Inter-universal geometry と ABC予想 (応援スレ) 44

[過去ﾛｸﾞ] Inter-universal geometry と ABC予想 (応援スレ) 44 (1002ﾚｽ)
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25: 現代数学の系譜 雑談 ◆e.a0E5TtKE  [sage] 2020/04/12(日) 13:21:52 ID:hAg37Ryy

>>23

つづき

Background
In March 2018 Peter Scholze and Jacob Stix travelled to Japan to visit Shinichi Mochizuki to discuss with him his claimed proof of the abc conjecture.
In documents released in September 2018, Scholze?Stix claimed the key Lemma 3.12 of Mochizuki’s third Inter-Universal Teichmuller Theory (IUTT) paper reduced to a trivial inequality under certain harmless simplifications, invalidating the claimed proof.2 Mochizuki agreed with the conclusion that under the given simplifications the result became trivial,
but not that the simplifications were harmless. However, Scholze and Stix were not convinced by the arguments as to why their simplifications drastically altered the theory, and we stand at an impasse.
The documents released by both sides3 include two versions of a report by Scholze?Stix, titled Why abc is still a conjecture, each with an accompanying reply by Mochizuki, as well as a 41-page article,
Report on discussions, held during the period March 15 ? 20, 2018, concerning Inter-Universal Teichmuller Theory (IUTCH).
This latter document, which shall be be referred to as ‘the Report’, is written in a style consistent with Mochizuki’s IUTT papers, and his other documents concerning IUTT.
As such, it can be difficult (at least for me) to extract concrete and precisely-defined mathematical results that aren’t mere analogies or metaphors. Rather than analogies,
one should strive to express the necessary ideas or objections in as precise terms as possible, and I argue that one should use category theory to clean up the parts of the arguments that are not actual number theory or arithmetic geometry.

つづく

http://rio2016.5ch.net/test/read.cgi/math/1586655469/25

26: 現代数学の系譜 雑談 ◆e.a0E5TtKE  [sage] 2020/04/12(日) 13:22:51 ID:hAg37Ryy

>>25

つづき

Conclusion
These notes have attempted to cast some of the examples proposed by Mochizuki to answer Scholze?Stix’s concerns in a more category-theoretic light. Ideally all discussions about the content of IUTT can be addressed in such precise terms,
rather than worry about things like “the risk that different people will “remember” different labeling appartuses [sic], which result in structurally non-equivalent mathematical structures”, Report (DfLb)
By replacing discussion of psychology and suggestive metaphors by rigorous definitions of all the categories in which objects live, and keeping track of forgetful functors, communication about IUTT can focus on the difficult mathematical content, rather than about whether or not objects need specific labels.

Addendum, 22 October
After giving the matter more thought, I came to the realisation that Mochizuki is using a subtly different definition of diagram than that which is commonly accepted. In the example ‘colimits and diagrams’ above, we had the diagram
(引用終り)
以上

http://rio2016.5ch.net/test/read.cgi/math/1586655469/26

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