Quanta Magazine tells that 30-year old Peter Scholze is now the youngest Fields medalist due to the revolution that he launched in arithmetic geometry (see this).
Scholze's work might be interesting also from the point of view physics, at least the physics according to TGD. I have already made a attempt to understand Scholze's basic idea and to relate it to physics. About the theorems that he has proved I cannot say anything with my miserable math skills.
The notion of perfectoid
Scholze introduces first the notion of perfectoid.
The tilt of the perfectoid
What Scholze introduces besides perfectoids K also what he calls tilt of the perfectoid: Kb. Kb is something between p-adic number fields and reals and leads to theorems giving totally new insights to arithemetic geometry
Characteristic p (p is now the prime labelling p-adic number field) means nx=0. This property makes the mathematics of finite fields extremely simple: in the summation one need not take care of the residue as in the case of reals and p-adics. The tilt of the p-adic number field would have the same property! In the infinite sequence of the p-adic numbers coming as iterated p:th roots of starting point p-adic number one can sum each p-adic number separately. This is really cute if true!
It seems that one can formulate the arithmetics problem in the tilt where it becomes in principle as simple as in finite field with only p elements! Does the existence of solution in this case imply its existence in the case of p-adic numbers? But doesn't the situation remain the same concerning the existence of the solution in the case of rational numbers? The infinite series defining p-adic number must correspond a sequence in which binary digits repeat with some period to give a rational number: rational solution is like a periodic solution of a dynamical system whereas non-rational solution is like chaotic orbit having no periodicity? In the tilt one can also have solutions in which some iterated root of p appears: these cannot belong to rationals but to their extension by an iterated root of p.
The results of Scholze could be highly relevant for the number theoretic view about TGD in which octonionic generalization of arithematic geometry plays a key role since the points of space-time surface with coordinates in extension of rationals defining adele and also what I call cognitive representations determining the entire space-time surface if M8-H duality holds true (space-time surfaces would be analogous to roots of polynomials). Unfortunately, my technical skills in mathematics needed are hopelessly limited.
TGD inspires the question is whether the finite cutoffs of Kb - almost perfectoids - could be particularly interesting physically. At the limit of infinite dimension one would get an ideal situation not realizable physically if one believes that finite-dimensionality is basic property of extensions of p-adic numbers appearing in number theoretical quantum physics (they would related to cognitive representations in TGD). Adelic physics involves all extensions of rationals and the extensions of p-adic number fields induced by them and thus also extensions of type Kb. I have made some naive speculations about why just these extensions might be physically of a special signiticance.
See the chapter Does M8-H duality reduce classical TGD to octonionic algebraic geometry?. See also the article Could the precursors of perfectoids emerge in TGD?