Faculty of Mathematics and Computer Science

Research Activities at the Faculty

The Faculty of Mathematics and Computer Science is broken down into the Mathematical Institute and the Institute of Computer Science. The Mathematical Institute is able to look back on more than 125 years of obligatory mathematical research. The Institute of Computer Science evolved (in 1989) out of the Computer Science Department founded by the Universität Leipzig and completed, in the context of the overall re-organization of the University in the beginning of the 90s, its own internal re-formation. Both Institutes are subdivided into sections in which one or more professors and staff members collaborate on related subject interests.

Mathematical Institute

The spectrum of research at the Mathematical Institute in the individual sections Algebra, Analysis, Didactics, Geometry, Functional analysis and Mathematical Physics, Numerical Mathematics and Scientific Computations, Optimization and Finance Mathematics, Economic Mathematics/Stochastics involves such areas as:

• arithmetic algebraic geometry with crypto-graphic applications, algebraic theory of singularities, computer algebra, constructive methods of commutative algebra, structure and arithmetic of Abelian varieties, discrete logarithmic problems, inverse Galois problem
• analytical, geometrical and numerical methods in hydrodynamics and continuum mechanics, eigenvalue theory elliptical operators, free boundary value problems in hydrodynamics, non-linear laws of conservation, mathematical theory of capillarity, existence problems and Regularity problems for variation on inequalities,
• Geometry and analysis on manifolds, curvature operators (Jacobi-Operator, skew-symmetric curvature operator), spectral geometry, twistors and spinors, symplectic geometry, Floer-homology, pseudo-holomorphic curves, periodical trajectories of Hamiltonian systems, Holonomy of affine connections und super-symmetric structures, conform parallel spinors on Kähler-manifolds, ambi-twistor spaces for Spin- minifolds,
• Operator-theory and operator-algebras, non-commutative geometry, modular-theory and quantum field theory, Lie-Groups,
• Multi-lattice procedures, efficient solvers for Integral equations,
• Phase transitions and entropy maximization, procedure optimization with stochastic data,
• Problems of prediction, interpolation and approximation, statistics und control of stochastic processes on financial markets and for insurances, credit-risk-modelling, stochastic decision processes, positive definite functions,
• Language and communication in the teaching of mathematics, computers in the teaching of mathematics.

For this purpose it was possible to obtain, in a series of projects, research grants from the Federal Ministry of Education and Research (BMBF), German Research Foundation (DFG), German Academic Exchange Service (DAAD), European Union (EU) and private funding.

Unfortunately, in 2006 a decision to cut positions, which had already been determined a few years earlier, at the Universität Leipzig was carried out. The chair for “Mathematical Physics” was, with the discontinuance of the position holder, not re-occupied and cancelled without replacement.

In research and teaching of the natural sciences there have existed, for many years, manifold interrelationships between the Mathematical Institute and the Max Planck Institute for Mathematics. Joint research seminars and workshops provide evidence thereof (e.g. the work-group for Mirror-Symmetry, the work-group for Micro-Structures, the advanced seminar Analysis, the work-group Neuronal Grids and Cognitive Systems, advanced seminar Mathematical Physics and advanced seminar Geometry); and equally important are the offerings of special lectures, presented by the Professors Jürgen Jost and Stefan Müller from the Max Planck Institute concerning current research areas for the students in the major study area of Mathematics.

One of the manifestations of the joint research work with the Max Planck Institute for Mathematics in the Natural Sciences is the annually organized work-shops in the areas of, among others, Differential Geometry and Stochastics. It should be mentioned as well that the much anticipated appointment of Dr. Angela Stevens as an Honorary Professor for ”Mathematics in the Life Sciences“ was a successful move.

Interdisciplinary research has always been, since the founding of the Mathematical Institute, highly regarded. Based on this tradition as well is the fact that an application was submitted, jointly with scientists of the Computer Science, Physics and the MPI Departments for the promotion of an Excellence Cluster: the “Felix Klein Centre for Mathematics and its Applications in the Natural Sciences“. This application has since then successfully passed the first round of selection. The final decision regarding the consummate cluster application will be made in October 2007.

The six year old Graduate Research Unit “Analysis, Geometry and their Connection to the Natural Sciences” is, as of this year, a section of the Graduate Centre “Mathematics/Computer Science Natural Science“ of the Research Academy Leipzig founded at the University. Currently, ten scholarship holders and one post-doc researcher are active within the College. It is the goal of the Graduate Research Unit to exhibit the reciprocal stimulation between Natural Sciences and Mathematics, and to make this usable in the education of doctoral candidates for Mathematics. Leipzig’s particular strength exists in its natural-science relationship to mathematical research, which is found in more than just the departments that are typically attributed to applied mathematics. Specifically, the natural science research targets are generated out of: Material Sciences and Hydro-Dynamics, Thermo-Dynamics, Statistic Mechanics, Classical Mechanics, String Theory, Image processing, Neuronal Grids and out of the General Theory of relativity. Of course, the research programme is a mathematical one; the methodology of variations, the interplay between geometries of various characteristics (Arithmetic, Riemanic, Symplectic), partial differential equations and global analysis as well as the close inter-relationships of Analysis, Numerical Mathematics and Mechanics characterize the research programme in many facets. Through the variety in the guest-speaker programme at the MPI for Mathematics in the natural sciences, excellent opportunities arise for doctoral candidates to come into direct contact with noted scientists.

Institute of Computer Science

The chair for “Technical Computer Science”, suddenly made vacant in 2005, was unfortunately not able to be occupied within the time frame of this research report. By means of appropriate substitution for the chair, however, the teaching responsibilities in this specialized area were comprehensively fulfilled.

The Institute of Computer Science has been taking part in the successfully-executed endeavours, to date, of the Excellence-Initiative. The two research areas, planned for the Felix-Klein-Zentrum (among others), Algorithms of Discrete Mathematics and Theory of Information are to be introduced through the Computer Science Institute. Currently, numerous EU projects, German Research Foundation (DFG) and Federal Ministry of Education and Research (BMBF) projects as well as industrial projects are being worked on at the Institute. A large portion of the fund-raising for grants has been received by the endowment professorship of the Deutsche Telekom AG for “Applied Telematics”.
At the Institute of Computer Science and within the time-frame of this report, the new study options Bachelor and Master of Sciences were implemented in light of the Europe-wide Bologna-Process mandates. Comprehensive and intensive work was necessary in order to guarantee enrolment for the winter semester 2006/2007 into these new study options.
The work in the Graduate College “Knowledge Representation” in the second funding time-frame was able to be successfully continued. The Graduate Research Unit has been a component of the Graduate Centre “Mathematics/Computer Science and Natural Science “ of the Research Academy founded at the University. Targets for research are Foundations, Methods and Applications for Technologies of Knowledge Representation. Colloquia were accomplished on a regular basis. One college doctoral candidate was able to successfully defend his dissertations.

There were generated out of the sections within the Institute for the 8th annual research report: 8 monographs, 35 articles in recognized subject magazines and 89 contributions in collection or proceeding volumes. The individual research results were introduced in a plethora of presentations in seminars, workshops and colloquia of the Institute.