Galaad is a joint project with Laboratoire J.A. Dieudonné
U.M.R. C.N.R.S. no 6621, University of Nice Sophia-Antipolis.

Our research program is articulated around effective algebraic geometry and its applications. The objective is to develop algorithmic methods for effective and reliable resolution of geometrical and algebraic problems, which are encountered in fields such as CAGD, robotics, computer vision, molecular biology, etc. We focus on the analysis of these methods from the point of view of complexity as well as qualitative aspects, combining symbolic and numerical computation.

Geometry is one of the key topics of our activity, which includes effective algebraic geometry, discrete and combinatorial geometry, differential geometry, computational geometry of semi-algebraic sets. More specifically, we are interested in problems of small dimensions such as intersection, singularity, topology computation, and questions related to algebraic curves and surfaces.

On one hand, we consider algebra, particularly the problems of resolution. We are involved in the design and analysis of new methods of effective algebraic geometry. Their developments and applications are central and often critical in concrete problems.

On the other hand, approximate numerical calculations, usually opposed to symbolic calculations, and the problems of certification are also at the heart of our approach. We intend to explore these bonds between geometry, algebra and analysis, which are currently making important strides. These objectives are both theoretical and practical. Recent developments enable us to control, check, and certify results when the data are known to a limited precision.

Finally our work is implemented in software developments. We pay attention to problems of genericity, modularity, effectiveness, suitable for the writing of algebraic and geometrical codes. The implementation and validation of these tools form another important component of our activity.

Our scientific activity is defined according to three broad topics: geometry, resolution of algebraic systems of equations, and symbolic-numeric links.

In order to solve an algebraic problem effectively, a preprocessing analyzing step is often mandatory. From such study, we will be able to deduce the method of resolution that is best suited to and thus produce an effective solver, dedicated to a certain class of systems. The effective algebraic geometry provides us tools for analysis and makes it possible to exploit the geometric properties of these algebraic varieties. For this purpose, we focus on new formulations of resultants allowing us to produce solvers from linear algebra routines, and adapted to the solutions one seeks to approach.

We are interested in the properties of solutions of polynomial equations,
which result from the geometry of monomials appearing in equations, and based
on Newton polytope associated to each polynomial. This toric
elimination theory (or sparse), introduced by I. M. Gelfand's group

The above-mentioned tools of effective algebraic geometry make it possible to analyze in detail and separately the algebraic varieties. On the other hand, traditional algorithmic geometry deals with problems whose data are linear objects (points, segments, lines) but in very great numbers. Combining these two approaches, we concentrate on problems where collections of piecewise algebraic objects are involved. The properties of such geometrical structures are still not well known, and the traditional algorithmic geometry methods do not always extend to this context.

The objects in geometrical problems are points, lines, planes, spheres, quadrics, .... Their properties are, by nature, independent from the reference one chooses for performing analytic computations. Which leads us to methods from invariant theory. In addition to the development of symbolic geometric computations that exploit these invariants, we are also interested in developing more synthetic representations for handling those expressions.

The analysis of singularities for a (semi)-algebraic set provides a better understanding of their structure. As a result, it may help us better apprehend and approach modeling problems. We are particularly interested in applying the theory of singularities to cases of curves silhouettes, shadows curves, moved curves, medial axis, self-intersections, appearing in algorithmic problems in CAGD and shape analysis.

In order to deduce the solutions of a system of polynomials in

We are interested in the ``effective'' use of duality, that is, the properties of linear forms on the polynomials or quotient rings by ideals. We undertake a detailed study of these tools from an algorithmic perspective, which yields the answer to basic questions in algebraic geometry and brings a substantial improvement on the complexity of resolution of these problems. Our focuses are effective computation of the algebraic residue, interpolation problems, and the relation between coefficients and roots in the case of multivariate polynomials.

The preceding work lead naturally to the theory of structured
matrices. Indeed, the matrices resulting from polynomial problems, such as
matrices of resultants or Bezoutians, are structured. Their rows and columns
are naturally indexed by monomials, and their structures generalize the Toeplitz matrices
to the multivariate case. We are interested in
exploiting these properties and the implications in solving
polynomial equations

When solving a system of equations, a first treatment is to transform it into several simpler subsystems when possible. We are interested in a new type of algorithms that combine the numerical and symbolic aspects, and are simultaneously more effective and reliable. For instance, the (difficult) problem of approximate factorization, the computation of perturbations of the data, which enables us to break up the problem, is studied. More generally, we are working on the problem of decompositing a variety into irreducible components.

