M Theory Lesson 65

As nosy snoopy noted, these crystal Calabi-Yau papers are really very interesting. I would like to know more about these random partitions. Okounkov has some notes here.

Recall that in Kapranov’s non-commutative Fourier transform for three coordinates $x$, $y$ and $z$, it is natural to represent monomials by cubical paths traced out on such melting crystal partitions. The two coordinate case goes back to Heisenberg’s original paper, as we have seen. In a modern guise, his sum rule arises in honeycombs, which look a bit like shadows of melting corners.

1 Response so far »

  1. 1

    Doug said,

    Hi Kea,
    These references may be related to 2003, Andrei Okounkov, ‘The uses of random partitions‘, especially section 5.2.2 Construction of the minimizer.

    I think I have referenced this paper before.
    Zur Izhakian, ‘Duality of Tropical Curves‘, 31 pages, 4 figures, 2005
    Some figures appear to link to honeycombs?
    Tropical is usually an alternative name for Min-Plus Algebra, but Max-Plus is discussed.
    http://arxiv.org/abs/math/0503691

    Zur Izhakian, ‘Tropical Varieties, Ideals and An Algebraic Nullstellensatz’, 27 pages, 2 figures, 2005
    http://arxiv.org/abs/math/0511059

    Daniele Alessandrini, ‘Amoebas, tropical varieties and compactification of Teichmuller spaces’, 41 pages, 2005
    “… every polynomial relation among trace functions on Teichmuller space may be turned automatically in a tropical relation among intersection forms over the boundary.”
    http://arxiv.org/abs/math/0505269

    Amoeba (mathematics): “In mathematics, an amoeba is a set associated with a polynomial in one or more complex variables. Amoebas have applications in algebraic geometry“ are also discussed in the above references.
    http://en.wikipedia.org/wiki/Amoeba_%28mathematics%29

    Zur Izhakian,2004, also has ideas about:
    Algebraic Curves in Parallel Coordinates – Avoiding the “Over-Plotting” Problem
    http://arxiv.org/abs/cs/0403005
    and
    New Visualization of Surfaces in Parallel Coordinates – Eliminating Ambiguity and Some “Over-Plotting”
    http://arxiv.org/abs/cs/0403004


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