Although I am not an expert on either of the subjects, I think it is save to say that such an application is highly unlikely.

First of all it is my understanding that Motives came up in connection with *perturbative* Quantum Field Theory. That is to say in perturbative Quantum Field Theory you come to a point where you have to calculate individual Feynman diagrams and this can be related to periods of certain motives. Usually the problem can be separated into some group theoretical part, where you calculate casimirs, gamma matrix traces and so on. In the end what you are left with is a sum of terms, whose denominators are products of something like $\frac{1}{k^2 - m^2}$, the precise structure is determined by momentum conservation at each vertex and the rule is that you have to integrate over each loop momentum.

If there are enough factors, the denominator has an interesting pole structure. Basically you have a bunch of intersecting hyperboloids (that part I have not thought carefully about). Now algebraic geometers like to think in geometrical terms and have their own names for this situation: You are calculating a period of some 'motive' (it should be just related to the poles of the denominator).

Of course physicists have done these integrals long before mathematicians developed an interest for them (again?). Basic tricks are to introduce Feynman parameters or use the schwinger representation. For example Marcolli uses this representation in papers she also talks about motives. Similiarly Connes and Kreimer seem to mainly clarify constructions, that had been known to people that calculated 5-th loop order QED diagrams (i.e. how to cut up divergent diagrams). Although I probably just don't understand the more sophisticated parts of their work.

Now it is usually not emphasized, but many partial differential equations can be treated perturbatively by feynman diagram methods. Essentially one only considers tree diagrams of a corresponding QFT. I suspect that these are the Wyld diagrams.

In any case the Millenium Problem asks for existence of certain solutions, given the fact that diagram methods are mostly a calculational tool, it is highly unlikely that they can be useful to it. Since the connection of QFT with motives is calculational, it is unlikely, that they are useful.

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