I've been reading your blog for a while. I sympathize some of your view abour mathematics, although I don't have necessary expertise to adequately form some "metamathematical" opinions.
I wonder what do you think about theory of motives(following Grothendieck, Beilinson and Voevodsky) and how it can potentially help algebraic/arithmetic geometry and number theory.
As far as I know, there have been some theory bulding on the matter recently(Voevodsky, Cisinski, Deglise), but the theory is appears to be far from the one envisioned by Grothendieck. Is thery of motives is still in its infancy? Or, maybe, it's not as useful as many people thought( I heard this opinion from some people).
I also heard an opinion that Grothendieck was himself dissatisfied with motives( in favor of anabelian geometry approach ) for dealing with some questions. Thought this is understandable, since it was a time before Voevodsky.
I really don't know answers to your questions. In fact, I am not sure that answers are possible at the moment.
Theory of motives is useful, as at the very least the work of Voevodsky shows. May be it is not as useful as people hoped, but this may change with time. As far as I know, the whole vision of Grothendieck is not realized. On the one hand side, mathematicians have no obligation to work along the lines suggested by Grothendieck. On the other side, the ideas of Grothendieck proved to be so fruitful that any of them deserves to be developed to the full extent. Even if he himself changed his opinion.
Dear Sowa,
ReplyDeleteI've been reading your blog for a while. I sympathize some of your view abour mathematics, although I don't have necessary expertise to adequately form some "metamathematical" opinions.
I wonder what do you think about theory of motives(following Grothendieck, Beilinson and Voevodsky) and how it can potentially help algebraic/arithmetic geometry and number theory.
As far as I know, there have been some theory bulding on the matter recently(Voevodsky, Cisinski, Deglise), but the theory is appears to be far from the one envisioned by Grothendieck.
Is thery of motives is still in its infancy? Or, maybe, it's not as useful as many people thought( I heard this opinion from some people).
I also heard an opinion that Grothendieck was himself dissatisfied with motives( in favor of anabelian geometry approach ) for dealing with some questions. Thought this is understandable, since it was a time before Voevodsky.
Василий Тертьяков:
ReplyDeleteSorry for the belated reply.
I really don't know answers to your questions. In fact, I am not sure that answers are possible at the moment.
Theory of motives is useful, as at the very least the work of Voevodsky shows. May be it is not as useful as people hoped, but this may change with time. As far as I know, the whole vision of Grothendieck is not realized. On the one hand side, mathematicians have no obligation to work along the lines suggested by Grothendieck. On the other side, the ideas of Grothendieck proved to be so fruitful that any of them deserves to be developed to the full extent. Even if he himself changed his opinion.