The behavior of a problem in the vicinity of a data can be interpreted in terms of deformations. Accordingly, the methods of homotopy consist in introducing a new parameter and in following the evolution of the solutions between a known position and the configuration one seeks to solve. This parameter can also be introduced in a symbolic manner, as in the techniques of perturbation of non-generic situations. We are interested in these methods, in order to use them in the resolution of polynomial equations as well as for new algorithms of approximate factorization.

The numerical problems are often approached locally.
However in many problems, it is significant to give global
answers, making it possible to certify calculations.
The symbolic-numeric approach combining the algebraic and analytical aspects,
intends to answer these problems. Especially, we focus on
certification of geometric predicates that are essential for the analysis
of geometrical structures

The sequence of geometric constructions, if treated in an exact way, often leads to a rapid complexification of the problems. It is then significant to be able to approximate these objects while controlling the quality of approximation. Subdivision techniques based on the algebraic formulation of our problems are exploited in order to control the approximation, while locating interesting features such as singularities. This approach combines geometrical, algebraic and algorithmic aspects.

According to an engineer in CAGD,
the problems of singularities obey the following rule:
less than 20% of the treated cases are
singular, but more than 80% of time is necessary to develop a code
allowing to treat them correctly. Degenerated cases are thus critical from both
theoretical and practical perspectives.
To resolve these difficulties, in addition to the
qualitative studies and classifications, we study methods of perturbations of symbolic systems, or adaptive methods based on
exact arithmetics. For example, we work on the computation of the
sign of expressions, and on approaches combining modular and approximate
computations, which speed up the exact answer

3D modeling is increasingly familiar for us (synthesized images, structures, vision by computer, Internet, ...). The involved mathematical objects have often an algebraic nature, which are then discretized for easy handling. The treatment of such objects can sometimes be very complicated, for example requiring the computations of intersections or isosurfaces (CSG, digital simulations, ...), the detection of singularities, the analysis of the topology, ...We propose the developments of methods for shape modeling that takes into account the algebraic specificities of these problems. We tackle questions whose answer strongly depends on the context of the application being considered, in direct relationship to the industrial contacts of CAGD we have.

Robotics and computer vision come with specific applications of the methods for solving polynomial equations. That is the case for instance, for the calibration of cameras, robots, computations of configurations and workspace. The resolution of algebraic problems with approximate coefficients is omnipresent.

The chemical
properties of molecules intervening in certain drugs are related to the
configurations (or conformations) which they can take. These molecules are
seen as mechanisms of bars and spheres, authorizing rotations around
certain connections, similar to robots series. Distance geometry thus
plays a significant role, for example, in the reconstruction from NMR
experiments, or the analysis of realizable or accessible
configurations. The methods we develop are well suited for solving such a
problem.

See synaps web site:

We consider problems handling algebraic data structures such as polynomials, ideals, ring quotients, ..., as well as numerical computations on vectors, matrices, iterative processes, ...etc. Until recently, these approaches have been separated: the software for manipulating formulas is often not effective for numerical linear algebra; while the numerically stable and effective tools in linear algebra are usually not adapted to the computations with polynomials.

We design the software synaps (SYmbolic Numeric APplicationS) for symbolic
and numerical computations with polynomials.
This powerful kernel contains univariate and multivariate solvers as well as several
resultant-based methods for projection operations.
Currently, we are developing a module that is related to factorization, which is
relevant to the separation of irreducible components of a
curve in

In this library, a list of structures and functions makes it possible
to operate on vectors, matrices, and polynomials in one or more variables.
Specialized tools such as lapack, gmp, superlu, rs, gb, ... are also
connected and can be imported in a transparent way.
These developments are based on C++, and attention is paid to the generic
structures so that effectiveness would be maintained.
Thanks to the parameterization of the code (template)
and to the control of their instantiations (traits, expression
template), they offer generic programming without losing effectiveness.

See axel web site:

We are developing a module called axel (Algebraic Software-Components
for gEometric modeLing) dedicated to algebraic methods for curves and
surfaces. Many algorithms in geometric modeling require a combination of
geometric and algebraic tools. Aiming at the development of reliable and
efficient implementations, axel is to provide a framework for such
combination of tools, involving symbolic and numeric computations.

