Project Description Sage (http://www.sagemath.org/) is a completely open source general purpose mathe- matical software system, which appeared under the leadership of co-PI Stein (University of Washington) and has developed explosively within the last ten years among research mathe- maticians. It is similar to Maple, MuPAD, Mathematica, magma, and (via the third-party packages Numpy and Scipy) Matlab, and is based on the popular Python programming lan- guage. Sage has gained strong momentum in the mathematics community far beyond its initial focus in number theory, in particular in the field of combinatorics. To date, Sage probably has the strongest support for combinatorics of all mathematical software systems. The additional features that we plan to implement as part of this project heavily rely on the fact that Sage has already a large hierarchy of abstract classes, in the form of categories and axioms. This makes it possible to implement morphisms and maps between various objects, which will be essential for our projects. In addition, with the development of SageMathCloud and thematic tutorials, Sage is becoming more and more appealing, accessible, and easier to use for teaching purposes. This project will work in close collaboration with OpenDreamKit (http://opendreamkit. org/), a recently funded Horizon 2020 European Research Infrastructure project, whose mission includes support for Sage by streamlining access, distribution, portability on a wide range of platforms, including High Performance Computers or cloud services, and to improve user interfaces for example through SageMathCloud. The total budget for OpenDreamKit is about 7.6 million Euros, which includes salary for an average of 11 full time developers for various open source software systems. Since this project will support Sage through new and improved user platforms, this will make our code highly visible in an interdisciplinary framework. 1 Results of Prior NSF Support We will report on grant NSF OCI-1147247 entitled \Collaborative Research: SI2-SSE: Sage- combinat: Developing and Sharing Open Source Software for Algebraic Combinatorics" 2012- 2015 with Daniel Bump, Gregg Musiker, Anne Schilling, and William Stein as PIs. Of course, numerous other NSF grants have supported the general development of Sage over the past 10 years (see http://sagemath.org/development-ack.html). Sage-Combinat is a subproject of Sage whose mission is \to improve Sage as an ex- tensible toolbox for computer exploration in (algebraic) combinatorics, and foster code sharing between researchers in this area". There is a long tradition of software packages for algebraic combinatorics. These have been crucial in the development of combinatorics since the 1960s. The originality of the Sage-Combinat project lies in successfully addressing the following si- multaneous objectives. It offers a wide variety of interoperable and extensible tools, integrated in general purpose mathematical software, as needed for daily computer exploration in algebraic combinatorics; it is developed by a community of researchers spread around the world and across 1 institutions; and it is open source and depends only on open source software. Developing Sage- Combinat has required both mathematical and computer science expertise. For example, great emphasis is placed on high-level programming techniques (object orientation and polymorphism, iterators, functional programming) to obtain concise, expressive, general, and easily maintained code. As part of this grant, we have co-organized and/or supported 8 Sage Days. Sage Days are usually one week workshops aimed to teach Sage to new users, introduce new developers to the development tools and walk them through the development process, and most of all collaboratively develop new features. Sage Days are extremely interactive with a lot of time devoted to design discussions and coding sprints. Most of the Sage Days that we organized had about 25-40 participants (depending on the topic), with the Sage Days in Paris having over 50 participants. • Sage Days 40: Algebraic Combinatorics, Institute for Mathematics and Its Applications (IMA), University of Minnesota, July 9{13, 2012 organized by Gregg Musiker, Franco Saliola, Anne Schilling, and Nicolas M. Thi´ery • Sage Days: Multiple Dirichlet Series, Combinatorics, and Representation Theory, ICERM, February 11{15, 2013 organized by Franco Saliola, Anne Schilling, and Nicolas M. Thi´ery • Sage Days 49: Free and Practical Software for (Algebraic) Combinatorics, June 17{21, 2013, Orsay (France) organized by Alejandro Morales, Anne Schilling, and Nicolas M. Thi´ery • Sage Days 54: Sage Developer Days, UC Davis, November 4{8, 2013 organized by Daniel Bump, Anne Schilling, and Travis Scrimshaw • Sage Days 60: Sage Days in Chennai (India), August 14{17, 2014 organized by Arvind Ayyer, Amritanshu Prasad, and S. Viswanath • Sage Days 64: Algebraic Combinatorics, UC Davis, March 17{20, 2015 organized by Daniel Bump, Anne Schilling, and Travis Scrimshaw • Sage Days 64.5: Cluster Algebras, University of Minnesota, June 1{5, 2015 organized by Gregg Musiker, Dylan