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## Sage Days 23: Number Theory and Computer Algebra |

Motivation The purpose of this workshop was to explore the state of the art in computational number theory and computer algebra, to investigate concrete open source implementations of relevant algorithms in Sage (http://sagemath.org), to explain to young people how they can contribute to Sage, and to identify directions for future work. Workshop The workshop was attended by 48 people from Australia, Belgium, Canada, France, Germany, Italy, the Netherlands, Sweden, United Kingdom, and the United States. Many were from European Research Training Network GTEM that organized the meeting. There was a particularly strong showing of junior researchers and graduate students. Program The main program of talks consisted of 7 keynote lectures that were each 50 minutes long, with 1-2 talks each day. There were also working sessions and coding sprints on the following topics: * Descent on Cyclic Covers of the Projective Line * ABC triples * Models for elliptic curves * Function field arithmetic * Hyperbolic geometry * Rational polynomials in FLINT * Factoring in ZZ[x] using FLINT, and computing Swinnerton-Dyer Polynomials * MPIR integer arithmetic developement * Linbox development * Design of the Sage Notebook (for classroom use) * Improve integer factorization in Sage * Solving conics * Counting representations of numbers of sums of squares * Porting functionality from ECHIDNA: computation of dimensions Atkin-Lehner spaces Each morning of the workshop we had status reports about the progress of each group, and many of the groups worked very hard on their projects, long into the night, both at the hotel lobby and at the Lorentz Center. Impressions The main impact of the workshop was that several young mathematicians that are also exceptional computer programmers, including Maarten Derickx and Jeroen Demeyer, learned what Sage is, and have subsequently become very involved in the development of Sage. In particular, they learned a wide range of skills and procedures that are needed to do Sage development, which are quite difficult to learn without a workshop. Robert Miller's tutorial on ``how to do Sage development'' was especially helpful in this regard. Another outcome of the workshop is that many participants became aware of whole classes of algorithms, problems, and coding techniques, which they hadn't been aware of before. The long list of coding sprint topics above gives a sense of this range of topics. There were also many informal talks, by Dan Bernstein and others, that went into much more detail about some of these topics. Also, the workshop resulted in new work on implementing the Lenstra-Stevenhagen finite field representation algorithm. Conclusion The goal of the workshop was to teach mathematicians about Sage and current computational issues in number theory and computer algebra, and get new people to contribute to Sage. The talks did a great job at conveying some of the state of the art in the above topics, and the working sessions and coding sprints succeeded and getting many talented software engineers involved in Sage development. Acknowledgment We thank the Lorentz Center for the excellent facilities and for the excellent local organization by workshop coordinator Pauline Vincenten. The meeting was supported by the European Commission under contract MRTN-CT-2006-035495 (GTEM) and by the Lorentz Center. [Back] |