https://ja.wikipedia.org/wiki/%E6%95%B0%E5%AD%A6%E3%83%96%E3%83%AC%E3%82%A4%E3%82%AF%E3%82%B9%E3%83%AB%E3%83%BC%E8%B3%9E 数学ブレイクスルー賞 ノーベル賞に数学部門がないこともあって、数学界では長らくフィールズ賞が最高権威の賞とされてきたが、これには40歳以下という年齢制限があり、賞金規模もノーベル賞には遠く及ばないものである。しかし近年ではクラフォード賞、ミレニアム賞、アーベル賞そして本賞と、優れた数学研究に高額賞金を与える学術賞が次々と創設されている。 受賞者 2022年 - 望月拓郎[2] https://breakthroughprize.org/Laureates/3/L3906 Takuro Mochizuki Kyoto University 2022 Breakthrough Prize in Mathematics For monumental work leading to a breakthrough in our understanding of the theory of bundles with flat connections over algebraic varieties, including the case of irregular singularities.
New Horizons in Mathematics Prize New Horizons in Mathematics Prizeは若手の研究者に授与される賞。賞金総額10万ドル。 2016年 ペーター・ショルツェ(辞退)
https://twitter.com/FumiharuKato/status/1514984048275619850 >@FumiharuKato >IUT理論における「宇宙」という用語が指し示すものが >望月さん独特のものであるのは本当です。その宇宙は >集合論の宇宙とも、グロンタンディーク宇宙とも >異なっていますので。 は何でそれに対するRobertsの質問 >@HigherGeometer >Replying to @FumiharuKato >Is there a formal definition of what Mochizuki means by the term? に答えないの？ Collasが頓珍漢な返答してるのに見て見ないふり？ 結局、「宇宙」という用語が何を意味しているのかさえ 誰も理解してないんじゃないの？ (deleted an unsolicited ad)
>>428 補足 >宇宙際Teichmuller理論 >[7] The Mathematics of Mutually Alien Copies: from Gaussian Integrals to Inter-universal Teichmuller Theory. PDF NEW !! (2020-12-23) >https://www.kurims.kyoto-u.ac.jp/~motizuki/Alien%20Copies,%20Gaussians,%20and%20Inter-universal%20Teichmuller%20Theory.pdf
この P3 （q-paramete） Let N be a fixed natural number > 1. Then the issue of bounding a given nonnegative real number h ∈ R?0 may be understood as the issue of showing that N ・ h is roughly equal to h, i.e., N ・ h “=〜” h [cf. §2.3, §2.4]. When h is the height of an elliptic curve over a number field, this issue may be understood as the issue of showing that the height of the [in fact, in most cases, fictional!] “elliptic curve” whose q-parameters are the N-th powers “qN ” of the q-parameters “q” of the given elliptic curve is roughly equal to the height of the given elliptic curve, i.e., that, at least from the point of view of [global] heights, qN “=〜” q [cf. §2.3, §2.4]. In order to verify the approximate relation qN “=〜” q, one begins by introducing two distinct - i.e., two “mutually alien” - copies of the conventional scheme theory surrounding the given initial Θ-data. Here, the intended sense of the descriptive “alien” is that of its original Latin root, i.e., a sense of abstract, tautological “otherness”.
>>490 >宇宙際Teichmuller理論 >[7] The Mathematics of Mutually Alien Copies: from Gaussian Integrals to Inter-universal Teichmuller Theory. PDF NEW !! (2020-12-23) >https://www.kurims.kyoto-u.ac.jp/~motizuki/Alien%20Copies,%20Gaussians,%20and%20Inter-universal%20Teichmuller%20Theory.pdf
上記より下記引用 ・Gaussian integral ∫ ∞ -∞ e-x2 dx = √π ・[archimedean and nonarchimedean] valuations ・Changes of universe as arithmetic changes of coordinates 関連
P6 § 1. Review of the computation of the Gaussian integral § 1.1. Inter-universal Teichm¨uller theory via the Gaussian integral The goal of the present paper is to pave the road, for the reader, from a state of complete ignorance of inter-universal Teichm¨uller theory to a state of general appreciation of the “game plan” of inter-universal Teichm¨uller theory by reconsidering the well-known computation of the Gaussian integral ∫ ∞ -∞ e-x2 dx = √π via polar coordinates from the point of view of a hypothetical high-school student who has studied one-variable calculus and polar coordinates, but has not yet had any exposure to multi-variable calculus.