We rely on external libraries, such as
cgalsynaps

We endeavour to provide data structures and functionalities related to curved objects (classes of basic objects provided with predicates and constructions). Currently, the module contains algorithms for computing intersection points of curves or surfaces, detecting self-intersection points of parameterized surfaces, implicitization, computation of topology, meshing implicit surfaces, ..., etc.

See multires web site:

The Maple package multires contains a list of functions related to the
resolution of polynomial equations. The prime objective is to illustrate
various algorithms on multivariate polynomials, and is not effectiveness
which is achieved in a more adapted environment as synaps. It provides methods to build
matrices whose determinants are multiples of resultants on certain varieties,
and solvers depending on these formulations, and based on eigenvalues and
eigenvectors computation. It contains the computations of Bezoutians in
several variables, the formulation of Macaulay

During the process of resolution of polynomial systems, computing normal forms in an algebra quotient is usually done by exact methods. This stage is immediately downstream from the process of modeling and precedes in most cases of numerical approximations for the solutions.

Computing a normal form indeed yields an exact representation for all
solutions of the given system. Traditional algebraic methods for solving
systems of polynomial equations do not perform well when polynomials have
approximate coefficients. Based on this fact and the work of synaps

We are now working on a criterion that allows the removal of the assumption of finite solutions in the system.

We present a new method to solve univariate algebraic equations. This method
is based on the integration of a special vector field synaps library for the
experiments. Based on the approach developed in

Collaborations with Marc Chardin, university of Paris VI, and Carlos D'Andrea, university of Berkeley CA.

We continued our efforts on the implicitization problem, which we began to
study a couple of years ago

In collaboration with Marc Chardin (University Paris VI), we detailed
and improved the method based on approximation complexes we had
introduced in

We collaborate with Carlos D'Andrea on the
irreducibility of multivariate subresultants. These subresultants were
introduced by Marc Chardin in

The third direction we investigated is to solve the implicitization
problem from the effective computation of residues by matrix formulation.
In general, Bezoutian matrices give a multiple of the implicit equation.
The unwanted factor in this equation can be removed (without factorization) using an
algorithm for the residue computation in the multivariate setting. The advantage of this
approach is to treat surfaces with base points without restrictive geometric hypotheses
on the zero-locus of base points. These results are published in

The Bezoutian is a fundamental tool which, surprisingly, appears in many
areas of constructive algebra. In a work presented at MEGA'03 conference

Self-intersections often occur as an unexpected result of CAGD operations such as offset, draft, fillet or sweep, and detecting them is a real challenge for many modeling tools. We developed a new sampling algorithm for finding self-intersections of parameterized surfaces relying on the definition of region in the parameter domain and proceeding in three steps: region/region intersections, neighbor regions intersections and regions self-intersections. This work is supported by the European project GAIA II, whose main target is the study of self-intersection locus. Depending on the kind of information wearied by the regions, several versions of the algorithm can be used, and each provides specific ways for handling self-intersection and intersections of regions. Each version shares a common way to encode the regions, the neighborhood information, and a bounding volume hierarchy upon. The algorithm is often faster than the generation of sample points, so that it is interesting when handling surfaces defined procedurally. An article has been submitted to the ACM Symposium on Solid Modeling and Applications 2004.

We proposed a sweeping plane algorithm for computing an arrangement of
quadrics in 3D vertical decomposition of the
arrangement, where cells are of constant size, and it is
output-sensitive in terms of the complexity of this decomposition.

Though this vertical decomposition has a much lower complexity than
the classical Collins' cylindrical decomposition, it is still
quite high. We are working on partial decompositions that
would have a better complexity.

We focused our study on geometric predicates needed by the algorithm (see Section ). In fact, the algorithm requires the manipulation of algebraic numbers of degree up to 256, whereas the description of the non-decomposed arrangement involves only numbers of degree 64. It is not clear whether an algorithm can fill this gap, but it seems that a partial decomposition may ameliorate the problem. Currently, we are working in this direction.

We consider the effective classification of Steiner surfaces, that
are parameterizations of degree 2 from

In a recent work

We are interested in applying techniques from differential geometry
in the study of geometric objects, such as curves and surfaces used in
Computer Aided Geometric Design, and performing robust algorithms for dealing with
singularities. In

We study a major step in factorization algorithms
that proceed via approximations (see for example

Collaborations with Mark Giesbrecht and George Labahn.

