CIMPAICTPUNESCOMESRMICINNPAKISTAN
Research School 
Local
Analytic Geometry 
(04
 13 February, 2012) 

Organized
by: 

 Centre International de Mathématiques
Pures et Appliquées(CIMPA)
 Abdus Salam School of Mathematical
Sciences(ASSMS)
 National Center for Mathematics
(NCM), GC University Lahore


Supported
by: 

 The Abdus Salam International
Centre for Theoretical Physics (ICTP)
 Higher Education Commission (HEC),
Pakistan


LIST
OF COURSES 

Sr. 
Speaker 
Course 
Abstract 
1. 
Barbu Berceanu 
Basics of local algebra 
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. 

2. 
Viviana Ene 
Basics of analytic
geometry 
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 CohenMacaulay spaces will also
be given. 

3. 
Jürgen
Herzog 
Singularities 
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 semiuniversal 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. 

4. 
Gerhard Pfister 
SINGULAR and applications 
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. 

5. 
Dorin Popescu 
Approximation Theorems 
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
madic 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. 

6. 
Peter Schenzel 
Standard Bases II 
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.


7. 
Marius Vladoiu 
Algorithms and Computing
in local algebra 
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. 

8. 
Alexandru Dimca 
Standard Bases I 
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 HilbertSamuelFunction, Computation
of the dimension of germs of analytic
spaces) will be given. 

Bibliography 


G.M. Greuel
/ G. Pfister: A Singular Introduction
to Commutative Algebra, Springer
2007

T. de Jong
/ G. Pfister: Local Analytic Geometry,
Vieweg 2000

C. Lossen/
G. Pfister (Eds.): Singularities
and Computer Algebra. LMS Lecture
Notes 324 , Cambridge University
Press (2006)

S. Cojocaru/
G. Pfister/ V. Ufnarovski (Eds.):
Computational Commutative and
Noncommutative Algebraic Geometry,
NATO Science Series III, Computer
and Systems Sciences 196 (2005)


Schedule


Please
click the link to download the schedule
for the whole event. 


International
Centre for Pure and Applied Mathematics 

CIMPA
(International Centre for Pure and
Applied Mathematics) is a nonprofit
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


Registration
Forms 



CIMPA
Downloads 

Gerhard
Pfister 

A Singular
Introduction to Commutative Algebra 



Voor Jeannette.
Nils en Noah 



SINGULAR
and Applications talk at the CIMPA School
Lahore 2012 (Part I) 



SINGULAR
and Applications talk at the CIMPA School
Lahore 2012 (Part II) 


Dorin
Popescu 

Lecture
1. Artin Approximation 


A Short
Report on CIMPA 

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), LahorePakistan. 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.