P7 § 1.3. Introduction of identical but mutually alien copies
P12 § 2. Changes of universe as arithmetic changes of coordinates § 2.1. The issue of bounding heights: the ABC and Szpiro Conjectures
In this case, the height of a rational point may be thought of as a suitable weighted sum of the valuations of the q-parameters of the elliptic curve determined by the rational point at the nonarchimedean primes of potentially multiplicative reduction [cf. the discussion at the end of [Fsk], §2.2; [GenEll], Proposition 3.4]. Here, it is also useful to recall [cf. [GenEll], Theorem 2.1] that, in the situation of the ABC or Szpiro Conjectures, one may assume, without loss of generality, that, for any given finite set Σ of [archimedean and nonarchimedean] valuations of the rational number field Q,
In particular, when one computes the height of a rational point of the projective line minus three points as a suitable weighted sum of the valuations of the q-parameters of the corresponding elliptic curve, one may ignore, up to bounded discrepancies, contributions to the height that arise, say, from the archimedean valuations or from the nonarchimedean valuations that lie over some “exceptional” prime number such as 2.
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 ex
528 名前：ample of this sort of situation is the functor between Galois categories of ´etale coverings induced by a morphism of connected schemes. 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, 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”! (引用終り)
>>495 追加 >宇宙際Teichmuller理論 >[7] The Mathematics of Mutually Alien Copies: from Gaussian Integrals to Inter-universal Teichmuller Theory. PDF NEW !! (2020-12-23) >https://www.kurims.kyoto-u.ac.jp/~motizuki/Alien%20Copies,%20Gaussians,%20and%20Inter-universal%20Teichmuller%20Theory.pdf
＜”宇宙”について＞ これ、望月氏の 宇宙 ”relationships between universes”の説明が、下記にあるけど 結構独特で、世間的には、ちょっとズレている気がする。「複数の宇宙の使用は、1960年代の数学」（下記）とかね 一方、（後述の）ちょうど1960年代に、数学基礎論で強制法が考えられて、「強制法は集合論の宇宙 V をより大きい宇宙 V* に拡大する」（下記）みたいな話がある だから、数学基礎論の強制法を知っている人（あるいは、いまどき ”universe”の数学的意味を検索した人）は、IUTの”Inter-universal”という語法に違和感を感じる気がする 代数系なり代数幾何にしろ、集合論や圏論としても、せいぜい集合と類までで収まるはず。（圏論でも、”局所的に小さい (locally small) ”で収まるはず）”宇宙”は、普通は出てこない
＜下記に対訳を作ってみた＞ ＜原文＞ P27 § 2.10. Inter-universality: changes of universe as changes of coordinates One fundamental aspect of the links [cf. the discussion of §2.7, (i)] ? namely, the Θ-link and log-link ? that occur in inter-universal Teichm¨uller theory is their incompatibility with the ring structures of the rings and schemes that appear in their domains and codomains. In particular, when one considers the result of transporting an ´etale-like structure such as a Galois group [or ´etale fundamental group] across such a link [cf. the discussion of §2.7, (iii)], one must abandon the interpretation of such a Galois group as a group of automorphisms of some ring [or field] structure [cf. [AbsTopIII], Remark 3.7.7, (i); [IUTchIV], Remarks 3.6.2, 3.6.3], i.e., one must regard such a Galois group as an abstract topological group that is not equipped with any of the “labelling structures” that arise from the relationship between the Galois group and various scheme-theoretic objects. 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.
＜原文＞ 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. 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.
＜原文＞ 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, 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”!
＜原文＞ 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. Then, from the point of view of the length two substring 01 on the left, the digit “1” of this substring may be specified by means of its “coordinate relative to this substring”, namely, as the symbol to the far right of the substring 01. In a similar vein, from the point of view of the length two substring 10 on the right, the digit “1” of this substring may be specified by means of its “coordinate relative to this substring”, namely, as the symbol to the far left of the substring 10. On the other hand, neither of these specifications via “substring-based coordinate systems”is meaningful to the opposite length two substring; that is to say, only the solitary abstract symbol “1” is simultaneously meaningful, as a device for specifying the digit of interest, relative to both of the “substring-based coordinate systems”.
（>>527） >宇宙際Teichmuller理論 >[7] The Mathematics of Mutually Alien Copies: from Gaussian Integrals to Inter-universal Teichmuller Theory. PDF NEW !! (2020-12-23) >https://www.kurims.kyoto-u.ac.jp/~motizuki/Alien%20Copies,%20Gaussians,%20and%20Inter-universal%20Teichmuller%20Theory.pdf
