U(M) Postdoc Algebraic Geometry Seminar, Winter 2014

The Postdoc Algebraic Geometry Seminar is having a themed semester on the algebraic geometry of cluster algebras. We are attempting to collect copies of our lecture notes here, as well as links to primary sources. These notes are presented `as is', and are undoubtedly incomplete with respect to proofs and references.

Lectures and lecture notes:

  • Greg Muller: Cluster algebras and algebraic geometry (3 lectures)
    • Lecture notes
    • Too many primary sources to list, but here are several good non-technical introductions.
      • S. Fomin, Total positivity and cluster algebras, 2010.
        Notes from an ICM lecture. The first half focuses on history, and the second focuses on cluster algebras from triangulated surfaces.
      • A. Zelevinsky, What is... a Cluster Algebra?, 2007.
        A "What is..." article. At 2 pages, it sketches the algebraic construction and briefly mentions some connections.
      • S. Fomin and A. Zelevinsky, Cluster algebras: Notes for CDM-03, 2004.
        Notes from a CDM lecture. Very readable introduction to the basic algebra and combinatorics. However, at a decade old, many stated conjectures have been resolved (but not all!).
    • Here are a couple of good technical introductions.
      • A. Berenstein, S. Fomin, and A. Zelevinsky, Cluster algebras III: Upper bounds and double Bruhat cells, 2005.
        Don't let the numeral fool you, this should be the first paper any algebraic geometer reads about cluster algebras (all four "Cluster Algebras N'"papers can be read independently). It establish several of the most basic algebraic and geometric lemmas about cluster algebras, introduces upper cluster algebras, and constructs the cluster structure on the coordinate ring of any double Bruhat cell. However, it does not use the quiver version of cluster algebras which has become more standard recently.
      • B. Keller, Cluster algebras and derived categories, 2012.
        Precise and thorough without losing the story. These notes focus on the quiver perspective, and build up through the various levels of generality. Derived categories only appear in the extensive final chapter, which attempts to collect all the most significant results from additive categorification.

  • Angélica Benito: Locally acyclic cluster algebras
    • Lecture notes (scanned)
    • Primary sources:
      • G. Muller, Locally acyclic cluster algebras, 2011.
        Locally acyclic cluster algebras are those which are "locally elementary" in a geometric sense. The concept is introduced, the main consequences are proven, and first examples are proven. The reader is warned that Theorem 8.3 and consequently much of Section 8 are incorrect.
      • A. Benito, G. Muller, J. Rajchgot, and K. Smith, "Singularities of locally acyclic cluster algebras" (to appear), 2014.
        This paper considers positive characteristic reductions of cluster algebras, and specifically Frobenius splittings. It establishes that locally acyclic cluster algebras have at worst canonical singularities.

  • Ian Shipman: Cluster ensembles and quantization
    • Lecture notes (scanned)
    • Primary sources:
      • V. Fock, and A. Goncharov, Cluster ensembles, quantization, and the dilogarithm, 2009.
        Cluster ensembles consider not just a cluster variety, but a dual variety with a natural Poisson structure. This is not the first place the authors introduce cluster ensembles, but it's one of the most accessible references.
      • V. Fock, and A. Goncharov, Moduli spaces of local systems and higher Teichmuller theory, 2003-2009.
        A vast paper with a wealth of ideas, including the motivational example of cluster ensembles: the "higher Teichmuller spaces". At 34 pages, the introduction itself is a substantial but worthwhile read.

  • Jenna Rajchgot: Poisson geometry and cluster algebras
    • Lecture notes
    • Primary sources:
      • M. Gehktman, M. Shapiro, and A. Vainshtein, Cluster algebras and Poisson geometry, 2003.
        This short but consequent paper defines and classifies compatible Poisson structures on a cluster algebra.
      • A. Berenstein, and A. Zelevinsky, Quantum cluster algebras, 2004.
        This paper defines and classifies compatible quantizations of a cluster algebra. Since every compatible Poisson structure quantizes in a unique compatible way, the data defining the two structures coincides (a `compatiblity matrix').

  • Tyler Foster: Grassmannians, the positroid stratification, and Postnikov diagrams (2 lectures)
    • Lecture notes
    • Primary sources:
      • A. Knutson, T. Lam, and D. Speyer, Positroid Varieties: Juggling and Geometry, 2011.
        This paper proves many combinatorial and geometric properties of positroid varieties, and serves as one of the most readable introductions to the subject.
      • A. Postnikov, Total positivity, Grassmannians and networks, 2003-2006
        This seminal unpublished paper introduces several constructions essential to the connection between positroids and cluster algebras. However, it is also incomplete, and one of the missing sections is a clear description of the conjectural cluster structure on the homogenous coordinate ring of a positroid variety.
      • G. Muller and D. Speyer, Cluster algebras of Grassmannians are locally acyclic, 2014.
        This short paper proves that the cluster algebras associated to positroids are locally acyclic. It also contains a concise description of this cluster structure, which is missing from the prior references.

  • Dusty Ross: TBD

  • Morgan Brown: TBD

Other resources:

  • Many more papers, preprints and links of note may be found at the Cluster Algebras Portal, maintained by S. Fomin.
  • A Java applet by B. Keller for mutating quivers and their associated clusters can be found here. If quiver mutation is still mysterious to you, fifteen minutes of playing with this applet will probably fix it.