** Date: **23 December 2002 - 11 January 2003

** Venue: **Harish-Chandra Research Institute, Allahabad

**Organizing Committee:**

- S. D. Adhikari HRI, Allahabad,
- W. Decker, University of Saarlands,
- S. A. Katre, University of Pune,
- R. A. Rao,TIFR Bombay,
- D. N. Sheth, S. P. College,
- J. K. Verma, IIT Bombay

** **

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

- constructive module theory, Ext, Tor, sheaf cohomology, Beilinson monads,
- computing radicals, primary decomposition and normalization,
- solving szstems of polynomial equations,
- algorithms for invariant theory,
- computing in local rings, invariants and classification of hypersurface singularities,
- Puiseux expansion, invariants of plane curve singularities, deformation theory.

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 downloaded.

- Notes of lectures and other material
- Five lectures on computational commutative algebra by Ravi Rao
- A quick Introduction to algebraic geometry and elliptic curves by D. S. Nagaraj B. Sury
- Elliptic Curves over Finite Fields by B. Sury
- Mordell-Weil Theorem by D. S. Nagaraj and B. Sury
- Macaulay II Tutorials by B. Adsul, C. D'Cruz and A. V. Jayanthan
- Notes on Groebner Bases by W. Decker and F. Shreyer [to be completed]
- Notes on invariant theory by J. K. Verma