Sage Days 64.5: Cluster Algebras

If you want to know who was physically participating, check out our group photo.

Here are a few more pictures of the group working, coding, climbing, and waving.

General Information

Sage Days 64.5: Cluster Algebras will be a five-day workshop to design and implement algorithms and computer exploration tools for research in cluster algebras and quiver representations. All software developed during the workshop will be made freely available as part of the free and open-source mathematics software system Sage (http://www.sagemath.org).

The workshop will consist of both talks on cluster algebras (focusing on features to be implemented) as well as on logistics of Sage development. In particular, students and researchers with background in cluster algebras are very welcome to attend and to get involved in computer explorations and development of Sage's cluster package. There will be ample time allotted for design discussions and coding sprints to implement the developed algorithms.

When and where?

June 1-5, 2015, at the IMA, University of Minnesota, in Minneapolis, MN, USA.

Schedule

All talks will be held in Lind 305, and all coffee breaks in Lind Hall 400.

The starting point is the core implementation of cluster algebras as merged in Sage-5.9, see #10298, with possible new features discussed in spring 2013 in Minneapolis.

Topics

• Topics and goals for the week will be discussed the first day of the workshop. Below is a list of possible topics:

Progress Report on Wednesday

* Salvatore, Dylan: Re-implementing ClusterSeeds using F-polys and g-vectors. Also, implementing Tropical Semifields

* Jesse, Chris - Add modularity so that one can compute certain cluster seed quantities without others

* Gregg, Chris, Jesse, Emily - Related to above, allowing other names for cluster variables. (Goal is to merge all three of these pieces together to have trac tickets rewriting ClusterSeed)

• Ticket numbers: #18615

* Simon - Working on F5 algorithm, have a ticket for review

* Anna, Emily - fixed bug in snake graphs and triangulation

* Aram, Vivianne - mutation analyzer, using functions to only mutate at certain type of vertices 18594

• Ticket numbers: #18594

* Greg, Harold, Jacob, Matt - implementing upper cluster algebra presentation algorithm but saturation is computing a quotient ideal that is time intensive.

• Ticket numbers: tba, 18619

* Christian, Vincent, Nathan W., Simon - Chevie interface for reflection groups. Almost ready for review.

• Ticket number :11187

* Vincent, Christian - Subword complexes. Almost ready for review.

• Ticket number: 11010

* Nathan W. - m-eralized Cambrian lattices

• Ticket number: 18597

* Chris and David - code for weakly separated sets and visualization

* Greg, Mandy, Eric, Trey - working with Greg's stereographic projection, modeling them internally where they intersect

• Ticket number: 16310

* Florian - added class for Quantum Tori

* Greg, Mandy - Scattering diagrams (visualization and algebraic computations of broken lines, consistent completions)

• Ticket number: ?

Topics from first feature meeting (6+ votes)

• Implementing g-vectors, c-vectors and F-polynomials without cluster mutation; possibly replace mutation
  • - Gregg Musiker - Dylan Rupel - Jesse Levitt - Greg Muller - Salvadore Stela - Harold - Natan
• Quantum cluster algebras
  • - Dylan - Florian - Mandy - Jesse Levitt - Chen
• Finite order scattering diagrams
  • - Salvatore Stella - Greg - Dylan - Mandy - Gregg - Erick - Jacob - Trey - Harold
• Flexibility to name cluster variables; perhaps label internally by g-vectors?
  • - Gregg Musiker - David Speyer - Christian - Panupong - Chris - Anna - Darlayne
• Considering mutation sequences of a restricted type: Sink/source, urban renewal, green
  • - Salvatore Stella - Aram - Matt - Eric - Gregg - Jan
• Presenting upper cluster algebras
  • - Harold - Greg - Jacob - Matt - Natan
• Interface to gap3/chevie; subword complexes, reflection groups, and sorting
  • - Christian - Vincent Pilaud - Simon - Aram - Salomon - Nathan - Viviane
• Path algebra arithmetic
  • - Simon - Christian - Salomon - Anna - Chen - Trey - Jacob - Eric

Important features

• Allow labellings of the vertices of quivers and cluster seeds (i.e., provide a dict that contains this information, and which is used all over to work with cluster seeds)
• in some situations, it might be nice to update variable names. E.g., for Plücker coords and SL[N].

• generalized associahedra from cluster complexes (Christian) - http://trac.sagemath.org/sage_trac/ticket/10817

• Check why variable_class() is so much slower compared with cluster_class(). This might be because Laurent polynomials are slow in sage. (Christian)

• Finish the implementation of finite reflection groups (Christian) -- http://trac.sagemath.org/ticket/11187

• Allowing options to work with g-vectors, d-vectors, c-vectors directly rather than with cluster variables
• Cluster algebras as algebras

Other features

• Quantum cluster algebras
• Generalized cluster algebras
• Visualize ideal triangulations
• Exotic Belavin-Drinfeld cluster algebras or cluster algebras from Double Bruhat Cells
• Urban Renewal
• more on Quiver Representations?

Other features, easy

• Include new Quiver mutation types for Gale-Robinson Quivers (Gregg) - http://trac.sagemath.org/sage_trac/ticket/14444

• Add sum of quivers/cluster seed as disjoint union