- to interest and educate graduate students in the Gröbner bases method, which by now has become one of the standard methods in computer algebra,
- to stimulate new research in this area by bringing together researchers in seminars and other events during the year,
- to stimulate the interaction among researchers in computer algebra, commutative algebra, software design, and applications, as related to the theory of Gröbner bases.

Intensive Course on Gröbner Bases, January 12-30, 1998:

The *Intensive Course on Gröbner Bases*, organized by Prof.
Bruno Buchberger and Prof. Franz Winkler, took place from January 12 through
30. The course was primarily designed as a comprehensive introduction for
ph.d. students and young postdocs. The 28 participants came from Austria,
Belarus, Czechia, Germany, Hungary, Iran, Italy, Japan, Moldavia, Norway,
Romania, Russia, Spain, Sweden, Ukraine, and USA.

List of lecturers and subjects:

*B. Buchberger* (Linz): Introduction to the theory and construction
of Gröbner bases, elementary applications such as ideal membership,
computation in residue class rings, solution of algebraic equations, linear
syzygies, Hilbert functions, non-linear syzygies, and inverse mappings.

*F. Winkler* (Linz): Introduction to general rewriting, reductions
relations, term rewriting, geometric applications of Gröbner bases,
such as implicitization and parametrization of algebraic varieties, computation
of dimension of a variety.

*F. Schwarz* (Bonn): Janet bases as an algebraic tool for investigating
partial differential equations, applications of Janet bases for vector
fields of Lie groups, for 2nd order ordinary differential equations.

*L. Robbiano* (Genova): Introduction to the theory of Hilbert functions
and the computation of Hilbert functions, how to compute Gröbner bases
with the aid of Hilbert functions, using Hilbert functions for computing
free resolutions.

*W.W. Adams* (College Park, MD): Gröbner bases over rings,
primality testing using Gröbner bases, primary decomposition of ideals
by Gröbner bases.

Conference ``33 Years of Gröbner Bases'', February 2-4, 1998:

The conference ``33 Years of Gröbner Bases'' was held from February 2 through 4. Prof. Bruno Buchberger and Prof. Franz Winkler (both RISC-Linz) were the chairmen of the conference. The 94 participants came from 14 countries.

The conference consisted of one day of tutorial lectures on various
applications of the theory of Gröbner bases, and two days of presentations
of original research. The proceedings are published as: B.Buchberger and
F.Winkler (eds.), *Gröbner Bases and Applications*, London Math.
Soc. LNS 251, Cambridge Univ. Press, 1998.

Titles of the tutorial lectures and research talks in alphabetic order of the authors follow.

Tutorial lectures:

*B. Buchberger* (Linz): Introduction to Gröbner Bases.

*F. Chyzak* (Rocquencourt): Gröbner Bases, Symbolic Summation
and Symbolic Integration.

*W. Decker and T. de Jong* (Saarbrücken): Gröbner Bases
and Invariant Theory.

*G.-M. Greuel and G. Pfister* (Kaiserslautern): Gröbner Bases
and Algebraic Geometry.

*S. Hosten and R. Thomas* (Fairfax, VA and College Station, TX):
Gröbner Bases and Integer Programming.

*H.M. Möller* (Dortmund): Gröbner Bases and Numerical
Analysis.

*L. Robbiano* (Genova): Gröbner Bases and Statistics.

*D. Wang* (Grenoble): Gröbner Bases Applied to Geometric Theorem
Proving and Discovering.

Original research talks:

*B. Amrhein and O. Gloor* (Bern and Tübingen): The Fractal
Walk.

*M.A. Borges and M. Borges* (Santiago de Cuba): Gröbner Bases
Property on Elimination Ideal in the Noncommutative Case.

*M.-J. Gonzalez-Lopez and L. Gonzalez-Vega* (Santander): Newton
Identities in the Multivariate Case: Pham Systems.

*M. Insa and F. Pauer* (Innsbruck): Gröbner Bases in Rings
of Differential Operators.

*J.B. Little* (Worchester, MA): Canonical Curves and the Petri
Scheme.

*H. Lombardi and H. Perdry* (Besançon): The Buchberger Algorithm
as a Tool for Ideal Theory of Polynomial Rings in Constructive Mathematics.

*K. Madlener and B. Reinert* (Kaiserslautern): Gröbner Bases
in Non-Commutative Reduction Rings.

*J.L. Miller* (Bowling Green, KY): Effective Algorithms for Intrinsically
Computing SAGBI-Gröbner Bases in a Polynomial Ring over a Field.

*Müller-Quade, R. Steinwandt and T. Beth* (Karlsruhe): An
Application of Gröbner Bases to the Decomposition of Rational Mappings.

*P. Nordbeck* (Lund): On some Basic Applications of Gröbner
Bases in Non-commutative Polynomial Rings.

*L. Robbiano and M.P. Rogantin* (Genova): Full Factorial Designs
and Distracted Fractions.

*T. Sauer* (Erlangen): Polynomial Interpolation of Minimal Degree
and Gröbner Bases.

*J. Schicho* (Linz): Inversion of Birational Maps with Gröbner
Bases.

*J. Snellman* (Stockholm): Reverse Lexicographic Initial Ideals
of Generic Ideals are Finitely Generated.

*Q.-N. Tran* (Linz): Parallel Computation and Gröbner Bases:
An Application of Converting Bases with the Gröbner Walk.

The following invited talks were also given at the conference.

*W.W. Adams* (College Park, MD): Some Applications of Computational
Algebra to Partial Differential Equations.

*B. Buchberger* (Linz): 33 Years of Gröbner Bases: The Early
Years.

*F. Schwarz* (Bonn): Janet Bases for Symmetry Groups.