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Viviane Pons

Maˆıtresse de Conf´erences, Universit´e Paris-Sud viviane.pons@lri.fr – @PyViv

Exp´erimentation math´ematique

et Combinatoire avec Sage

(2)

Combinatoire

Combinatoire

Etude des ensembles finis ou d´´ enombrables de structures discr`etes : graphes, arbres, permutations, mots binaires...

Diff´erentes approches, diff´erentes questions

I Combinatoire ´enum´erative

I Combinatoire bijective

I Combinatoire alg´ebrique

I Combinatoire analytique

(3)

Combinatoire

Combinatoire

Etude des ensembles finis ou d´´ enombrables de structures discr`etes : graphes, arbres, permutations, mots binaires...

Diff´erentes approches, diff´erentes questions

I Combinatoire ´enum´erative

I Combinatoire bijective

I Combinatoire alg´ebrique

I Combinatoire analytique

Viviane Pons (U-PSud) Exp´erimentation math´ematique January 22, 2018 2 / 18

(4)

Combinatoire

Exemple : arbres binaires

Questions:

I Combien d’arbres binaires `an noeuds?

Pour 11 : 58786

I A quoi ressemble un arbre binaire al´eatoire ?

I Quels autres objets

combinatoires sont li´es aux arbres binaires ?

I Quelles structures alg´ebriques ?

(5)

Combinatoire

Exemple : arbres binaires

Questions:

I Combien d’arbres binaires `an noeuds?

Pour 11 : 58786

I A quoi ressemble un arbre binaire al´eatoire ?

I Quels autres objets

combinatoires sont li´es aux arbres binaires ?

I Quelles structures alg´ebriques ?

Viviane Pons (U-PSud) Exp´erimentation math´ematique January 22, 2018 3 / 18

(6)

Combinatoire

Exemple : arbres binaires

Questions:

I Combien d’arbres binaires `an noeuds?

Pour 11 : 58786

I A quoi ressemble un arbre binaire al´eatoire ?

I Quels autres objets

combinatoires sont li´es aux arbres binaires ?

I Quelles structures alg´ebriques ?

(7)

Combinatoire

Exemple : arbres binaires

Questions:

I Combien d’arbres binaires `an noeuds? Pour 11 : 58786

I A quoi ressemble un arbre binaire al´eatoire ?

I Quels autres objets

combinatoires sont li´es aux arbres binaires ?

I Quelles structures alg´ebriques ?

Viviane Pons (U-PSud) Exp´erimentation math´ematique January 22, 2018 3 / 18

(8)

Combinatoire

Exemple : arbres binaires

Questions:

I Combien d’arbres binaires `an noeuds? Pour 11 : 58786

I A quoi ressemble un arbre binaire al´eatoire ?

I Quels autres objets

combinatoires sont li´es aux arbres binaires ?

I Quelles structures alg´ebriques ?

(9)

Combinatoire

Exemple : arbres binaires

Questions:

I Combien d’arbres binaires `an noeuds? Pour 11 : 58786

I A quoi ressemble un arbre binaire al´eatoire ?

I Quels autres objets

combinatoires sont li´es aux arbres binaires ?

I Quelles structures alg´ebriques ?

Viviane Pons (U-PSud) Exp´erimentation math´ematique January 22, 2018 3 / 18

(10)

Combinatoire

Exemple : arbres binaires

Questions:

I Combien d’arbres binaires `an noeuds? Pour 11 : 58786

I A quoi ressemble un arbre binaire al´eatoire ?

I Quels autres objets

combinatoires sont li´es aux arbres binaires ?

I Quelles structures alg´ebriques ?

(11)

Combinatoire

Quelles motivations ?

I algorithmiques : analyse d’algorithmes, g´en´eration al´eatoire

I alg´ebriques : g´eom´etrie alg´ebrique, th´eorie des repr´esentations

Quelles m´ethodes ?

Viviane Pons (U-PSud) Exp´erimentation math´ematique January 22, 2018 4 / 18

(12)

Combinatoire

Quelles motivations ?

I algorithmiques : analyse d’algorithmes, g´en´eration al´eatoire

I alg´ebriques : g´eom´etrie alg´ebrique, th´eorie des repr´esentations

Quelles m´ethodes ?

(13)

Combinatoire

Exploration `a la main !

Viviane Pons (U-PSud) Exp´erimentation math´ematique January 22, 2018 5 / 18

(14)

Combinatoire

Listons les arbres binaires...

(15)

Combinatoire

Exploration par ordinateur

Viviane Pons (U-PSud) Exp´erimentation math´ematique January 22, 2018 7 / 18

(16)

Combinatoire

Exploration par ordinateur

(17)

SageMath

SageMath is a free open source mathematics software

I Created in 2005.

