[過去ﾛｸﾞ] Inter-universal geometry と ABC予想 (応援スレ) 49 (1002ﾚｽ)
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551(2): 現代数学の系譜 雑談 ◆yH25M02vWFhP 2020/10/18(日)12:54 ID:ZLSkSSTT(16/27)調 AAS
>>542
>そうか、この[Alien]っていうのが、重要な論文なんだね〜（＾＾
"universe"の説明が詳しいね
以下抜粋する
（参考）
http://www.kurims.kyoto-u.ac.jp/~motizuki/Alien%20Copies,%20Gaussians,%20and%20Inter-universal%20Teichmuller%20Theory.pdf
[7] The Mathematics of Mutually Alien Copies: from Gaussian Integrals to Inter-universal Teichmuller Theory. PDF
NEW !! (2020-04-04)
(抜粋)
Contents
§ 2. Changes of universe as arithmetic changes of coordinates
§ 2.10. Inter-universality: changes of universe as changes of coordinates
Ｐ28
It is precisely this state of affairs that results in
the quite central role played in inter-universal Teichm¨uller theory by results in
[mono-]anabelian geometry, i.e., by results concerned with reconstructing
various scheme-theoretic structures from an abstract topological group that “just
happens” to arise from scheme theory as a Galois group/´etale fundamental group.
In this context, we remark that it is also this state of affairs that gave rise to the term
“inter-universal”: That is to say, the notion of a “universe”, as well as the use of
multiple universes within the discussion of a single set-up in arithmetic geometry, already
occurs in the mathematics of the 1960’s, i.e., in the mathematics of Galois categories
and ´etale topoi associated to schemes. On the other hand, in this mathematics of the
Grothendieck school, typically one only considers relationships between universes ? i.e.,
between labelling apparatuses for sets ? that are induced by morphisms of schemes, i.e.,
in essence by ring homomorphisms. The most typical example of this sort of situation
is the functor between Galois categories of ´etale coverings induced by a morphism of
connected schemes.
つづく
552(1): 現代数学の系譜 雑談 ◆yH25M02vWFhP 2020/10/18(日)12:54 ID:ZLSkSSTT(17/27)調 AAS
>>551
つづき
By contrast, the links that occur in inter-universal Teichm¨uller
theory are constructed by partially dismantling the ring structures of the rings in their
domains and codomains [cf. the discussion of §2.7, (vii)], hence necessarily result in
much more complicated relationships between the universes ? i.e., between the labelling apparatuses for sets ? that are adopted in the Galois categories that occur in the domains and codomains of these links, i.e., relationships that do not respect the various labelling apparatuses for sets that arise
from correspondences between the Galois groups that appear and the respective
ring/scheme theories that occur in the domains and codomains of the links.
That is to say, it is precisely this sort of situation that is referred to by the term
“inter-universal”. Put another way,
a change of universe may be thought of [cf. the discussion of §2.7, (i)] as
a sort of abstract/combinatorial/arithmetic version of the classical notion
of a “change of coordinates”.
In this context, it is perhaps of interest to observe that, from a purely classical point of
view, the notion of a [physical] “universe” was typically visualized as a copy of Euclidean
three-space. Thus, from this classical point of view,
Ｐ29
a “change of universe” literally corresponds to a “classical change of the coordinate system ? i.e., the labelling apparatus ? applied to label points in
Euclidean three-space”!
Indeed, from an even more elementary point of view, perhaps the simplest example of the
essential phenomenon under consideration here is the following purely combinatorial
phenomenon: Consider the string of symbols
010
? i.e., where “0” and “1” are to be understood as formal symbols.
つづく
557: 現代数学の系譜 雑談 ◆yH25M02vWFhP 2020/10/18(日)13:03 ID:ZLSkSSTT(22/27)調 AAS
>>551 補足
"universe"の説明:
”http://www.kurims.kyoto-u.ac.jp/~motizuki/Alien%20Copies,%20Gaussians,%20and%20Inter-universal%20Teichmuller%20Theory.pdf
[7] The Mathematics of Mutually Alien Copies: from Gaussian Integrals to Inter-universal Teichmuller Theory. PDF
NEW !! (2020-04-04)
(抜粋)
Contents
§ 2. Changes of universe as arithmetic changes of coordinates
§ 2.10. Inter-universality: changes of universe as changes of coordinates
Ｐ28
It is precisely this state of affairs that results in
the quite central role played in inter-universal Teichm¨uller theory by results in
[mono-]anabelian geometry, i.e., by results concerned with reconstructing
various scheme-theoretic structures from an abstract topological group that “just
happens” to arise from scheme theory as a Galois group/´etale fundamental group.
In this context, we remark that it is also this state of affairs that gave rise to the term
“inter-universal”: That is to say, the notion of a “universe”, as well as the use of
multiple universes within the discussion of a single set-up in arithmetic geometry, already
occurs in the mathematics of the 1960’s, i.e., in the mathematics of Galois categories
and ´etale topoi associated to schemes. On the other hand, in this mathematics of the
Grothendieck school, typically one only considers relationships between universes - i.e.,
between labelling apparatuses for sets - that are induced by morphisms of schemes, i.e.,
in essence by ring homomorphisms. The most typical example of this sort of situation
is the functor between Galois categories of ´etale coverings induced by a morphism of
connected schemes.”
(引用終り)
"universe"は
ラベルに関連した ある種のGalois category ってことでしょうか？（＾＾；
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