Advanced
School on
Specific Algebraic Manipulators
Barcelona, September
1215,
2007
Symbolic manipulation is a common tool in many domains of
science. For small and medium size computations different tools which
can be used to obtain the desired results are available. However, faced
to longer computations (e.g., dealing with a power series in 6
variables truncated to order 30, without any symmetry reduction, having
near 2 million coefficients) they are useless. This lead to some
scientists, mainly in Astronomy, to develop Algebraic Manipulators with
specific purposes, for a concrete family of problems at the very
beginning of the Computer Science. Since then a strong activity has
been developed and programs able to deal efficiently with millions of
terms have been produced. The first aim of the course is to present the
basic concepts: data structure, basic and higher level operations, to
use fully literal,
rational or floating point (including multiple precision) coefficients
and how to use them to be able to solve a variety of problems. The
second aim is to encourage participants to write their own SAM to solve
the problems in which they are interested, independently of the source
of these problems. Problems in algebraic geometry have a lot in common
with the analysis of singularities done in the study of differential
equations and discrete transformations. The impact of new processor
architectures and the possibility of distributed tasks will be also
discussed.
The course will
take place at Room B1, located at
the ground level of the Court of Sciences (Pati de Ciencies), in
the right part of the historical building of the Universitat
de Barcelona (the address is at the end of this page). We will provide
wifi access for the participants.
How to arrive
Lecturers
M. Gastineau (IMCCE, Observatoire de Paris), A. Giorgilli
(U. di Milano), À. Jorba (U. Barcelona), À. Haro (U.
Barcelona),
U. Locatelli (U. Roma 2), R. Ramírez (U. Politècnica
Catalunya),
P. Roldán (U. Politècnica Catalunya), C. Simó (U.
Barcelona)
Tentative contents of the lectures
 M. Gastineau
Storage of multivariate polynomials to speed up computations and reduce
memory footprint.
Algorithms for full and truncated multiplication. Impact of the
ordering.
The substitution problem and related problems. Some applications.
Advantages of multicore and multiprocessor, use of distributed
computing, memory management.
 A. Giorgilli  U.
Locatelli
Construction of first integrals and normal forms. Lie transforms.
Representation of polynomials, algebra on polynomials, solution of the
homological equation.
Applications: HenonHeiles problem and triangular libration points.
Exponential stability estimates.
Representation of trigonometric polynomials. The case of many zero
coefficients. Constructive expansions and analytical estimates for the
Kolmogorov normal form.
Applications: The forced pendulum. The general problem of three bodies.
Expansion of the perturbing function. Construction of KAM tori.
Obtaining rigorous results: interval arithmetics, iteration of the
estimates, matching of the estimates produced by computer with a
statement of the KAM theorem. See this for some material related to the lectures.
 A. Haro
Automatic differentiation tools for manipulation of Formal series.
Applications:
 computation of invariant KAM tori and its normal form for
area preserving maps
 computation of invariant tori and its
invariant manifolds for quasiperiodic skew products
 A. Jorba
Construction of algebraic manipulators in C/C++. Use of extended
precision and intervalar arithmetics. Applications to Celestial
Mechanics.
 R. RamírezRos
Multiple precision computation of exponentially small splittings.
Examples.
 P. Roldán
Analysis of sequential algorithms to compute normal forms.
Parallelisation schemes: rough and fine granularity. Implementation and
performance.
 C. Simó
Computation of invariant manifolds in generic and degenerate cases. On
the Gevrey properties of the coefficients. Applications to restricted
threebody problems and to saddleparabolic points.
Slides and links
Rafael RamírezRos: http://www.ma1.upc.edu/~rafael/research.html
Pau Roldán: http://www.ma1.upc.edu/~roldan/20070914_SAM/
LaskarGastineau: TRIP: a general computer algebra system dedicated to celestial mechanics
Michael Gastineau: Multiplication of polynomials ,
Parallel computations and memory management ,
Storage of multivariate polynomials
Àngel Jorba: Two sessions
Tentative
schedule

Wednesday

Thursday

Friday

Saturday

09:0010:00

Gastineau

Gastineau

Gastineau

Gastineau

10:0011:00

Gastineau

GiorgilliLocatelli

Gastineau

Ramírez

11:0011:30

Coffee
break 
Coffee
break 
Coffee
break 
Coffee
break 
11:3012:30

GiorgilliLocatelli

GiorgilliLocatelli

GiorgilliLocatelli

Roldán

12:3013:30

GiorgilliLocatelli

Simó

GiorgilliLocatelli

Simó 
13:3015:30





15:3017:00

Jorba

Jorba

15:3016:30 Ramírez

16:3017:00
Coffee break

17:0018:00 Roldán

18:0019:00 Simó



17:0017:30

Coffee
break 
Coffee
break 

17:3019:00

Haro

Haro


Preregistration
To attend you must preregister first. To do it, click
here.
As the attendance is
limited, the preregistration will be closed when the course is full.
Registration
The registration fee is of 50€ for researchers belonging to the
Consolider project iMATH, and 100€ for other researchers. Details will
be emailed to preregistered participants.
Grants
There are some grants to help with the local expenses to young
students. If you are interested, mark the appropriate box when
preregistering, send your CV (in pdf format) to the address sam07imub.ub.es, and ask a
senior researcher to email a support letter to the same address,
including the phone of the supporter.
Note that finantial help will be granted with the following order of
priority: young researchers belonging to teams included in iMATH,
other researchers belonging to these teams, other spanish
participants, nonspanish participants.
The deadline to ask for a
grant is July 13. Granted students will be
notified by email on July 20.
Accomodation
Contact to: investigadorsresa.es, www.residenciainvestigadors.es,
www.resa.es
Make sure that the reservation is done well in advance. Otherwise it
can be full.
Organizers
The organizers are Àngel Jorba and Carles Simó, with
the collaboration of Ariadna Farrés and Montse Navarro. For more
information, write them to the address sam07 at imub.ub.es (this
address is
not clickable, you should type it in your email program; this has been
done to minimize the amount of spam we receive).
Support
The course is supported by iMATH (Consolider project from the Spanish
Minsitry of Education and Science), IMUB and the research teams on
Dynamical Systems of the UB.
IMUB 
Institut de
Matemàtica 
Gran Via de les Corts
Catalanes 585, 
08007 Barcelona, Spain 


Phone: (+34)
93
402 13 85 
Fax:
(+34)
93 403 59 63 

Last modified:
Aug 30, 2007