LP2], linked the Bäcklund transformation to a well-known complex algebraic construction, the envelopes and tangents of complex curves [LP1], and discussed dressing of Willmore surfaces and its connection to the Darboux transformation [L4].

Moreover, it turned out that a generalization of the classical Darboux transformation allows to define and to give a geometric interpretation of the spectral curve of a conformal torus [BLPP]. As an application of this ansatz, we have a unified view of the integrable system theory for CMC [CLP], Hamiltonian stationary Lagrangians [LR] and Willmore surfaces [GL]. Generically, the Darboux transformation preserves these surface classes. However, at singular points of the spectral curve the Darbouxtransformation may change geometric properties, e.g. break the mean curvature property [L3]. If the surface is given by a harmonicity condition, then Darboux transforms are generically given by parallel sections of the associated family of flat connections of the harmonic map. In particular, an induced transformation on harmonic maps can be defined, and we show in [BDLQ] that this transformation is indeed the simple factor dressing of the harmonic map. An interesting special case are minimal surfaces in 3-space, [LM1]. Here the simple factor dressing is the well-known Lopez-Ros deformation applied to all rotations of the surface. The Darboux transformation of a minimal surface is in general not minimal but a Willmore surface in the 4-sphere [LM2]. Isothermic surfaces, that is surfaces which allow a conformal curvature line parametrisation, also allow to introduce a complex spectral parameter and to generalise classical transformations. In the case when the isothermic surface has non-vanishing constant mean curvature, the integrable systems arising from it when viewed as a CMC surface, as isothermic surface and as constrained Willmore surfaces are strongly linked [L5]. In case of a minimal surface, we obtain minimal ''bubbletons'' [L6]. We also obtain in collaboration with Cho, Ogata a generalisation of the classical Bianchi permutability by introducing Sym-type Darboux transforms [CLO1].

An important tool in my study of surfaces is their visualisation: it allows both to demonstrate a geometric situation and to experiment to obtain new results in surface theory and its applications.

Geometrically inspired discretisation of surfaces play an important role in providing fast and efficient numerical methods to solve PDEs. I'm currently investigating certain discrete closed curves in collaboration with Cho and Ogata which appear from physics, to find efficient tools to solve the underlying differential equations, [CLO2].

Applications of discrete geometry are plentiful in machine learning, shape optimisation, computer graphics, architecture, engineering, chemistry. My current engagement within a SPRINT collaboration explores how to improve machine learning algorithms via geometric information given by discrete curves. Within the Leverhulme Trust funded research network, discrete geometry plays an important role to approach questions on possible shape of surfaces and provides new efficient algorithms, used for example in computer graphics.

With I. Tyukin and A. Gorban, we have created a Visual Intelligence Lab in the Department of Mathematics with various visualisation tools (e.g., touchscreen computer, 3D projector). This provides equipment for interdisciplinary researchc, e.g. a joint project with Apical on image processing and vision systems, as well as an interdisciplinary project with Aldo Rona (Engineering and Alstom) investigating optimal shapes of turbines and with Andrew Hudson (Chemistry) on aerosol coalescence. Here are links to my picture gallery and to some of my labs. I'm proud that some of my pictures found their way onto the EWM webpage (even though some are unfortunately slightly distorted).

Before turning to surface theory, I've worked on isoparametric manifolds: I provided a constructive geometric method to decide whether an isoparametric manifold is homogeneous [L1] by using algebraic tools (root systems, Jordan algebras and representation theory) as much as geometric ones (canonical connections). This ansatz allows to investigate infinite dimensional isoparametric submanifolds (see PhD thesis of K. Weinl, Uni Augsburg) and to give a unified proof of the inhomogeneity of the FKM examples.

For a quick look at my research profile please see scholar.google.co.uk.