# Katrin Leschke

** **** address:** Department of Mathematics

University of Leicester

Leicester, LE1 7RH

*CH203*

**office:**

*see my time table (or sent me an email for an appointment)*

**office hours:**

*+44(0116) 252 5237*

**phone:****k.leschke@le.ac.uk**

*email:*### Teaching

This year, I will be on study leave in the first semester and teach in the second one "Curves and Surfaces".

### Admin

I'm Head of Student support (this includes organisation of houses).

I'm a member of DMG, MCP, SSC, LTC, RC, ISET, appeals panels, co-director of the VisLab.

I'm the webmaster of EWM and as such, member of the standing panel of EWM.

### Grants

I have been awarded an international network grant Minimal surface: integrable systems and visualisation by The Leverhulme Trust.

### 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.

The focus in my recent research is in surface theory, where I study conformal maps into the 4 sphere. 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, [LM]. 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.

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. 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 algebra

s 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.

### Workshops/Conferences

Differential Geometry Workshop at the BMC 2011 in Leicester (with Mark Haskins, Imperial College).

LMS Midlands Regional Meeting, June 11th 2013, in Leicester (with Ian McIntosh, University of York). Workshop on "Advances in Surface Theory", June 12-14 2013 in Leicester (with Ian McIntosh, University of York).

### Students

Current PhD students: L. Collard (with A. Hudson, Chemistry), H. Obaida (co-supervisor), H. Kadhim (co-supervisor).

Diplom students (at Universität Augsburg): A. Gouberman, Title of Thesis: "Darboux-Transformationen des Clifford-Torus"

Former student projects: M. Pirashvili (QHG), J. Habdank (discrete minimal surfaces),Pearce Dedmon (Visualisation of Wente tori and touch screen viewer)

Former MMath projects: Z. Sapi Mkwawa (minimal surfaces), N. Lomas (Wente torus), K. Reed (discrete surfaces), H. Raistrick (Delaunay surfaces), L. Fernandez (CMC & aerosols), Chris Cuthbert (The history of the Willmore conjecture).