Sage Days 16: Barcelona, Spain  Computational Number Theory
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Sage Days 16 will take place on 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.
Official webpage (note: schedule below is more uptodate)
Projects
Mailing lists
Participants list (open): http://groups.google.com/group/sagedays16
Organizers list (closed): http://groups.google.com/group/sagedays16org
Pictures
Schedule
All video is here. I made this video by reencoding HD video to "iPhone" video using Handbrake. Each video file is at most 200mb, and will play fine with VLC (and on the iPhone, of course). See also Sage on Blip TV for all these videos in an easy to view format.
Sunday, June 21 

19:00 
Meet informally in the lobby of the Resedentia 

21:00 
From the Resedentia, go to dinner 

Monday, June 22 

9:00 
Meet with Jordi Quer at the Residencia lobby, take the train together to CRM 

10:3011:30 
William Stein 
'''Sage: Unifying Mathematical Software''', video part 1, video part 2 
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''' video part 1, video part 2 
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''' video part 1, video part 2 
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 video part 1, video part 2 

FME 

18:45 
Leave from Residencia to UPC 

19:00 
Coding Sprint Organization at UPC video introductions 

Tuesday, June 23 

FME 

10:3011:30 
Jordi Guàrdia 
'''New ideas for computing integral bases''' video part 1, video part 2 
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 into 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''' video part 1 video part 2 
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 
Clément Pernet & Majid Khonji 
'''Computing exactly with unsafe resources: fault tolerant exact linear algebra and cloud computing''' video part 1 video part 2 
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 represents nowadays the best perspectives to access large and cheap computing power, but based on unreliable resources. Fault tolerant techniques are therefore developed 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 development of algorithm based fault tolerant protocols. In particular, the Chinese Remaindering Algorithm, offering an embarrassingly easy parallelization, can be adapted to work as an error correcting code and tolerate errors. We will present and 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''' video part 1 video part 2 
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''' video part 1 video part 2 
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''' video part 1 video part 2 
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 / Status Report 

Thursday, June 25 

FME 

10:3011:30 
David Loeffler 
'''Padic modular forms in Sage''' video part 1, video part 2 
I will give a quick introduction to padic modular forms, which are a generalisation of classical modular forms. I will first give a quick introduction to the theory, and then describe a few algorithms that can be used to compute them, and give an example of one of these which has been implemented in Sage since 3.4.1. Finally I will talk a little about some issues in inexact padic linear algebra that come up in the process. 

11:3012:00 
Coffee Break 

12:0013:00 
David Kohel 
ECHIDNA: Open source Magma extensions for Sage sws, PDF video part 1 video part 2 
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 video part 1 video part 2 
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 Organizer / Status Report 

Friday, June 26 

FME 

10:3011:30 
Rainer SchulzePillot 

Ordinary theta series count (in their Fourier coefficients) the number of ways in which the integral positive definite quadratic form Q(x)=x^t A x in m variables represents an integer n. Siegel theta series count instead the number of ways in which a given positive semidefinite g x g  matrix T can be represented as X^t A X with an integral m x gMatrix X. In the same way in which ordinary theta series give modular forms on the upper half plane for congruence subgroups of SL_2(Z) the Siegel theta series give modular forms on a g(g+1)/2dimensional space H_g for the symplectic group Sp_g(Z) and its congruence subgroups. Some computations for these have been done by Scharlau, Schiemann, and myself about 10 years ago; since then nothing much has happened apart from isolated computations of examples  maybe it's time to start another systematic attack on the subject. 

11:3012:00 
Coffee Break 

12:0013:00 
Emmanuel Thomé 

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 

<2> 14:3015:30 <2> Maite Aranes  '''Manin symbols over number fields (pdf)''' Sage worksheet, and video
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 // Status Report 

Saturday, June 26 

FME 

10:30 
Coding Sprint wrapup video part 1, video part 2 video part 3 
Organizers
Michael Abshoff, Martin Albrecht, John Cremona, Jordi Quer, William Stein, Enrique GonzálezJiménez, Joaquim Puig, Gonzalo Tornaría, Robert Miller.
Participants
 Michael Abshoff, Technische Universität Dortmund
 Martin Albrecht, University of London (Room C010 at Residencia)
 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 (Room C010 at Residencia)
 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 (Room C113 at Residencia)
 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