I remember the buzz in Germany's math graduate community, when word of Scholze's perfectoid spaces began to go around. Reading groups spawned everywhere trying to get a grasp on the technicalities:
Work in number theory very often deals only with one of two fundamentally different settings.
* Either objects where multiplication with any natural number can be inverted ('characteristic 0', examples for such objects are the rational or complex numbers or "something inbetween the two"),
* or objects where a certain prime number p has a special role; namely, the multiplication by p is the 0-map. This sounds horrible, but it actually has a great implication: (a+b)^p = a^p + b^p, because the middle binomial coefficients are multiples of p. This makes x -> x^p a multiplicative and additive (!) map, the FROBENIUS.
Scholze introduced a way to pull the Frobenius map over to characteristic 0. He could do this 'tilting' in towers and in this way compared the theory of towers in characteristic 0 and p. For details, see his famed answer here [1].
Very soon it became clear, that this tool had remarkable applications and his thesis explored only one of them: a proof of the monodromy-weight conjecture in characteristic 0 by tilting results in characteristic p.
This result alone made the characteristic 0 neck hair stand up :-)
[1] https://mathoverflow.net/questions/65729/what-are-perfectoid...
Work in number theory very often deals only with one of two fundamentally different settings.
* Either objects where multiplication with any natural number can be inverted ('characteristic 0', examples for such objects are the rational or complex numbers or "something inbetween the two"),
* or objects where a certain prime number p has a special role; namely, the multiplication by p is the 0-map. This sounds horrible, but it actually has a great implication: (a+b)^p = a^p + b^p, because the middle binomial coefficients are multiples of p. This makes x -> x^p a multiplicative and additive (!) map, the FROBENIUS.
Scholze introduced a way to pull the Frobenius map over to characteristic 0. He could do this 'tilting' in towers and in this way compared the theory of towers in characteristic 0 and p. For details, see his famed answer here [1].
Very soon it became clear, that this tool had remarkable applications and his thesis explored only one of them: a proof of the monodromy-weight conjecture in characteristic 0 by tilting results in characteristic p.
This result alone made the characteristic 0 neck hair stand up :-)
[1] https://mathoverflow.net/questions/65729/what-are-perfectoid...