Winter School on Arithmetic and Computational Algebraic Geometry
Harish-Chandra Research Institute, Allahabad
23 December 2002 - 11 January 2003
Arithmetic aspects: The school is an introduction to arithmetic
and computational aspects of algebraic geometry, with particular
emphasis on practical exercise
sessions with the computer algebra systems SINGULAR and Macaulay 2.
The school will begin with arithmetic aspects of algebraic geometry
leading to a proof of Riemann Hypothesis for elliptic curves.
We will closely follow the book by Daniel Bump for this part.
There will be 45 lectures in this part by active researchers
in the interconnected areas of commutative algebra, algebraic geometry and
number theory. This part will conclude with expository talks by Prof. D.
Prasad and C. S. Dalawat on the conjectures of Weil and Swinnerton-Dyer.
The organizers strongly feel that by introducing the three
subjects together to graduate students, it will be possible to explain the
beautiful connections being discovered these days among these subjects.
Computational aspects: The computational aspects will be introduced
begining on Jan 1.
We will start with Buchberger's algorithm to compute Groebner
bases and syzygies, and basic applications to computational problems
arising from the geometry-algebra dictionary, ideal and radical
membership, ideal interesections, ideal quotients, saturation,
homogenization, elimination, Hilbert functions.
Prof. W. Decker (University of Saarland, Germany) and Dr. C. Lossen
(University Kaiserslautern, Germany) will then begin with lectures and
tutorials on more advanced topics, such as
They will quickly introduce the MAPLE packages CASA (to compute,
for instance, parametrizations of rational curves) and Schubert (to
compute objects and invariants arising in intersection
theory and thus enumerative geometry), and they will demonstrate SURF, a
package for visualizing curves and surfaces. Finally, they will apply the
computational techniques to problems coming from actual research in
algebraic geometry. This could mean problems suggested
by the participants, or problems suggested by the lecturers, for instance,
constructing and classifying varieties of low codimension, in particular,
surfaces in projective 4-space.
Members are invited to send/raise questions in computational
commutative algebra, number theory & algebraic geometry which Professor
Wolfram Decker and Dr. Christoph Lossen will answer in the Workshop.
The goal of this Workshop is to familiarize the students and
researchers about the scope of the freewares SINGULAR and Macaulay 2.
Some idea of this can be got from the School link
http://www.math.uni-sb.de/~ag-decker. You may also refer to the
publications of Professor W. Decker, from where his paper
with Professor F.O. Schreyer on 'Computational Algebraic Geometry Today'
which appeared in Applications of Algebraic Geometry to Coding Theory,
Physics and Computation, 65-119, Kluwer Academic Publishers, 2001, can be
- constructive module theory, Ext, Tor, sheaf cohomology, Beilinson
- computing radicals, primary decomposition and normalization,
- solving szstems of polynomial equations,
- algorithms for invariant theory,
- computing in local rings, invariants and classification of
- Puiseux expansion, invariants of plane curve singularities,