We develop effective solutions for the problem of sparse interpolation of a multivariate black box polynomial in floating point arithmetic. We also inquire into some techniques designed for exact computations in the setting of floating point, such as early termination and post check. Based on our observation of the similarity between the exact Ben-Or/Tiwari interpolation and the classical Prony's method for interpolating a sum of exponential functions, we exploit the generalized eigenvalue reformulation of Prony's method. Our methods are implemented in Maple, and we now investigate the sensitivity of our solutions in collaboration with Mark Giesbrecht and George Labahn.

As an application, through black box polynomial interpolations, we may dramatically increase the efficiency of computing the determinant of the non-singular maximal principal minor in a Bezoutian matrix that is for a set of multivariate polynomials. We further investigate the exploitation of the sparsity in the decomposition of a Bezoutian.

Our work on certified predicates started a few years ago. We first
studied the case of predicates for arrangements of circular
arcs

Deducing the geometry or topology of a curve or surface from its algebraic definition is an important task we have to perform efficiently when dealing with geometric problems such as arrangement computations, structure analysis, ..., etc. In order to avoid incoherent computation, this task often requires certificated methods.

In relation with the ECG and GAIA project, we consider first
the problem of computing the topology of a curve defined as the
intersection of two surfaces. The method that
we developed computes the critical values for the projection along a plane
direction, regular points in-between these critical points and connect the
branches according to the regularity of the end points. The approach
depends on univariate and multivariate polynomial solvers. A first prototype
has been implemented based on the synaps library. See

The problem of meshing a surface given by an implicit equation has also been
studied. We develop a new method, which allows us to guarantee the topology
in the smooth part and give a model of singularity elsewhere, while
producing a number of linear pieces related to the geometry of the
object. We use Bernstein bases to represent the function in a box and
subdivide this representation according to a generalization of Descartes
rule, until the problem in each box boils down to the case where either the
implicit object is proven to be homotopic to its linear approximation in the
cell or the size of the cell is smaller than

The work on the normalization of the language C++ has been pursued. Two
contributions in the form of technical reports to the ISO/CEI JTC1/SC22/WG21
committee, have been proposed. The first one is on template aliasing in C++

The visit of B. Stroustrup in summer was the occasion of further work
on the parameterisation of code in C++, leading in particular to the
definition of the ``concept'' mechanism and to a proposition on template
argument checking

See the CGAL-like Curved
Kernel web site
axel web site

The theoretical work on certified predicates (see
Section ) allowed us to work on an efficient
implementation and to improve last year's first preliminary
code

as

: a kernel with circular arcs and all the predicates needed to interface with the CGAL arrangement algorithms; a concept RootOf_2 for algebraic numbers of degree 2 and some models of this concept; a demo showing the computation of an arrangement of circular arcs, using the CGAL arrangements and this kernel. in A. Kakargias' internship report

: some experimental studies, comparing several number types to support the predicates. as

: a prototype implementation of a complete set of comparisons between algebraic numbers of degree from 1 to 4; experimental results and applications to geometric predicates.

See the synaps web site

The synaps library benefits for the work of N. Chleq (DREAM engineer) for
the configuration and documentation process. The tools and techniques used
now to install the package follow more standard rules, relying on the
autoconf, automake tools. An automatic test suite as been settled down to
reinforce the robustness of the implementation. Documentation is now
generated automatically, also by classical tools. Besides, the existing
implementation has been extended and improved for applications in
geometric problems.

See the cgal web site:

The package ``3D Triangulation'' of CGAL is maintained in
collaboration with Sylvain Pion (Geometrica team). It was released in
CGAL 3.0 in November, 2003.

In

We also investigated other applications to CAGD through the
European project ECG. In

The self-intersection algorithm developed by Jean-Pascal Pavone has been integrated in the ThinkDesign software of the Think3 company as a COM component.

Collaboration with Opale.

In the context of the COLORS Forme, for simulation and optimization purposes,
we consider the question of compact encoding of wing shapes.
Thanks to the basic properties of control, degree elevation, ..., the
representation of parametric curves in the Bernstein basis offers very powerful
tools for geometric optimization.
Based on B-spline representation, we extended this approach
and obtained more flexibility and a smoother
behavior in numerical simulation. This has been experimented during the
internship of M. Celikbas, on an academic case, which is a test step in the
global project on wing shape optimization. See

In molecular biology, biochemistry and structural biology, knowing the structures of proteins is the key to learn their functions. Understanding the functions of proteins is one of the main goals in numerous ongoing genomic projects. In these projects, more and more protein structures are elucidated through two major experiments: X-ray crystallography and Nuclear Magnetic Resonance (NMR). Even if the proportion of solved structures using NMR is only 15% today, this technique has made a lot of improvements during the past few years.

