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John C. Baez

  1. G2 and the Rolling Ball.

    Mon, 05/14/2012 - 15:01 — SomNambul
    Authors: John C. Baez, John Huerta
    Subjects: Differential Geometry
    Abstract

    Understanding the exceptional Lie groups as the symmetry groups of simpler
    objects is a long-standing program in mathematics. Here, we explore one famous
    realization of the smallest exceptional Lie group, G2. Its Lie algebra acts
    locally as the symmetries of a ball rolling on a larger ball, but only when the
    ratio of radii is 1:3. Using the split octonions, we devise a similar, but more
    global, picture of G2: it acts as the symmetries of a 'spinorial ball rolling
    on a projective plane', again when the ratio of radii is 1:3.

  2. Higher-Dimensional Algebra VII: Groupoidification.

    Tue, 09/01/2009 - 12:47 — SomNambul
    Authors: John C. Baez, Alexander E. Hoffnung, Christopher D. Walker
    Subjects: Quantum Algebra
    Abstract

    Groupoidification is a form of categorification in which vector spaces are
    replaced by groupoids, and linear operators are replaced by spans of groupoids.
    We introduce this idea with a detailed exposition of "degroupoidification": a
    systematic process that turns groupoids and spans into vector spaces and linear
    operators. Then we present three applications of groupoidification. The first
    is to Feynman diagrams. The Hilbert space for the quantum harmonic oscillator
    arises naturally from degroupoidifying the groupoid of finite sets and
    bijections.

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