address: Department of Mathematics
University of Leicester
Leicester, LE1 7RH
office: CH203
office hours: see my time table (or sent me an email for an appointment)
phone: +44(0116)2525237
email: k.leschke@mcs.le.ac.uk

Teaching

This semester I teach Complex Analysis (MA3121) and a supervision in pure Maths (as last semester).

Admin

I'm the Study Abroad Tutor: I'm the contact person for students with a BSc/MMath (Europe/USA) degree, and for International students (ERASMUS). I'm also a member of the Research Committee, and I organize the Internal Pure Maths Seminar. Moreover, I try to coordinate the Pure Maths webpages.

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 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 quaternionified complex analysis.

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], and linked the Bäcklund transformation to a well-known complex algebraic construction, the envelopes and tangents of complex curves [LP1].

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 conformal torus [BLPP]. As an application of this ansatz, we have a unified view of the integrable system theory for CMC (with Carberry, Pedit), Hamiltonian stationary Lagrangians [LR] and Willmore surfaces [GL]. However, the Darboux transformation at singular points of the spectral curve may change geometric properties, e.g. break the mean curvature property [L3].

In my PhD thesis "Homogeneity of  isoparametric submanifolds"  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.

Students

Phd student (jointly supervised with M. Georgoulis): A. Saad

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

Projects: M. Pirashvili (QHG), Z. Sapi Mkwawa (minimal surfaces), J. Habdank (discrete minimal surfaces)