The usage of NMR is mainly hampered by the difficulty in analyzing the data collected from these experiments. NMR gives two types of information: correlations between atom nuclei through chemical bounds and correlations between atom nuclei through space. The second type of information is critical in solving the 3D structure of proteins. As described by Crippen and Havel in 1988, distance geometry is the mathematical basis for a geometric theory of molecular conformation. As the natural approach to solve this problem, distance geometry gives a clear representation of what a protein structure is and what are the relations between atom nuclei distances.

In the GALAAD project, a set of programs have been developed to generate and analyze these data automatically in testing and developing various distance geometry tools for this problem. Based on preliminary results, although with basic distance geometry tools we are not able to solve the problem with such noisy and missing data provided by NMR experiments, we are not that far. Further research is required for improvements and to correctly tackle this problem.

The project ``Géométrie de Distances et Génomique Structurale à haut débit par RMN'' aims at deducing the conformation of molecules in Euclidean space, from partial information on the relative distances between atoms or the radicals (group of atoms). This information is obtained by experiments called NMR or Nuclear Magnetic Resonance. During his post-doctorate, partly in our team, Antoine Marin has been working on typing techniques for the amino acids starting from their chemical shifts, regrouping of the residues observed on the spectra in the sequence of protein, and rebuilding of the 3D-structure starting by matrix methods from information of distance.

See
ecg project web site

INRIA (Geometrica and Galaad) is
coordinating the European project:

- Acronym: ecg, number IST-2000-26473

- Title: Effective Computational Geometry for Curves and Surfaces.

- Specific Programme: IST

- RTD (FET Open)

- Start date: may 1st 2001 - Duration: 3 years

- Participation of Inria as coordinating site

- Other participants:

ETH Zürich (Switzerland),

Freie Universität Berlin (Germany),

Rijksuniversiteit Groningen (Netherlands),

MPI Saarbrücken (Germany),

Tel Aviv University (Israel)

- Abstract: Efficient handling of curved objects in Computational Geometry. Geometric algorithms for curves and surfaces, algebraic issues, robustness issues, approximation.

Monique Teillaud is the technical coordinator of the project within the Board and the
members, as well as the communication with the Project
Officer in Brussels.

She is also maintaining the public web site of the project, together with internal web sites with restricted access, and the mailing lists for internal communication (the members mailing list contains 66 addresses).

Steve Oudot (Geometrica team) is maintaining the php
scripts for submission of technical reports, originally written by Menelaos
Karavelas (Prisme team). He also wrote new scripts for
deliverables submission.

Olivier Ruatta settled a MySQL server that is used for the
database of technical reports.

See the GAIA II project
web site

In collaboration with the university of Nice UNSA, the Galaad team
is involved in the European project gaia:

- Acronyme : gaia II, number IST-2001-35512

- Title : Intersection algorithms for geometry based IT-applications using approximate algebraic methods

- Specific program of the project : IST

- Type of project : FET-Open

- Beginning date : 1st of july 2002 - During : 3 years

- Participation mode of INRIA : participant via the UNSA

- Partners list :

SINTEF Applied Mathematics, Norvegia

Johannes Kepler University, Austria

UNSA, France

Université de Cantabria, Spain

Think3 SPA, Italy and France

University of Oslo, Norvegia

- Abstract of the project : Detection and treatment of intersections and self-intersections, singularity analysis, classification, approximate algebraic geometry and applications to CAG.

Collaboration with A. Dickenstein and C. D'Andrea (Univ. of Buenos Aires).

Collaboration with the Mathematics Department of the University of Buenos Aires in Argentina, in the framework of an ECOS-Sud project for 3 years (2001-03). This project supports 2-week visits of researchers (this year M. Elkadi visited Buenos Aires) and longer stays of PhD students (this year, G. Cheze spent one month in Buenos Aires) or Postdocs. It is titled "Robust solution of algebraic systems and applications in computer-aided design" and concerns elimination theory, resultant matrices, but also newer research subjects related to applications in modeling and CAGD.

