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mitchell porter
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Peter Woit's anti-string-theory blog, "Not Even Wrong", also follows the controversy around Shinichi Mochizuki's claimed proof of the abc conjecture. Lately the number theorist Kirti Joshi has posted a few times about his attempt to reconstruct the controversial part of Mochizuki's argument in an altered theoretical framework. I spotted something about Joshi's work and tried to point it out, but apparently the comment isn't being allowed through, so I'll mention it here.
Mochizuki's theoretical framework is called "Inter-Universal Teichmüller Theory". Roughly speaking, Teichmüller Theory studies the moduli spaces of Riemann surfaces, and I think Mochizuki is referring to Grothendieck Universes, which are sets large enough to model most of what one wishes to do in mathematics, insofar as it can be based on set theory. (It might be easier to characterize a Grothendieck Universe by what it doesn't contain: it doesn't contain the very largest cardinals, nor does it encompass any "large categories", these being categories which are the size of a proper class.) There are generalizations of classical Teichmüller Theory to other kinds of objects, notably p-adic objects. Mochizuki's "inter-universal" theory is meant to compare certain algebraic objects in a way that is impossible in Grothendieck's theory of schemes, by detaching the additive and multiplicative parts of their algebra from each other.
Mochizuki's claimed proof of abc, as I understand it, starts by translating the proposition about a+b=c into a claim about elliptic curves (this is standard). Then he considers a lot of objects associated with such a curve, including a particular vector space; then (in physics language) he parallel-transports this ensemble of objects through a series of universes, by mapping the curve onto its counterpart in each universe, then the next, and so on, until he returns to the original instance of the curve. These mappings are somewhat underdetermined, due to three distinct "indeterminacies"... Then we see what has happened to the volume of a region of the vector space associated with the elliptic curve, in the course of this odyssey; and the abc inequality is deduced from this.
The dispute over Mochizuki's proof centers on the validity of this procedure. His critics Scholze and Stix claim that it cannot work as advertised; Mochizuki says their criticism only applies to an oversimplified strawman of his theory... Joshi, meanwhile, has developed an Arithmetic Teichmüller Theory, based on perfectoid spaces defined by Scholze, in which he thinks that Mochizuki's method can be more transparently reproduced. It seems neither Mochizuki nor Scholze accepts this, but Joshi is just getting on with his reconstruction of the proof anyway, in a series of papers.
Here is what I wanted to point out: In a progress report, "Mochizuki’s Corollary 3.12 and my quest for its proof", Joshi talks about the counterpart in his theory, of Mochizuki's three indeterminacies. He writes:
This is all that I wanted to point out - the fascinating fact that some math which is absolutely central to string theory, has shown up in an attempt to re-derive Mochizuki's proof of the abc conjecture. This shouldn't be surprising given that versions of Teichmüller Theory play a role both in string theory and in Mochizuki's work... It would be very interesting if the connection went even deeper, e.g. if Mochizuki's "inter-universal" mappings truly correspond to an identifiable kind of duality or gauge transformation in string theory.
Mochizuki's theoretical framework is called "Inter-Universal Teichmüller Theory". Roughly speaking, Teichmüller Theory studies the moduli spaces of Riemann surfaces, and I think Mochizuki is referring to Grothendieck Universes, which are sets large enough to model most of what one wishes to do in mathematics, insofar as it can be based on set theory. (It might be easier to characterize a Grothendieck Universe by what it doesn't contain: it doesn't contain the very largest cardinals, nor does it encompass any "large categories", these being categories which are the size of a proper class.) There are generalizations of classical Teichmüller Theory to other kinds of objects, notably p-adic objects. Mochizuki's "inter-universal" theory is meant to compare certain algebraic objects in a way that is impossible in Grothendieck's theory of schemes, by detaching the additive and multiplicative parts of their algebra from each other.
Mochizuki's claimed proof of abc, as I understand it, starts by translating the proposition about a+b=c into a claim about elliptic curves (this is standard). Then he considers a lot of objects associated with such a curve, including a particular vector space; then (in physics language) he parallel-transports this ensemble of objects through a series of universes, by mapping the curve onto its counterpart in each universe, then the next, and so on, until he returns to the original instance of the curve. These mappings are somewhat underdetermined, due to three distinct "indeterminacies"... Then we see what has happened to the volume of a region of the vector space associated with the elliptic curve, in the course of this odyssey; and the abc inequality is deduced from this.
The dispute over Mochizuki's proof centers on the validity of this procedure. His critics Scholze and Stix claim that it cannot work as advertised; Mochizuki says their criticism only applies to an oversimplified strawman of his theory... Joshi, meanwhile, has developed an Arithmetic Teichmüller Theory, based on perfectoid spaces defined by Scholze, in which he thinks that Mochizuki's method can be more transparently reproduced. It seems neither Mochizuki nor Scholze accepts this, but Joshi is just getting on with his reconstruction of the proof anyway, in a series of papers.
Here is what I wanted to point out: In a progress report, "Mochizuki’s Corollary 3.12 and my quest for its proof", Joshi talks about the counterpart in his theory, of Mochizuki's three indeterminacies. He writes:
Anyone who has studied string theory will recognize the name of Miguel Virasoro. The reason there is a connection to string theory, is that the Riemann surfaces of the original Teichmüller Theory, show up in string theory as the space-time surfaces that a string traces out as it propagates. The Virasoro algebra of transformations is one of the basic algebraic entities in string theory, and string theory path integrals are integrals over Teichmüller space.
This is all that I wanted to point out - the fascinating fact that some math which is absolutely central to string theory, has shown up in an attempt to re-derive Mochizuki's proof of the abc conjecture. This shouldn't be surprising given that versions of Teichmüller Theory play a role both in string theory and in Mochizuki's work... It would be very interesting if the connection went even deeper, e.g. if Mochizuki's "inter-universal" mappings truly correspond to an identifiable kind of duality or gauge transformation in string theory.