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Local Analytic Geometry

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Report by Raza Choudary

Report by Michel Waldschmidt

A joint CIMPA-ICTP research school

Objectives :

Local analytic geometry is concerned with germs of zero sets of analytic functions, that is, the study of such sets in the neighbourhood of a point.

The school intends to introduce PhD students and young researchers to local analytic geometry and to enable them to receive the required knowledge for doing significant research in this area of mathematics.

In this school we plan to give introductions to topics related to local analytic geometry and computational algebra. A basis for the courses will be the book of Theo de Jong and Gerhard Pfister : ’Local Analytic Geometry’. The aim of the school is to give an overview of local methods in geometry and their applications in various fields of mathematical and non-mathematical sciences. We also want to give a course in which it is shown how the existing computer algebra systems can be used to handle non-trivial examples in the above mentioned theories and how the algorithms based on Gröbner bases are implemented in these systems.

Organizing committee :

- A.D. Raza Choudary (Abdus Salam School of Mathematical Sciences, Pakistan)
- Gerhard Pfister (University of Kaiserslautern, Germany)
- Alexandru Dimca (University of Nice, France)

Date and location :

February 4-13, Abdus Salam School of Mathematical Sciences Lahore – Pakistan

Scientific program :

A common basis for the knowledge in local algebra will be given. 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, the local parametrization theorem and the normalization of germs of analytic spaces including Serre’s R1 and S2 Criterion. Computational methods in local analytic geometry will be presented. The basis is the concept of standard bases, a generalization of Gröbner bases to local rings, especially to power series rings. This includes Grauert’s division theorem as a basis for the theory of standard bases. Charaterizations, properties and applications of standard bases will be discussed. An introduction to singularity theory is planned. The basics (finite determinacy of hypersurface singularities, simple singularities, Morse Lemma) will be explained. We 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. On top of this we will have an excursion to deformation theory. Grauert’s theorem about the existence of a semi-universal deformation is one highlight of the course.

  1. Barbu Berceanu : Basics of Local Algebra
  2. Alexandru Dimca : Standard Bases I
  3. Viviana Ene : Basics of Analytic Geometry
  4. Jürgen Herzog : Singularities
  5. Gerhard Pfister : SINGULAR and Applications
  6. Dorin Popescu : Approximation Theorems
  7. Peter Schenzel : Standard Bases
  8. Marius Vladoiu : Algorithms and Computing in Local Algebra

lecture notes

Deadline for registration :

December 7, 2011

Application procedure and Online registration only for applicants not from Pakistan.

Applicants from Pakistan must contact the local organizer : A.D. Raza Choudary

Voir en ligne : School’s local website