Apparently (https://twitter.com/stewartbrand/status/1635057392814821376) Bing's AI search thinks that "the full theory of motives remains elusive". My impression was that the current version of motivic (stable) homotopy theory did everything that Grothendieck was hoping for, but I have not read Grothendieck's original work on this so I am not sure. Is my impression correct?
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asked Mar 13, 2023 at 8:26
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13$\begingroup$ There are still a number of basic conjectures by Grothendieck related to motives which are still unproven, such as his standard conjectures, as well as his conjecture on periods. $\endgroup$– Boaz MoermanCommented Mar 13, 2023 at 8:34
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2$\begingroup$ chatGPT provides a different answer : "Yes, mathematicians take Grothendieck's theory of "motives" very seriously. In fact, the theory of motives is one of the most active areas of research in algebraic geometry and number theory today. (...)" $\endgroup$– Philippe GaucherCommented Mar 13, 2023 at 10:47
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12$\begingroup$ The methods of Voevodsky (and others) produce a triangulated category, which is morally the derived category of an abelian category of mixed motives. However, we don't actually know how to construct to constructed a t-structure with good properties etc. This is related to the standard conjectures mentioned above. So yeah the subject is very far from done. $\endgroup$– Donu ArapuraCommented Mar 13, 2023 at 12:08
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2$\begingroup$ Arguably, a necessary condition for the completion of a "full theory of motives" would be someone claiming the million dollars for the Hodge conjecture. $\endgroup$– Timothy ChowCommented Mar 13, 2023 at 22:31
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