Peter Scholze and the Future of Arithmetic Geometry (2016)
I remember the buzz in Germany&#x27;s math graduate community, when word of Scholze&#x27;s perfectoid spaces began to go around. Reading groups spawned everywhere trying to get a grasp on the technicalities:<p>Work in number theory very often deals only with one of two fundamentally different settings.<p>* Either objects where multiplication with any natural number can be inverted (&#x27;characteristic 0&#x27;, examples for such objects are the rational or complex numbers or &quot;something inbetween the two&quot;),<p>* 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 -&gt; x^p a multiplicative and additive (!) map, the FROBENIUS.<p>Scholze introduced a way to pull the Frobenius map over to characteristic 0. He could do this &#x27;tilting&#x27; in towers and in this way compared the theory of towers in characteristic 0 and p. For details, see his famed answer here [1].<p>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.<p>This result alone made the characteristic 0 neck hair stand up :-)<p>[1] <a href="https:&#x2F;&#x2F;mathoverflow.net&#x2F;questions&#x2F;65729&#x2F;what-are-perfectoid-spaces" rel="nofollow">https:&#x2F;&#x2F;mathoverflow.net&#x2F;questions&#x2F;65729&#x2F;what-are-perfectoid...</a>