I http://www.sagemath.org/

I Mission: Creating a viable free open source alternative to Magma, Maple, Mathematica and Matlab.

Viviane Pons (U-PSud) Exp´erimentation math´ematique January 22, 2018 8 / 18

(18)

SageMath Source and language

I the main language of Sage is python(but there are many other source languages: cython, C, C++, fortran)

I the source is distributed under the GPL licence.

(19)

SageMath Sage and libraries

One of the original purpose of Sage was to put together the many existent open source mathematics software programs: Atlas, GAP, GMP, Linbox, Maxima, MPFR, PARI/GP, NetworkX, NTL, Numpy/Scipy, Singular, Symmetrica,...

Sage is all-inclusive: it installs all those libraries and gives you a common python-based interface to work on them.

On top of it is the python / cython Sage library it-self.

Viviane Pons (U-PSud) Exp´erimentation math´ematique January 22, 2018 10 / 18

(20)

SageMath Development model

Development model

I Sage is developed by researchers for researchers: the original philosophy is to develop what youneed for your research andshare it with the community.

I The first full-time Sage developer has been recruited in January 2016 at Paris-Sud university by the OpenDreamKit project.

I The code is maintained through a git repo with a system of tickets and peer-review for new contributions:

http://trac.sagemath.org/

(21)

SageMath Development model

The Sage community

268 developers in 182 different places.

Mailing lists, Sage days

Viviane Pons (U-PSud) Exp´erimentation math´ematique January 22, 2018 12 / 18

(22)

OpenDreamKit

Funding?

I Runs without permanent funding

I Benefit from local funding sources

I Since 2015: TheOpenDreamKit project

(23)

OpenDreamKit

OpenDreamKit 2015 – 2019

Open Digital Research Environment Toolkit for the Advancement of Mathematics

A project funded by the Horizon 2020 European Research Infrastructures Work Programme.

I A budget of 7.6 millions run by 18 partner institutions in 7 countries

I Provides funding for many Open source mathematical software and projects: SageMath, GAP, PARI, Singular, Jupyter, LMFDB...

I Pays for full time developers, events, infrastructures

Viviane Pons (U-PSud) Exp´erimentation math´ematique January 22, 2018 14 / 18

(24)

OpenDreamKit

OpenDreamKit 2015 – 2019

Open Digital Research Environment Toolkit for the Advancement of Mathematics

A project funded by the Horizon 2020 European Research Infrastructures Work Programme.

I A budget of 7.6 millions run by 18 partner institutions in 7 countries

I Provides funding for many Open source mathematical software and projects: SageMath, GAP, PARI, Singular, Jupyter, LMFDB...

I Pays for full time developers, events, infrastructures

(25)

OpenDreamKit

OpenDreamKit 2015 – 2019

Open Digital Research Environment Toolkit for the Advancement of Mathematics

A project funded by the Horizon 2020 European Research Infrastructures Work Programme.

I A budget of 7.6 millions run by 18 partner institutions in 7 countries

I Provides funding for many Open source mathematical software and projects: SageMath, GAP, PARI, Singular, Jupyter, LMFDB...

I Pays for full time developers, events, infrastructures

Viviane Pons (U-PSud) Exp´erimentation math´ematique January 22, 2018 14 / 18

(26)

OpenDreamKit

OpenDreamKit 2015 – 2019

Open Digital Research Environment Toolkit for the Advancement of Mathematics

A project funded by the Horizon 2020 European Research Infrastructures Work Programme.

I A budget of 7.6 millions run by 18 partner institutions in 7 countries

I Provides funding for many Open source mathematical software and projects: SageMath, GAP, PARI, Singular, Jupyter, LMFDB...

I Pays for full time developers, events, infrastructures

(27)

OpenDreamKit

Follow us...

I http://opendreamkit.org/

I Twitter: @OpenDreamKit

Viviane Pons (U-PSud) Exp´erimentation math´ematique January 22, 2018 15 / 18

(28)

OpenDreamKit Usage

What can I do with Sage?

Number theory, combinatorics, graph theory,...

Most things: Calcul Math´ematique avec Sage

(29)

OpenDreamKit Usage

How can I use it?

I native install on Linux, Mac, and Windows

I online on CoCalc.com

Viviane Pons (U-PSud) Exp´erimentation math´ematique January 22, 2018 17 / 18

(30)

OpenDreamKit Usage

Demo !

Références

Télécharger maintenant ( PDF - 30 Page - 2.23 MB )

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