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Algebraic Geometry and Computer Algebra

Details | Lectures | The Group | Exemplary Schedule | Winter term 1999/2000 | Summer term 2000

Algebraic Geometry and Computer Algebra

Worldwide there are but few universities where the relation between theoretical research in algebraic geometry, complex analysis, singularity theory and computer algebra together with industrial applications is as well developed as in Kaiserslautern.

This fact is mirrored in teaching: courses are offered in these fields guiding the student to interesting problems in theory and applications.

Accompanying lectures in algebra, topology and differential geometry at the same time open wider perspectives on mainstream mathematics.

Details

Algebraic Geometry is one of the oldest of the classical mathematical disciplines. Traditionally, it deals with the study of algebraic varieties, that is, the zero-sets of (systems of) polynomial equations, in particular, with their geometry. There are deep connections with various other areas of mathematics and theoretical physics, such as complex analysis, number theory, topology, differential geometry or string theory. This interaction makes algebraic geometry a highly dynamic subject and relevant for many "real world" problems.

The development of algebraic geometry throughout the past centuries occurred in waves and was influenced by different schools, each of them using a different language. In the second half of this century, the foundations of algebraic geometry have been completely reformulated, introducing the language of schemes, vector bundles and sheaf cohomology. This has led to many new results and a deeper understanding of classical ones; it has allowed simplification of proofs as well as substantial generalisation of results.

The main task of algebraic geometry, the investigation of the global and the local structure of varieties brought forth the discipline of singularity theory. It is concerned with the study of the local behaviour of varieties at non-smooth points (which are decisive for the structure of a variety and resist a numerical treatment) and has applications, for instance, in the study of dynamical systems.

Current research in algebraic geometry as performed in Kaiserslautern includes, for example, the construction and the study of moduli spaces, i.e., varieties whose points represent the objects of interest, e.g., singularities, varieties of a certain class, vector bundles, sheaves, etc.

Moebius strip inside a vector bundle

In algebraic geometry, the computer has become an increasingly important tool for complicated calculations. Computer algebra is concerned with the development of efficient algorithms and their implementation. It emerged in the 1970's and gained popularity via computer algebra systems such as Macsyma, Reduce, Maple, Mathematica, MuPAD, etc. These systems have become indispensable in research, teaching and industrial development.

The computer algebra system SINGULAR, which has been developed in Kaiserslautern, has been designed for computations in commutative algebra, algebraic geometry and singularity theory. Its main features include the computation of standard bases, syzygies and the factorisation of multivariate polynomials over several ground fields. Its implementation of the Buchberger algorithm is one of the fastest worldwide. SINGULAR offers a C-like programming language and includes many libraries for applications in singularity theory and algebraic geometry. In joint projects with microelectronics, SINGULAR is also used for dealing with problems concerning analog circuits.

The Kaiserslautern research groups in algebraic geometry, singularity theory and computer algebra are prominent members of German and European networks and maintain research contacts worldwide. They are actively participating in the network EAGER for projective algebraic geometry, which consists of approximately 50 nodes in Western Europe. In singularity theory, the Kaiserslautern group is a member of the European Singularity Net ESN. The Computer Algebra group leads the European POSSO-project and participates in the German DFG-Schwerpunkt "Effiziente Algorithmen für diskrete Probleme und ihre Anwendungen".

There are but a few universities in the world where the theoretical research in algebraic geometry is linked as well with industrial applications as it is in Kaiserslautern. This fact is reflected in teaching. The programme "Mathematics International" with its specialisation "Algebraic Geometry and Computer Algebra" offers various courses in the above fields leading the student to interesting theoretical and practical problems. Moreover, the students may also participate in the activities of the Centre of Computer Algebra, present at the Department of Mathematics. Additional lectures in algebra, topology, differential geometry, etc. provide background knowledge and guarantee a broad education in contemporary mathematics. Moreover, there is an abundance of seminars, tutorials and colloquia, including talks by guest speakers, which provide a stimulating environment for research.

The Group

Prof. Dr. G.-M. Greuel, Prof. Dr. G. Pfister, Prof Dr. G. Trautmann, PD Dr. K. Wirthmüller, Dr. C. Lossen, Dr. B. Kreußler, Dr. J. Zintl and assistants.

Lectures

The following list is a selection of the offered lectures within the specialization "Algebraic Geometry and Computer Algebra". There may be some fluctuations due to a variety of reasons.

Exemplary Schedule for students who started in the winter term 1997/98

This is an exemplary schedule for the study programme Mathematics International within the scope of the specialization "Algebraic Geometry and Computer Algebra".

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