Интересно было бы выяснить, применимы ли в физике элементарных частиц те абстрактные теории, которые
были созданы Mochizuki. В частности,
As motivation, the method of Mochizuki to settle Szpiro’s Conjecture (and hence ABC) is to encode the key arithmetic invariants of elliptic curves in that conjecture in terms of “symmetry” alone, without direct reference to elliptic curves. One aims to do the encoding in terms of group-theoretic data given by (arithmetic) fundamental groups of specific associated geometric objects that were the focus of Grothendieck’s anabelian conjectures on which Mochizuki had proved remarkable results earlier (going far beyond anything Grothendieck had dared to conjecture). The encoding mechanism is addressed in the appendix; it involves a lot of serious arguments in algebraic and non-archimedean geometry of an entirely conventional nature (using p-adic theta functions, line bundles, Kummer maps, and a Heisenberg-type subquotient of a fundamental group).
Mochizuki’s strategy seems to be that by recasting the entire problem for Szpiro’s Conjecture in terms of purely group-theoretic and “discrete” notions (i.e., freeing oneself from the specific context of algebro-geometric objects, and passing to structures tied up with group theory and category theory), one acquires the ability to apply new operations with no direct geometric interpretation. This is meant to lead to conclusions that cannot be perceived in terms of the original geometric framework.
To give a loose analogy, in Wiles’ solution of Fermat’s Last Theorem one hardly ever works directly with the Fermat equation, or even with the elliptic curve in terms of which Frey encoded a hypothetical counterexample. Instead, Wiles recast the problem in terms of a broader framework with deformation theory of Galois representations, opening the door to applying techniques and operations (from commutative algebra and Galois cohomology) which cannot be expressed directly in terms of elliptic curves. An analogy of more relevance to Mochizuki’s work is the fact that (in contrast with number fields) absolute Galois groups of p-adic fields admit (topological) automorphisms that do not arise from field-theoretic automorphisms, so replacing a field with its absolute Galois group gives rise to a new phenomenon (“exotic” automorphisms) that has no simple description in the language of fields.
To be more specific, the key new framework introduced by Mochizuki, called the theory of Frobenioids, is a hybrid of group-theoretic and sheaf-theoretic data that achieves a limited notion of the old dream of a “Frobenius morphism” for algebro-geometric structures in characteristic 0. The inspiration for how this is done apparently comes from Mochizuki’s earlier work on p-adic Teichmuller theory (hence the “Teichmuller” in “IUT”). To various geometric objects Mochizuki associates a “Frobenioid”, and then after some time he sets aside the original geometric setting and does work entirely in the context of Frobenioids. Coming back to analogues in the proof of FLT, Wiles threw away an elliptic curve after extracting from it a Galois representation and then worked throughout with Galois representations via notions which have no meaning in terms of the original elliptic curve.
Насколько у физиков принято следить за передовыми работами в математике?
