The aim of the lecture is to obtain a common basis for the knowledge in local algebra. This includes Weierstraß preparation theorem and its applications (implicit function theorem, Hensel's Lemma, Newton's Lemma,Noether normalization). An application will be the dimension theory of germs of analytic spaces.

The basics of analytic geometry will be explained. This includes germs of analytic spaces, Nullstellensatz and the local parametrization theorem. We will also have the normalization of germs of analytic spaces including Serre's R1 and S2 Criterion. An outlook to Cohen-Macaulay spaces will also be given.

The lectures give an introduction to singularity theory. The basics (finite determinacy of hypersurface singularities, simple singularities, Morse Lemma) will be explained. On top ot this we will have an excursion to deformation theory. Grauert's theorem about the existence of a semi-universal deformation is the highlight of the course. We will see how to use Singular to compute it. Some research problems will be discussed at the end.

The computer algebra system SINGULAR has had its 25th birthday. The lectures will explain the interplay between computer algebra and its applications and illustrate the development during the last 25 years. We will see that needs in algebraic geometry and singularity theory always forced the development of the system. Deep theorems in group theory could be proved using the system. New applications to electrical engineering will be explained. We will also see applications to coding theory, partial differential equations. SINGULAR can also solve a Sudoku. Problems for further research will be discussed.

The course will give an overview on known approximation theorems, i.e. given a formal solution of a set of analytic equation then there are analytic solutions converging to the formal ones in the m-adic topology. The approaches of M. Artin, H. Grauert, H. Hironaka and D. Popescu will be explained. Several applications from different areas of mathematics will be given.

As global features of standard bases techniques there are applications in classical algebraic geometry. To this end elimination techniques for the construction of applications like Veronese embeddings, Segre products, joins and secant varieties and tangent varieties are considered. As a practical application there are SINGULAR proceedures for several of these geometric constructions. The main focus of this part makes participants familiar with basic constructions in classical algebraic geometry. A focus for applications are varieties of low degrees and their classification.

Computational methods in local analytic geometry will be explained. The basis is the concept of standard bases. We will discuss how to compute Weierstraß division approximately, Noether normalization etc. Elimination of variables will play an important role. We will discuss the following question: What do we compute geometrically when we eliminate? We will compare local versus global properties and see for example how the degree of a projective variety and the multiplicity of a singular point are related and can be computed.

The concept of Gröbner bases will be generalized to local rings, especially formal and convergent power series rings. This includes Grauert's division theorem as a basis for the theory of standard bases. Charaterizations and properties of standard bases will be discussed. Applications (computation of the Hilbert-Samuel-Function, Computation of the dimension of germs of analytic spaces) will be given.

CIMPA (International Centre for Pure and Applied Mathematics) is a non-profit international organization established in Nice (France) in 1978. Its aim is to promote international cooperation in higher education and research in mathematics and related subjects, particularly computer science, for the benefit of developing countries. To visit its website please click here

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The main purpose of the School was to introduce the PhD students and young researchers to local analytic geometry, its methods and applications in mathematics and some other disciplines of science.

The School was organized at Abdus Salam School of Mathematical Sciences (ASSMS), Lahore-Pakistan. A total of 49 PhD students and young researchers participated in the School. Among these, 14 were females. Most of female participants were from Pakistan. The participating foreign students and researchers came from Indonesia, Iran, Nepal, Philippines, Uzbekistan and Vietnam.

For detailed Report, please click the link.