Research Activities at the Faculty

Reports of the Institutes and Departments

Research Activities at the Faculty

Mathematical Institute

At the Mathematical Institute, the Chair of Theoretical Mathematics and the Chair of Mathematics in the Sciences (both C4 positions) were filled during the period under review, with Professor Annette Huber-Klawitter and Professor Matthias Schwarz joining the Institute. Professor Huber-Klawitter is the first female Professor ever to join the Mathematical Institute. The application procedures for the Chair of Numerical Mathematics and Scientific Computing (C4) and the Chair of Discrete Mathematics (C4) were still underway. Four doctorates and six Habilitation theses were successfully completed in the year 2000.

Research at the Mathematical Institute is carried out in the Departments of Algebra, Analysis, Didactics, Geometry, Functional Analysis/Mathematical Physics, Numerical Mathematics/Scientific Computing, Optimisation, and Mathematical Economy/Stochastics. It deals with areas such as:

• Arithmetic geometry, algebraic theory of singularities, computer algebra, constructive methods in commutative algebra
• Analytical, geometrical and computational methods in hydrodynamics and continuum mechanics, eigenvalue theory of elliptic operators, free boundary problems in hydrodynamics, non-linear conservation laws, mathematical theory of capillarity, existence and regularity problems for variational inequalities
• Geometry and analysis on manifolds, curvature operators (Jacobi operators, skew-symmetric curvature operators), spectral geometry, twistors and spinors, periodic orbits of Hamiltonian systems, holonomy of affine connections and supersymmetric structures, conformally parallel spinors on Kähler manifolds, ambitwistor spaces on spin manifolds
• Operator theory and operator algebras, non-commutative geometry, modular theory and quantum field theory, Lie groups
• Multigrid methods, efficient solvers for integral equations
• Phase changes and entropy maximisation, optimisation for stochastic data
• Problems of prediction, interpolation and approximation, statistics and control of stochastic processes for financial markets and insurance, stochastic decision theory, positive definite functions
• Language and communication in mathematical education, computers in mathematical education

Various research activities were financially supported by the DFG (National German Research Society), the DAAD (German Academic Exchange Agency) and the EU.

Close ties in both research and education have been nurtured in recent years between the Mathematical Institute and the Max Planck Institute of Mathematics in the Sciences. Examples of co-operation include joint research seminars and workshops (for instance on analysis, neural networks/cognitive systems, mathematical physics, geometry) as well as the range of special lectures on current research topics presented by Professors Jürgen Jost, Stefan Müller and Eberhard Zeidler from the Max Planck Institute, all of which are part of undergraduates' training. In November 2000 the conference "Prospects in Geometry", at which internationally recognised experts reported on current developments in differential geometry and neighbouring fields, was hosted by the Max Planck Institute for the fifth time. The conference was organised by the Max Planck Institute (Professor Matthias Schwarz) and the Mathematical Institute (Professor Hans Bert Rademacher) with the support of the Science Theory Centre.

The research work of the Mathematical Institute was acknowledged by the Function Theory Conference being held in Leipzig in 2000 - an annual conference which has been held in various places in Germany ever since 1984. This was the first time it was hosted in Saxony. The invitation from the organisers (Professors Bernd Fritzsche and Bernd Kirstein) was taken up to by 70 participants from eight countries, including prestigious speakers such as P. L. Duren (Ann Arbor), V.E. Katsnelson (Weizmann Institute, Rehovot), A. Bultheel (Leuven), O. Njåstad (Trondheim), J.M. Djukarev (Kharkov) and D. Alpay (Beer Sheva).

In October and December 2000, the Mathematical Institute organised colloquia to honour two scholars of outstanding merit: Professor Herbert Beckert (tying in with his 80th birthday) and Professor Erich Kähler, who sadly passed away in 1999. Their scientific achievements and personal dedication were acknowledged in special papers by the following professors: Robert Finn from Berkley ("Reminiscences, Observations and Anticipation"), Stefan Hildebrandt from Bonn ("Two-dimension or Parametric Variation Problems"), Konrad Gröger from Berlin ("Sobolev-Campanato Spaces as Aids in Investigating Boundary Problems"), Horst Schumann from Leipzig ("The Finite Basis of Padua Fortress"), Rolf Berndt from Hamburg ("Kähler's Whole Differentials") and Armin Uhlmann from Leipzig ("Basic Assumptions of Quantum Physics Presented Algebraically").

