Katrin Leschke

Reader

address: Department of Mathematics 
University of Leicester office
Leicester, LE1 7RH
office: CH203
office hours: see my time table (or sent me an email for an appointment)
phone: +44(0116) 252 5237
email: k.leschke@le.ac.uk

Personal details

 

Biography

I am a reader in Mathematics. I obtained my Diplom and PhD on topics in geometry at TU Berlin. Before joining the University of Leicester in 2007, I hold research positions at TU Berlin, the University of Massachusetts in Amherst and the University of Augsburg where I was awarded my habilitation. My research expertise is in surface theory, integrable systems and visualisation. I'm  leading an international network team working on minimal surfaces "m:iv"which is funded by the Leverhulme Trust which investigates links between minimal surfaces, integrable systems and visualisation. The m:iv network has organised 4 international conferences, several mini-workshops, seminars and supported research visits from network partners. I currently also work on applications of surface theory in machine learning, engineering and chemistry. I lead an interdisciplinary, international's research team "Maths Meets Arts Tiger Team" at the University of Leicester which investigated how  the general public can be engaged with modern mathematics and ran the first "Maths Meets Arts Festival" in July 2020. This project, and is predecessor,  were additionally supported by the ISSF, funded by the Wellcome Trust.

Qualifications

  • Diplom, TU Berlin PhD, University of Leicester

  • PhD, TU Berlin
  • Habilitation, University of Augsburg
  • Fellow of Higher Education Academy

 

I am currently Head of Postgraduate Research, as as such member of the research committee and the CSE PGR team. I'm also a member of appeals panels and co-director of the VisLab, and am on the PAP of CSE. Elsewhere, I'm the webmaster of EWM, and as such an elected member of the standing panel of EWM. I'm an elected member of the SLAM committee of the LMS. I was on the Scientific Committeee for the international conference PADGE 2017 in Leuven and on the ITLR 2017 panel of the Department of Mathematics, University of Warwick. I organised the IWD 2017 event of the Department of Mathematics at the University of Leicester.

Websites

Minimal surface: integrable systems and visualisation, my personal webpagemathscinetscholar.google.co.uk.

Recent grants

Minimal surfaces
The international network grant

I have been awarded an international network grant  Minimal surface: integrable systems and visualisation by The Leverhulme Trust. I have received funding for Public Engagement project "Minimal surfaces - artists views" from the ISSF, funded by the Wellcome Trust.

In 2020 I lead the University Tiger Team "Maths Meets Arts" funded by the Leicester Institute of Advanced Studies.

 

Teaching

I am a Fellow of the HEA.

I have obtained a wide range of pedagogic experience by teaching at undergraduate and postgraduate level in Germany, the USA and the UK. I consider teaching as an integral part of my work: experiences in teaching enhance my views in research, and conversely, new approaches in research lead to more meaningful teaching. My teaching expertise lies in the area of geometry and visualisation but I also made significant contributions to the curriculum of the Calculus and Analysis stream.

In teaching, student engagement is essential to student success. To facilitate discovery-lead learning, I developed modules to include substantial project work and innovative use of technology for visualisation of mathematical concepts. I received a certificate of recognition for the quality of my application for FHEA through the PEERS Experiential Route.

Lectures: Linear Algebra 1 (MA1112), Calculus and Analysis I (MA1012), Geometry and Visualisation (MA3152), Curves and Surfaces (MA3152), Proofs and Number Systems (MA1103), Complex Analysis (MA3121), Minimal surfaces (MA7155), Investigations in Mathematics (MA2510), MMath Project (MA4504), Analytic Geometry (MA0001 - DUT)

Projects: M. Pirashvili (Quaternionic Holomorphic Geometry), J. Habdank (Minimal surfaces and finite elements), Pearce Dedmon (Visualisation of Wente tori and touch screen viewer),

LMS bursary URB: A. Parkar (Lopez-Ros deformation of the gyroid)

MMath ProjectsZ. Sapi Mkwawa (Minimal surfaces), N. Lomas (Wente tori), K. Reed (Discrete sur- faces), H. Raistrick (Delaunay surfaces), L. Fernandez (CMC surfaces & aerosols), Chris Cuthbert (The history of the Willmore conjecture), Dominic Lusk (Goursat transfermations of minimal surfaces), Richard Burton (Remote Visual/Optical Interface for 3D surface visualisation), Aneesa Parkar (Triply periodic minimal surfaces), Josh Aberdeen (Minimal surfaces).

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 engineering and chemistry.

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.

ParticNodoid Bubbletonular 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) systems of PDEs, e.g. [L2].

Examples are Bäcklund and Darboux transformations which have been extensively studied in the past. In a series of 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 of the harmonic conformal Gauss map is the well-known Lopez-Ros deformation applied to all rotations of the surface. This 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 [CLO].

Lopez-Ros deformation of the Scherk surface
Lopez-Ros deformation of the Scherk surface

 

 

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 discrete closed curves in collaboration with Cho and Ogata. 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.

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 research, e.g. a  joint project with Apical on image processing and vision systems and a SPRINT project with XCAM in machine learning, as well as an interdisciplinary project with Aldo Rona (Engineering and Alstom)  investigating optimal shapes of turbines [KRLG, KROL, KRLGG, OKRLG] 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.

PhD students: L. Collard (2018, with A. Hudson, Chemistry), H. Obaida (co-supervisor, 2017), H. Kadhim (co-supervisor).

External Examiner: J. Perez (University of Granada, evaluator externo), M. Moruz (University of Valenciennes, rapporteur), Francesca Tripaldi (King's College London).

Internal Examiner: L. Barbosa Torres.

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Contact details

Department of Mathematics
University of Leicester
University Road
Leicester LE1 7RH
United Kingdom

Tel.: +44 (0)116 229 7407

Campus Based Courses

Undergraduate: mathsug@le.ac.uk
Postgraduate Taught: mathspg@le.ac.uk

Postgraduate Research: pgrmaths@le.ac.uk

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