The year 2003, last year of the collaboration was marked by the organization of a Cimpa Graduate School at Buenos Aires, in July 2003, on Systems of Polynomial Equations. The organizers were A. Dickenstein (U. Buenos Aires) and I. Emiris (U. Athens, Greece). Among the lecturers, there were A. Galligo and B. Mourrain. The School was followed by the 1st latin-american workshop on "Systems of polynomial equations", in which spoke M. Elkadi and G. Cheze.

The Team of Geometric and Algebraic Algorithms at the National
University of Athens, Greece, has been associated with Galaad for a
period of 3 years (2003-05). See its web site

This bilateral collaboration is titled
CALAMATA (CALculs Algebriques, MATriciels et Applications). The Greek
team (

The focus of this project is the solution of polynomial systems by matrix methods. Our approach leads naturally to problems in structured and sparse matrices. Real root isolation, either of one univariate polynomial or of a polynomial system, is of special interest, especially in applications in geometric modeling, CAGD or computational geometry. We are interested in computational geometry, actually, in what concerns curves and surfaces. The framework of this work is the European project ECG.

In 2003, 6 members of the Greek team visited Inria, either for week-long visits or for longer visits (from one to 3 months). Three Inria researchers visited Athens for one week. We also mention the participation of members of both teams in international or national conferences : the meeting of Computational geometry at Dagstuhl (Germany), the Cimpa Graduate School in Argentina, and the French conference on Computational geometry.

Agnes Szanto works at the departement of mathematics of the North Carolina State University.

This is a project on overdetermined algebraic systems funded by a U.S. grant of one year. A visit of O. Ruatta to the North Carolina State University is planned.

The objective of this investigation is to develop and implement highly efficient algorithms for the solution of over-constrained polynomial systems with finitely many, possibly multiple roots over the complex numbers, when the input is given with inexact coefficients. We refer to such problems as inexact degenerate systems. Both ``resultant based'' and analytic iterative methods are considered to tackle this problem using the large number of already existing works. The researchers will address the problem of the definition of ``nearly consistent'' systems, with computational methods generalizing the S.V.D. of the linear case. The complexity is one of the central issues of this research since we want efficient methods. Implementations will be available via Java applets on the internet.

We continued to organize a (almost) weekly seminar called ``Table
Ronde''. The list of talks is available at

Special issue of ``Theoretical Computer Science'' on Algebraic Numeric algorithms. Guest editors: I. Emiris, B. Mourrain et V.Y. Pan.

Monique Teillaud is a member of the

Editorial Committee.cgal

B. Mourrain is in charge, with Thierry Vieville, of the ``Formation par la recherche'' at INRIA Sophia-Antipolis,

M. Teillaud and J.P. Técourt are member of the ``Comité de centre'' at INRIA Sophia-Antipolis.

More information on our activities can be found at

L. Busé attended the international conference ACA 2003 hosted in North Carolina State University at Raleigh, USA. He gave a talk based on both works

. G. Chèze attended the CIMPA Graduate School at Buenos Aires in July 2003 on ``Systems of Polynomial Equations.'' He gave a talk at the workshop on ``Systems of Polynomial Equations.'' In September, he went to Strasbourg and Nancy for one week in order to work with M. Mignotte, P. Zimmermann and G. Hanrot.

M. Elkadi attended CIMPA Graduate School at Buenos Aires in July 2003 on ``Systems of Polynomial Equations.'' He gave a talk at the workshop on ``Systems of Polynomial Equations.'' He also attended the meeting ``Effectivity Problems V'' in Diamante (Italy) and gave a talk.

G. Dos Reis gave a talk at the Journée de Calcul Formel (Luminy, january) and presented a paper at the MEGA conference (june, Kaiserslautern).

A. Galligo gave a talk at the MEGA Conference (Kaiserslautern, Germany, June). He also gave a talk at the Journées de Calcul Formel (Marrakech, Maroc, April). He taught a course at the CIMPA Graduate School at Buenos Aires in July on ``Absolute polynomial factorization.''

