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    • CommentRowNumber1.
    • CommentAuthorEric
    • CommentTimeFeb 18th 2012


    I see a lot of fresh names to this forum (that’s great!) so I thought I’d take a moment to highlight some of its capability. This forum is more than a “dumb forum” where we can start threads and discuss stuff in simple ascii and a bit of html. You can actually do and discuss serious maths here. This forum is already fully itex capable and allows for wiki markup. Putting items in double brackets links back to the nLab.

    This is great because the nLab is also more than a simple wiki. In case you are not aware, there is now an official Publications of the nLab which contains peer reviewed wiki articles using the same syntax this forum uses. A perfect match for the very subject matter of Math 2.0. It is all meshed beautifully and Andrew has done a awesome job bringing everything together so coherently. As a sample, I copy some material below from sigma-model – exposition of classical sigma-models:

    (This is almost publication ready already)

    Exposition of classical σ\sigma-models

    We survey, starting from the very basics, classical field theory aspects of σ\sigma-models that describe dynamics of particles, stringss and branes on geometric target spaces.

    The Newtonian particle

    With hindsight, the earliest σ\sigma-model ever considered was also the very origin of the science of physics:

    In order to describe the motion of matter particles in space, Isaac Newton wrote down a differential equation with the famous symbols

    F=ma. F = m a \,.

    More in detail, this is meant to describe the following situation:

    • write X= 3X = \mathbb{R}^3 for the Cartesian space of dimension 3; think of this as a model for physics space;

    • write Σ=\Sigma = \mathbb{R} for the Cartesian space of dimensional 1; think of this as the abstract trajectory of a point particle;

    • write γ:ΣX\gamma: \Sigma\to X for a smooth function; think of this as an actual trajectory of a point particle in XX;

      write furthermore

      • v=γ˙Hom(TΣ,TX) v = \dot \gamma \in Hom(T \Sigma, T X) for the derivative of γ\gamma; think of this as the velocity of the particle; and

      • a=γ¨ a = \ddot \gamma for the second derivative, the acceleration of the particle (strictly speaking this is the covariant derivative with respect to the trivial connection on the (canonically trivialized) tangent bundle on 3\mathbb{R}^3, see below for the fully fledged discussion).

    We call then the collection of all smooth functions

    Conf=C (Σ,X) Conf = C^\infty(\Sigma, X)

    the configuration space of a physical model of a point particle propagating on XX.

    • CommentRowNumber2.
    • CommentAuthorEric
    • CommentTimeFeb 18th 2012

    PS: Clicking “Source” in the upper right corner of any comment displays the itex Source of that comment (a great feature!), which may be helpful to see what I’m talking about in the above comment.

    • CommentRowNumber3.
    • CommentAuthorScott Morrison
    • CommentTimeFeb 18th 2012

    (off topic reminder to use your full name, couldn’t find an email address for you, sorry :-)