The Graduate College "Analysis, Geometry, and Interaction with Natural Sciences", under which grants have so far been awarded to eight scholars, was launched in May 2000. The programme is designed to ascertain the mutual relations between mathematics and the sciences, and to use them in training doctoral students. This is an area where nowadays much needs to be done owing to the extensive separation of mathematicians from other scientists in graduate courses. The research programme is roughly divided into an analytical and a geometrical section – subjects which have a strong history at the Mathematical Institute. Leipzig's particular strength lies in the relations with the sciences maintained by more than just those disciplines generally considered to belong to applied mathematics. This strength has been underlined by the establishment of the Max Planck Institute of Mathematics in the Sciences, the second Max Planck Institute in Germany (after Bonn) to specialise in mathematics. Research focuses on aspects of material science, hydrodynamics, thermodynamics and statistical mechanics, classical mechanics, image-processing, neural networks, and the general theory of relativity. The research programme is characterised by variational methods, the interplay between geometry, global analysis and partial differential equations, and the close links between analysis, numerical mathematics and mechanics. One important aspect of the programme is to familiarise doctoral students with more than just one application and with more than just one mathematical methodology. The broad visiting scholars' programme at the Max Planck Institute of Mathematics in the Sciences provides doctoral students with excellent opportunities to come into direct contact with prominent scientists. An important role is played by the joint teams staffed by personnel from both the University of Leipzig and the Max Planck Institute.

Institute of Computer Science

In the year 2000, the Institute of Computer Science continued to flourish. This is apparent from the development in the numbers of scholars, personnel and students at the Institute. At the start of 2000, the Institute employed 65 scientific and non-scientific personnel – a figure which had risen to 86 by the end of December. The number of students has grown much faster, with the number of students enrolled on the Diplom degree course in Computer Science rising from 516 in 1999 to a record 679 in 2000. This places Computer Science among the top ten degree courses with the largest numbers of students at the University of Leipzig for the first time.

In the winter semester 2000/2001, Professor Klaus Peter Fähnrich started work at the Chair of Application-specific Information Systems. Following Professor Rünger's departure to take up an appointment at Chemnitz in April 2000, she was initially covered for by Dr Claudia Leopold from Friedrich Schiller University in Jena, and so all ten regular professorships are currently filled.

Two more chairs are to be set up at the Institute of Computer Science. These are a Foundation Chair of Applied Telematics (e-business) financed by Deutsche Telekom and the Chair of Bioinformatics funded by the DFG. Appointment procedures are already underway and are set to be completed in 2001. It should be emphasised that only five applications throughout Germany for the establishment of a bioinformatics chair have been approved, the Leipzig application being the only successful one from Eastern Germany.

The Institute's outstandingly positive overall development was accentuated at the Computer Science Faculty Council, whose members comprise faculties and departments of universities and university colleges which run Diplom degree courses in Computer Science under the general examination regulations for computer science and universities, and which are entitled to supervise doctoral and Habilitation theses in Computer Science. Before the Institute of Computer Science was accorded full membership in December 2000, it merely enjoyed visiting status at the Faculty Council.

Outstanding scientific events during the year included the three Habilitation degrees awarded to Dr Stefan Lange, Dr Christian Wolff and Frank Wolter. At this point, the acknowledgement granted to the scientific achievements of Dr Ralf over the span of many years, who was appointed a supernumerary professor at the Institute of Computer Science, and the appointment of Dr Klaus Hänssgen to Leipzig University of Applied Sciences as professor of information systems and multimedia technologies should be mentioned.

The work of the Graduate College "Knowledge Representation" was successfully continued. At present 18 researchers are enrolled on the programme. In addition to weekly colloquia, in 2000 a workshop, a spring school and two conferences dealing with different topics of the graduate programme, and a report colloquium were all held. The good work performed under the programme was corroborated by the DFG's decision in 2000 to extend its funding.

The installation of air conditioning in the rooms on the first floor of the main building brought about a marked improvement in working conditions. Another computer pool comprising 64 workstations (Unix and Microsoft Windows NT) was installed in the air-conditioned rooms. A new computer pool is to be set up for the Department of Application-specific Information Systems, enabling the Institute to meet students' needs in terms of both quantity and quality.