Sage Days 64.5: Cluster Algebras

The official webpage of the meeting is at the IMA website.

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 discussen in spring 2013 in Minneapolis.

Demo from Monday morning's Talk:

IMA-Demo.sagews

IMA Demo (Notebook)

Topics

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

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

Other features

• Scattering diagrams (visualization and algebraic computations of broken lines)
• 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 (Christian/Gregg) - http://trac.sagemath.org/sage_trac/ticket/14443

• Include new Quivers/cluster seed for square and triangular products of quivers (as in Keller's work on periodicity) (Christian/Gregg) - http://trac.sagemath.org/sage_trac/ticket/14443

• Implement snake graph / matrix multiplication formulas for cluster algebras from surfaces and friezes. (Gregg) - http://trac.sagemath.org/ticket/16310 Also see Dupont's sage code on friezes from http://www.math.jussieu.fr/~dupontg/files/strings.sage

Other features, hard

• Finish mutation type checking for affine D_n (try to dove-tail with student describing all affine B/C/etc. quiver types)

• Allow inputs of cluster algebras from surfaces (interface from ideal triangulations/bistellar flips as from John Palmeri or Robert Lipshitz) (Gregg?) - look at http://www.math.columbia.edu/~lipshitz/research.html#Programming

• Work with the improved graph editor (once it exists?)
• Weiwen Gu Decomposition algorithm - and then use this to classify cluster algebra from surface by genus, number of marked/boundary points, etc. (Emily/Al/Gregg)
• Quantum cluster algebras - with Dylan Rupel
• Work with cluster algebras as "algebras" and find bases or express elements in bases
• d-vectors (using inductive formulas from Cluster Algebras IV?)

• Jim Starx code for Quiver Representations and abilities to work with Auslander-Reiten translates, etc. (#12630)

• LP algebras (Thomas Lam and Pasha Pylyavskyy with Gregg?)
• Q-system and T-system calculations as in Di Francesco and Kedem's work

Changes to the compendium

• Convert the compendium into an rst file and provide it as a thematic tutorial.

• Change intro to describe downloading only sage >= 5.9

• Adding F-polynomials, G-vectors, C-vectors, etc. updates to principal_extension/restrictions, ...
• Mutation type names (eg. V, W, ...) need to be updated
• Generalized associahedra from cluster algebra in new ticket so that compendium accurately discusses these features

• check if all class sizes are correct - see #14048 by Hugh Thomas and Frederic Chapoton

• cut out groups_of_mutations() and probably other names need to be deleted or updated

Participants

• Darlayne Addabbo, University of Illinois at Urbana-Champaign
• Eric Bucher, Louisiana State University
• Xueqing Chen, University of Wisconsin
• Man Wai Cheung, University of California, San Diego
• Aram Dermenjian, École Polytechnique
• Salomon Dominguez de la Cruz, National Autonomous University of Mexico (UNAM)
• Christopher Fraser, University of Michigan
• Ana Garcia Elsener, National University of La Plata
• Florian Gellert, Universität Bielefeld
• Jan Geuenich, Rheinische Friedrich-Wilhelms-Universität Bonn
• Emily Gunawan, University of Minnesota
• Simon King, Friedrich-Schiller-Universität
• Tri Lai, University of Minnesota
• Jesse Levitt, Louisiana State University
• Jacob Matherne, Louisiana State University
• Matt Mills, Wayne State University
• Greg Muller, University of Michigan
• Gregg Musiker, University of Minnesota
• Vincent Pilaud, École Polytechnique
• Viviane Pons, Université de Paris XI (Paris-Sud)
• Nathan Reading, North Carolina State University
• Dylan Rupel, Northeastern University
• Stella Salvatore, Università di Roma "La Sapienza"
• David Speyer, University of Michigan
• Christian Stump, Freie Universität Berlin
• Kurt Trampel, Louisiana State University
• Panupong Vichitkunakorn, University of Illinois at Urbana-Champaign
• Harold Williams, University of Texas, Austin
• Nathan Williams, University of Quebec

Organizers

• MUSIKER, Gregg (University of Minnesota)
• RUPEL, Dylan (Northeastern University)
• STELLA, Salvatore (Università di Roma "La Sapienza")
• STUMP, Christian (Freie Universitat Berlin)