B. Mourrain was in charge of a session at the Journées Nationales de Calcul Formel (Luminy, January), and gave talks at the Dagsthul workshop on Verification and Constructive Algebra (Dagsthul, Germany, January), the Journées de Calcul Formel (Marrakech, Maroc, April), ECG workshop (Berlin, Germany, June), MEGA Conference (Kaiserslautern, Germany, June), the Rencontres Mondiales des Logiciels Libres (Metz, July), at the Surfaces Symposium (Treilles, July), COMPASS conference (Kefermarkt, Austria, September), SIAM conference on Geometric Design (Seattle, USA, November). He also taught a course at the CIMPA Graduate School in Buenos Aires in July on ``Systems of Polynomial Equations,'' and was invited to give a talk at the 5th Real Number and Computer conference (Lyon, September) and the workshop of the AS Constraint (Strasbourg, December).

J.P. Pavone attended the COMPASS conference (Kefermarkt, Austria, September).

O. Ruatta was an invited lecturer for Real Algebraic and Analytic Geometry summer school (June 2003 in Rennes, France) and attended the R.A.A.G. meeting for young researchers (Septembre 2003 in Coma-Ruga, Spain).

Monique Teillaud gave the following talks in conferences and workshops: 14th ACM-SIAM Sympos. Discrete Algorithms (SODA), Baltimore (January),

Perturbations and Vertex Removal in a 3D Delaunay Triangulation''; Journées Nationales de Calcul Formel, Luminy (January),Étude de prédicats géométriques; Dagstuhl Seminar 03121, Computational Geometry, invited talk (March),Arrangement of Quadrics in 3D; ECG Workshop on Applications Involving Geometric Algorithms with Curved Objects, Saarbruücken (September),Sweeping an arrangement of quadrics. She also attended: the ECG General Workshop, Berlin (June), the Journées de Géométrie Algorithmique, Giens (September), Workshop on Geometric Compression, Sophia Antipolis (November).Jean-Pierre Técourt attended the 19th European Workshop on Computational Geometry, March 24-26, University of Bonn, Germany.

G. Chèze: Practical sessions, Introductive course to C language, Algorithmic and Computational Sciences, UNSA, (78h). Mathematics Applied to Social Sciences, first year of the DEUG.

M. Elkadi: Courses in Algebra (Deug MI2) and differential calculus (Licence MASS).

A. Galligo : Course of mathematics and computer algebra in DEUG MASS1 at UNSA (192h).

B. Mourrain : DEA Algorithmique, Paris VI (10h). Maîtrise Math-Info, UNSA (40h).

J.P. Pavone : Computer Sciences at the IUT of Nice (96h).

O. Ruatta : Pratical sessions and exercices Algorithmic et programmation for the first year of the DEUG at UNSA (tronc commun 50h then Math-Info 20h). Exercices of Symbolic Computation for the DEUG MASS at UNSA (40h). Introduction to Unix-Linux for MST Geologie-Geophysique at UNSA (15h). Languages and tools for the web for MST Geologie-Geophysique at UNSA (18 h).

Jean-Pierre Técourt : Computer Sciences (Java) in Deug Tronc Commun, UNSA (64h).

Monique Teillaud : Geometric Computation, ESSI (École Supérieure en Sciences Informatiques) (20h). Computational Geometry, UNSA (Université de Nice Sophia Antipolis), Maîtrise d'informatique (8h). Geometric Computation, ISIA (Institut Supérieur d'Informatique et d'Automatique) (11h). Computational Geometry, École doctorale I2S (Module Imagerie), Montpellier (3h).

Guillaume Chèze, Factorisation de polynômes à plusieurs variables, UNSA.

Thi Ha LE, Classification and intersections of some parametrized surfaces and applications to CAGD (UNSA).

Jean-Pascal Pavone, Étude de la géométrie des surfaces paramétrées utilisées en CAO, UNSA.

Jean-Pierre Técourt, Algorithmique des courbes et surfaces implicites, UNSA.

See the

Mehmet Celikbas (INSA Toulouse),

Surface de Bezier et B-spline pour l'optimisation géométrique, from June 20th to September 20th.Lionel Deschamps, (DEA SIC Image Vision -ESSI),

Maillage de surfaces implicites, from April 1st to September 30th.Athanasios Kakargias (Athens National University),

Experimenting with the Curved Kernel, from June 29th to September 28th.Abder Labrouzy (ISIMA Clermont-Ferrand)

Intersection de courbes et surfaces algébriques, from April 7th to September 5th.Nicolas Martin (ESSI, 2nd year),

Arrangements of evolving circles in the plane, from June 30th to September 22th.