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:<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 ('characteristic 0', examples for such objects are the rational or complex numbers or "something inbetween the two"),<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 -> 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 '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].<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://mathoverflow.net/questions/65729/what-are-perfectoid-spaces" rel="nofollow">https://mathoverflow.net/questions/65729/what-are-perfectoid...</a>
> “To this day, that’s to a large extent how I learn,” he said. “I never really learned the basic things like linear algebra, actually — I only assimilated it through learning some other stuff.”<p>This was interesting to me. My math profs always admonished us to ensure foundations are completely watertight before advancing to the next thing in tiny increments. I've absorbed this to the point where I perhaps get stuck filling in inconsequential gaps at roughly the same level, kitting out base camp as fully as possible but postponing the ascent.<p>I have no dreams of becoming a professional mathematician, but maybe I'd have quicker insights and more creative ideas if I tried this approach too, tackling something impossible and working backwards.
"I never really learned the basic things like linear algebra, actually — I only assimilated it through learning some other stuff."<p>The hacker attitude seems to be the way to go even in fundamentally anti-hacker environments.