### Schedule for Sage Days 49 (June 17-21, 2013)

**For the main Sage Days 49 page go** here

#### Monday

Morning Session:

09h00 : Welcome, Sage installation, coffee

10h00 : Introduction to Sage*, Tom Denton (Slides!)

10h45 : Tutorial I:

Using the Sage notebook and navigating the help system, Mathieu Guay-Paquet (UQAM)11h30 : Talk I:

Find me a basis that...., Mike Zabrocki (York)

Abstract. This talk will start with an introduction to combinatorial Hopf algebras (CHAs). Ironically, one of the main open problems related to this area is to give a good definition of a CHA. A question that I think that Sage will help us answer is "what is the fundamental basis of a combinatorial Hopf algebra?" slides available for talk

Lunch Break: 12h30 - 14h30

Afternoon Session, 14h30-17h30:

- 14h30 :
Round table and discussion of programming projectsAnne Schilling (UC Davis)- 15h30 : Coffee Break
- 16h00 : Coding Sprints
- 17h00 : Coding Sprints

#### Tuesday

Morning Session:

09h00 : Talk II:

Combinatorial Actions, Orbit Averages, and Sage Implementation, Tom Roby (U. Connecticut) slides!! (open in Adobe Acrobat Reader to see animations)

Abstract. We consider a variety of combinatorial actions on finite sets whose orbits have unexpected properties. Starting with simple examples such as cyclic rotation of binary strings, we generalize to actions on Young tableaux and order ideals of partially ordered sets. We identify a particular phenomenon calledhomomesyappearing in many unrelated combinatorial contexts: namely that the average value of some natural statistic over each orbit is the same as the average over the entire set. Viewing these actions as products oftoggle operationsallows us to see how some of these actions are related and to extend much of this picture more broadly. In particular, we can generalize the operations of rowmotion and promotion (in Striker and Williams' terminology) on order ideals in a poset to (1) the order polytope of a poset (the continuous piecewise-linear category), and (2) to the collection of maps from a poset to rational functions in $|P|$ variables (the birational/geometrical) setting.For this latter setting, Darij Grinberg has developed Sage code that (1) computes the rowmotion operator; (2) checks whether this operator appears to have finite order; (3) helps check for homomesy of particular statistics under the action of the operator. Although we have proofs of homomesy for a number of special cases, much remains to be done even at the combinatorial level of order ideals on posets. For the two other categories almost everything we have discovered via computer-based computations is still conjectural.

This talk largely discusses recent work with Jim Propp, including ideas and results and code from Arkady Berenstein, David Einstein, Darij Grinberg, Shahrzad Haddadan, Jessica Striker, and Nathan Williams.

10h00 : Coffee Break

10h30 : Tutorial II:

Programming in Python and Sage, Viviane Pons (Marne-la-Vallée) (download tutorial, save and open through sage notebook)11h30 : Talk III:

Computing Ext algebras with Sage. An F_5 algorithm for path algebra quotients(Slides), Simon King (Friedrich Schiller U. Jena)

Abstract.Basic algebras are finite dimensional path algebra quotients and naturally occur in, e.g., representation theory of finite groups. We implement Ext algebras of basic algebras in Sage, together with an F5 algorithm.Given minimal projective resolutions for the simple modules of a basic algebra, one computes the Ext algebra by dealing with homogeneous algebraic relations in free associative algebras. Here, we use “Letterplace” from Singular. Our wrapper in Sage revealed some severe memory leaks.

Minimal projective resolutions for modules over basic algebras can be obtained by standard basis or linear algebra methods. The currently best implementation (for quivers with a single vertex) is David Green’s heady Buchberger algorithm, which is part of the optional p group cohomology spkg. We wish to replace it by a more efficient method: F5.

Faugère’s famous F5 algorithm computes standard bases for modules over polynomial rings. It uses a “signature”, that keeps track how elements were constructed, and is efficient by exploiting commutativity to avoid redundant constructions. We provide a non-commutative F5 algorithm, replacing the commutativity relations by the quotient relations in a basic algebra.

Given a module over a basic algebra, the output of the F5 algorithm is not just a standard basis but a so-called signed standard basis of the module. Our main result is: If a negative degree monomial ordering is used, then the signed standard basis provides enough information to read off the Loewy layers of the module. This information is not provided in an unsigned standard basis. Hence, the F5 algorithm could not easily be replaced by any other algorithm for the computation of standard bases.

The first Loewy layer provides a minimal generating set. Thus, the non-commutative F5 algorithm yields more information than Green’s algorithm and in addition is theoretically more efficient.

Lunch Break: 12h30 - 14h30

Afternoon Session, 14h30-17h30:

- 14h00 : FindStat Demo (Chris Berg, Viviane Pons, Travis Scrimshaw)
- 15h30 : Coffee Break
- 17h00 : Status Reports

#### Wednesday

Morning Session:

- 09h00 : Tutorial III
version control and contributing to sageChris Berg (UQAM)Guides:

- How to Referee Sage Trac Tickets by William Stein
- How to contribute to Sage by Sébastien Labbé
- Introduction to Sage Development by Mike Hansen
- Short step-by-step checklist for reviewing a patch by Franco Saliola
- Sage Developer's Guide:
- Walking Through the Development Process
- Review a patch
Videos:

- Contributing to Sage : Who, What and How: video of a talk by William Stein
Related thematic tutorials:

- 10h00 : Coffee Break
- 10h30 : Tutorial IV:
coercion and category framework, Simon King Tutorial worksheet, original webpage

Lunch Break: 12h30 - 14h30

Afternoon Session, 14h30-17h30: exercises and coding sprints with coffee break and status reports

- 15h30 : Coffee Break
- 17h00 : Status Reports

#### Thursday

Morning Session:

09h00 : Talk IV:

Numerical evaluation of D-finite functions: NumGfun and beyond, Mark Mezzarobba (RISC, Johannes Kepler U. Linz) Slides Maple worksheet

Abstract.D-finite (aka holonomic) functions are complex analytic solutions of linear ODEs with polynomial coefficients. The class of D-finite functions encompasses most elementary functions (exp, ln, sin, sinh...) as well as many common special functions (e.g., Bessel functions and generalized hypergeometric functions). Its nice algebraic and computational properties make it possible to develop a unified framework to deal with these functions in a computer algebra system, instead of developing ad hoc code for every single function.My talk will focus on the multiple precision numerical evaluation of D-finite functions. I will present NumGfun, a Maple package that provides a guaranteed evaluator for general D-finite functions as well as some other features related to the rigorous symbolic-numeric manipulation of such functions. I also plan to discuss some of the underlying algorithms and say a few words about applications to analytic combinatorics, the reasons that made me use Maple for this work, what I dislike about it and what role Sage may play as an alternative.

10h00 : Coffee Break

10h30 : Open Presentations

Lunch Break: 12h30 - 14h00

- 14h00 : Digiteo Seminar,
Le génie mathématique, du théorème de quatre couleurs à la classification des groupes, Georges Gonthier (Microsoft Research)

Afternoon Session,

- 15h30 : Coffee Break
- 17h00 : Status Reports

#### Friday

Morning Session:

- 09h00 : Open Presentations
- 10h00 : Coffee Break
- 10h30 : Open Presentations

Lunch Break: 11h30 - 13h30

Afternoon Session, 14h30-17h30: exercises and coding sprints with coffee break and status reports

- 15h00 : Final Status Reports

### Open Presentations

Open presentations are quick (5 to 15 minutes) presentations done by the participants. It can be demonstrations of projects done during the week. Or it can be about anything of interest to the participants including software useful for teaching or research.