# What is Motivic mathematics and how is it used in physics?

In a few videos I've seen where he discusses the new approach to calculating the super Yang Mills scattering amplitudes, Nima Arkani-Hamed sometimes alludes to the use of Motivic methods as being relevant. (For example in the last few seconds of this presentation).

I would be interested if someone could give even a superficial hint of what motivic mathematics is and how it is applied in physical problems.

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The Wikipedia article has a bit of information, but not something I can make sense of... good question. – David Z Jun 3 '12 at 19:13
@DavidZaslavsky yes, I couldn't understand that either. I found this reference with some physics examples included. However, it seems inescapable that there's a lot of unfamiliar mathematics to wade through to understand this. – twistor59 Jun 4 '12 at 8:02
see the answers in physicsoverflow.org/8964 – Arnold Neumaier Sep 11 '14 at 7:42

Number Theory studies zeta functions (and zeta functions turn out to be useful in Physics as a way to regularise a divergent formula, aka re-normalisation). "The" zeta function is Riemann's, which is, for any complex number s, the sum of the reciprocals of the integers raised to the s-th power, $$\sum _{n=1}^\infty {1 \over n^x}.$$ This diverges for s=1. But the function is complex-analytic and has an analytic continuation to the entire complex plane, denoted by $\zeta (s)$. For example, $\zeta (-1) = - \frac 1{12}$ or something like that. So it provides a way of "regularising" 1+2+3+4+5+6+7+...+n+ .... as equal to -1/12.
In the twentieth century, generalisations of the Riemann zeta function were formulated, first for number fields, and then for elliptic curves $E$, and then for any algebraic curve, or even surface, etc.. (Hasse and Weil then proved that these satisfied "functional equations" relating $\zeta_E(s)$ to $\zeta_E(1-s)$ and so these had analytical continuations to the entire complex s-plane even though their defining formulas diverged. So they provide different ways of regularising different things. In particular, zeta functions are used to regularise infinite determinants.