Sage Days 16: Barcelona, Spain  Computational Number Theory
Sage Days 16 will take place June 2227, 2009, the week after MEGA 2009. The event will be organised by the CRM (http://www.crm.cat) and the OSRM of the UPC (http://wwwfme.upc.edu/osrm/), and will take place at the FME, in the campus of the UPC, in Barcelona.
Mailing lists
Participants lists (open): http://groups.google.com/group/sagedays16
Organizers list (closed): http://groups.google.com/group/sagedays16org
Schedule
Monday, June 22 

10:3011:30 
William Stein 
Sage: open source mathematical software 
This will be an overview talk about Sage, which explains the history and motivation for the project, demos some key features of Sage, and discusses where we are going next. It will be accessible to people in all research areas and assumes no prior experience with Sage. 

11:3012:00 
Coffee Break 

12:0013:00 
Henri Cohen 
Experimental methods in number theory and analysis 
In this talk, I would like to give a number of examples of numerical experiments coming from number theory and analysis, mention the tools used to perform them, and show how they sometimes can lead to interesting and deep conjectures. 

14:3015:30 
Àngel Jorba 
Developing tailored software for specific problems 
We will discuss the advantages and inconveniences of developing software (in a general purpose language like C) for concrete problems. I will also mention the results of a pool done by the Spanish project "iMath" on the use of computational resources of the mathematical research groups in Spain. 

15:3016:00 
Coffee Break 

16:0017:00 
Round Table 

FME 

19:00 
Coding Sprint Organization 

Tuesday, June 23 

FME 

10:3011:30 
Jordi Guàrdia 
New ideas for computing integral bases 
The determination of the ring of integers of a number field is one of the main tasks of computational algebraic number theory. The use of higher Newton polygons provides a new insight in the problem, leading to a fast method to compute integral bases, discriminants and prime ideal factorization in number fields. 

11:3012:00 
Coffee Break 

12:0013:00 
William Stein 
How to use Sage to compute with Elliptic Curves 
I will explain how to use Sage to define elliptic curves over various fields, do arithmetic on them, and compute standard invariants. Then I'll talk about elliptic curves over finite fields, and how to count points and compute the group structure. Next, I'll talk about elliptic curves over number fields and Sage's implementation of Tate's algorithm. Finally, I'll discuss computing the invariants in the BSD conjecture for elliptic curves over QQ. 

13:0014:30 
Lunch 

14:3015:30 
Clement Pernet & Majid Khonji 
Computing exactly with unsafe ressources: fault tolerant exact linear algebra and cloud computing 
In several ways, challenges in computational mathematics (including computational number theory, graph theory, cryptanalysis, ...) involve large linear algebra computations over Z or Q. Distributed, peertopeer or Cloud computing represent nowadays the best perspectives to access to a large and cheap computing power, but based on unrealible ressources. Fault tolerant techniques are therefore developped in order to increase the confidence in the computations, or even to certify it. In the case of exact computations, the algebraic properties of the problems are well suited for the developpment of algorithm based fault tolerant protocols. In particular, the Chinese Remaindering Algorithm, offering an embarassingly easy parallelization, can be adapted to work as an error correcting code and tolerate errors. We will present an demonstrate these algorithms and protocols in the case of a distributed computation of the determinant of a matrix over Z. 

15:3016:30 
Martin Albrecht 
How to get started developing Sage 
In this talk, we will try to highlight a few interesting and relevant bits and pieces for getting into Sage development. We will give an overview of how Sage is structured and step through the Sage development process. The talk is meant to be fairly interactive with people asking questions etc. 

Free evening 
Sant Joan festivity 

Wednesday, June 24 

FME 

13:0014:00 
William Stein 
Modular forms and modular abelian varieties in Sage 
I will survey the capabilities in Sage for computing dimensions of modular forms spaces, congruence subgroups, modular symbols, modular forms, Brandt modules, overconvergent modular forms, halfintegral weight forms, and modular abelian varieties. I will discuss both what is in Sage, and what is missing. 

14:3015:30 
Christian Eder 
Faugere's F5 Algorithm: variants and implementation issues 
In this talk we shortly recall main properties of Gröbner bases used for their computations. After an introduction on Faugere's F5 Algorithm we examine its points of inefficiency, especially the reduction process, and present the variant F5C improving these. The benefits of this improvement are explained and represented in detail. Moreover some hints implementing F5's data structures are given and the positive effects of F5C on these are shown. In the end we give some insight into the implementation of F5's reduction process in an F4ish manner, i.e. using symbolic preprocessing. 

