Advanced School on Specific Algebraic Manipulators (SAM07)

Barcelona, September 12-15, 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 multi-core 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: Henon-Heiles 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 quasi-periodic skew products
  
  
• A. Jorba
  Construction of algebraic manipulators in C/C++. Use of extended precision and intervalar arithmetics. Applications to Celestial Mechanics.


• R. Ramírez-Ros
  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 three-body problems and to saddle-parabolic points.
  

Slides and links
Rafael Ramírez-Ros: http://www.ma1.upc.edu/~rafael/research.html
Pau Roldán: http://www.ma1.upc.edu/~roldan/2007-09-14_SAM/
Laskar-Gastineau: 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
09:00-10:00	Gastineau	Gastineau	Gastineau	Gastineau
10:00-11:00	Gastineau	Giorgilli-Locatelli	Gastineau	Ramírez
11:00-11:30	Coffee break	Coffee break	Coffee break	Coffee break
11:30-12:30	Giorgilli-Locatelli	Giorgilli-Locatelli	Giorgilli-Locatelli	Roldán
12:30-13:30	Giorgilli-Locatelli	Simó	Giorgilli-Locatelli	Simó
13:30-15:30
15:30-17:00	Jorba	Jorba	15:30-16:30 Ramírez 16:30-17:00 Coffee break 17:00-18:00 Roldán 18:00-19:00 Simó
17:00-17:30	Coffee break	Coffee break
17:30-19:00	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 i-MATH, and 100€ for other researchers. Details will be e-mailed 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 e-mail 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 i-MATH, other researchers belonging to these teams, other spanish participants, non-spanish participants.

The deadline to ask for a grant is July 13. Granted students will be notified by e-mail on July 20.

Accomodation

Contact to: investigadorsresa.es, www.residencia-investigadors.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. For more information, write them to the address sam07 at imub.ub.es (this address is not clickable, you should type it in your e-mail program; this has been done to minimize the amount of spam we receive).

Support

The course is supported by i-MATH (Consolider project from the Spanish Minsitry of Education and Science), IMUB and the research teams on Dynamical Systems of the UB.