Rupel, Salvatore Stella, and Christian Stump • Sage Days 65: Crystal bases and Hopf algebras, Loyola University Chicago, June 8{12, 2015 organized by Mark Albert, Aaron Lauve, and Peter Tingley In addition to the Sage Days, many new features were implemented as part of this project including rigged configurations, new models for crystal bases, puzzles (used to count Littlewood{ Richardson coefficients), various tableaux that arise in the combinatorics of the affine Grassman- nian, branching rules for Lie algebras, integrable representations of affine Lie algebras and the category framework. We would like to highlight in particular http://trac.sagemath.org/ticket/14102, which implements nonsymmetric Macdonald polynomials. As far as we are aware, there is no other mathematical software system which currently computes nonsymmetric Macdonald polynomials in this generality. This ticket was a truly collaborative effort carried out by many people at the ICERM program \Automorphic Forms, Combinatorial Representation Theory and Multiple Dirichlet Series" http://icerm.brown.edu/sp-s13/ in the Spring of 2013. Mathematical input 2 about the algorithms and definitions were given by Bogdan Ion, Siddhartha Sahi, Arun Ram, Dan Orr, and Mark Shimozono followed by several weeks of implementation of the infrastructure by Nicolas Thi´eryand Anne Schilling, and cross checking all details and testing the code with Mark Shimozono and Bogdan Ion. We would also like to highlight http://trac.sagemath.org/ticket/15805 which imple- ments methods for integrable representations of affine Lie algebras. These are important in mathematical physics and number theory. In 1990, Kass, Moody, Slansky and Patera published a two-volume work [21] on representations of affine Lie algebras, including tables of their rep- resentations and branching rules. The first volume reprises the basic theory, and the second volume contains the tables. In the introduction, they write: We present a vast quantity of numerical data in tabular form, this being the only source for such information. The computations are tedious and not particularly straightforward when it is necessary to carry them out individually. We hope the appearance of this book will spur interest in a field that has become, in barely 20 years, deeply rewarding and full of promise for the future. It would indeed be gratifying if these tables were to appear to the scientists of 2040 as obsolete as the dust-gathering compilations of transcendental functions appear for us today because of their availability on every pocket calculator. The patch was merged in Sage-6.7 and all computations in this book can be done now Sage. In general, a considerable effort was made to make the code accessible by documenting and explaining it in books and tutorials. Lam, Lapointe, Morse, Schilling, Shimozono and Zabrocki wrote a book on \k-Schur functions and affine Schubert calculus" which includes many examples on how to use k-Schur combinatorics in Sage. This book is freely available on the arXiv as well as on the Springer website. In addition, Thi´eryand his coauthors wrote a book \Calcul math´ematiqueavec Sage" (in French with partial translations to English) about Sage, see http://sagebook.gforge.inria.fr/. Bump included an appendix on Sage in the second edition of his text Lie groups. In the summer of 2012, Schilling and Zabrocki did a major overhaul of the symmetric function code which also added new functionality related to Hopf algebras. This was done in a 15,000 line patch and also includes a new tutorial on symmetric functions. As part of the class MAT146 on Algebraic Combinatorics at UC Davis, Schilling started writing a Sage tutorial for undergraduate students in relation to Richard Stanley's new book \Algebraic Combinatorics: walks, trees, tableaux and more", published by Springer in 2013, but also freely available online http://www-math.mit.edu/~rstan/algcomb/index. html. The thematic tutorial has been incorporated into Sage. Furthermore, the thematic tutorial Lie Methods and Related Combinatorics in Sage by Bump and Schilling has been kept up to date with documentation of new features. Here are some further highlights regarding the outcome during the first two years of the grant (see also the annual reports on this grant for more details): • A total of 287 patches written under the UC Davis side of this project were merged into main Sage. Many more patches are currently being developed. The patches can be viewed at http://trac.sagemath.org/. • During 2012 and 2013, Bump added major substantial speedups and enhancements to the Lie code in Sage (implemented in the WeylCharacterRing class) which allows one to cal- culate tensor products and branching rules for characters of irreducible representations of 3 Lie groups; these speedups, together with a database of maximal subgroups that interfaces with the branching rule code, make Sage a powerful tool for Lie group computations. • 17 papers/preprints were written by Schilling, her collaborators and students as part of the UC Davis side of the project; 2 books were written which include Sage examples.