16:00 
Coding Sprint 

Thursday, June 25 

FME 

10:3011:30 
David Loeffler 
TBA 
TBA 

11:3012:00 
Coffee Break 

12:0013:00 
David Kohel 
ECHIDNA: Open source Magma extensions for Sage 
I will present the open source GPL repository of Magma code: 

13:0014:30 
Lunch 

14:3015:30 
Robert Miller 
Fast compiled graphs in Sage 
There will be a demonstration and advertisement of new developments in graph theory in Sage. In particular, compiled Sage graphs have finally reached the same level of functionality as NetworkX graphs, the slower Python implementation. 

16:00 
Coding Sprint 

Friday, June 26 

FME 

10:3011:30 
Gonzalo Tornaria 
TBA 
TBA 

11:3012:00 
Coffee Break 

12:0013:00 
Emmanuel Thomé 
Multiplication of binary polynomials 
Multiplying binary polynomials is an elementary operation which occurs as a basic primitive in several contexts, from computer algebra to coding theory and cryptography. We study here a variety of algorithms for this operation, with the intent of obtaining satisfactory speeds for a wide range of possible degrees. We look into "low level" aspects related to microprocessorspecific optimizations, and higher level algorithms such as of course the Karatsuba and ToomCook approaches, but also two different FFT algorithms. Several improvements are presented. We provide comparisons of the timings obtained with those of the NTL library. The software presented can, as of NTL 5.5, be hooked into NTL as an addon. 

13:0014:30 
Lunch 

14:3015:30 
Maite Aranes 
Manin symbols over number fields 
I will discuss results about cusps and Manin symbols over a number field K, which should be useful in the computation of spaces of cusp forms for GL(2, K) via modular symbols. I will also present ongoing work on implementations of both of these in Sage. 

16:00 
Coding Sprint 

Saturday, June 26 

FME 

10:30 
Coding Sprint wrapup 
Organizers
Michael Abshoff, Martin Albrecht, John Cremona, Jordi Quer, William Stein, Enrique GonzálezJiménez, Joaquim Puig, Gonzalo Tornaría.
Participants
 Michael Abshoff, Technische Universität Dortmund
 Martin Albrecht, University of London
 Maite Aranes, University of Warwick
 Tomasz Buchert, Adam Mickiewicz University
 Michal Bulant, Masaryk University
 Gabriel Cardona, Universitat de les Illes Balears
 Wouter Castryck, Leuven
 Henri Cohen, Bordeaux
 Francesc Creixell, UPC
 Christian Eder, TU Kaiserslautern
 Burcin Erocal, RISC, JKU  Linz
 Julio Fernández, UPC
 Imma Gálvez, UAB
 Enrique GonzálezJimenez, Universidad Autónoma de Madrid
 Josep González, UPC
 Jordi Guàrdia, UPC
 Xavier Guitart, UPC
 Amir Hashemi, Isfahan University of Technology (Iran)
 Nikolas Karalis, National Technical University of Athens
 Hamish IveyLaw, SydneyMarseille
 David Kohel, Institut de Mathématiques de Luminy
 Joan Carles Lario, UPC
 Offray Vladimir Luna Cárdenas, Javeriana (Colombia)
 David Loeffler, University of Cambridge
 Robert Miller, University of Washington
 Antonio Molina, Addlink Software Científico
 Enric Nart, UAB
 Sebastian Pancratz, University of Oxford
 Clement Pernet
 Joaquim Puig, UPC
 Jordi Quer, UPC
 Anna Río, UPC
 Víctor Rotger, UPC
 Bjarke Roune, University of Aarhus
 Utpal Sarkar, HP (+UPC)
 Diana Savin, Ovidius University (Romania)
 Rainer SchulzePillot, Universitaet des Saarlandes
 Mehmet Sengun, University of DuisburgEssen
 Jaap Spies, Holland
 William Stein, University of Washington
 Emmanuel Thome, INRIA Lorraine
 Andrew Tonks, London Metropolitan University
 Gonzalo Tornaría, Universidad de la República (Uruguay)
 Eulàlia Tramuns, UPC
 Montrserrat Vela, UPC
Preston Wake, McMaster
 Christian Wuthrich, University of Nottingham
 Brian Wyman, Univ of Michigan