Wednesday, February 2, 2011

Motive Madness

In his lectures on motives, Manin begins with the central question:
If numbers are similar to polynomials in one variable over a finite field, what is the analogue of polynomials in several variables? Or, in more geometric terms, does there exist a category in which one can define absolute Descartes powers $\textrm{Spec} Z \times \cdots \times \textrm{Spec} Z$?
Here $\textrm{Spec} Z$ refers to the so called spectrum of the integers $Z$. It is thought of as a space built out of prime elements. But everyone knows that prime spaces should be described in terms of knots and tangles. Bits and trits are associated to spaces of prime dimension. If we built spaces from bits and trits and other $p$-dits, where there are always $d = p + 1$ MUBs, we would be building spaces from knots. A Cartesian product takes strings of prime elements. But let's not forget that dits have ever increasing dimensions, and that $\omega$ categories are the most natural spaces to create out of paths.


  1. I like this. Our conventions on what are the variables and the unknowns in polynomials comes to mind- that and as coefficients these numbers in matrices may themselves be represented in the various binary codes and tangles.

    Kea, where do you think these sorts of theories are heading? I keep thinking of an old book which said mass was just a knot made of space. Made of nothing was my first reaction to it. Can I not ask of what strings are made?

    Thank you for such interesting links, and for the people who seem to share your wisdom.

    The PeSla

  2. Thanks for visiting, ThePeSla. And I like your coining of the term wildcard.

    Where are these theories heading? Well, I'm not sure which set of theories you have in mind, because to me there is only one big theory at any given time, although people may use different languages to describe it. And this current theory most certainly has the power that is required of it, even if you or I struggle to understand it. The question then is: what is holding it back? How much more constraint can it stand?


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