http://www.math.titech.ac.jp/~taguchi/bib/ List of (pre)Publications [22] A relation between some finiteness conjectures on Galois representations --- a brief introduction to the Fontaine-Mazur Conjectures, ( Proceedings of the Number Theory Camp held at Pohang Unversity of Science and Technology, January, 2004, pp.34--43 )
http://www.math.titech.ac.jp/~taguchi/bib/camp.pdf A relation between some finiteness conjectures on Galois representations ? a brief introduction to the Fontaine-Mazur Conjectures Yuichiro Taguchi ( Proceedings of the Number Theory Camp held at Pohang Unversity of Science and Technology, January, 2004, pp.34--43 ) 0013現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2020/03/18(水) 08:09:32.65ID:vfR9jLHlhttp://www.math.titech.ac.jp/~taguchi/nihongo/index.html Yuichiro TAGUCHI 田口雄一郎(東京工業大学) http://www.math.titech.ac.jp/~taguchi/nihongo/bunsho.html 数学関係の文章 アーベル多様体と数論 ( 九州大学公開講座 「現代数学入門」 ( 2013年 7月 28日 ) の講演ノート ) 類体論 (「整数論札幌夏の学校」 ( 2006年8月28日 ) に於ける講義ノート ) 有理点の整数論 ( 高校生 ( または一般の方 ) 向け講義ノート ) Fermat の最終定理を巡る数論 ( 『日本の科学者』 vol.40, no.3 ) Artin 導手の誘導公式 ( 2001年度 日本数学会 秋季大会 代数学一般講演アブストラクト集 ) Mod p Galois 表現について ( 特に像が可解の場合 ) ( RIMS講究録 1154 ) abc予想の話 ( 昔、北大理学部 HP の「サイエンストピックス」に掲載されたもの ) Fontaine-Mazur予想の紹介 ( RIMS講究録 1097 ) Fermatの最終定理 ( Wilesによる証明の一般向け解説 ) eとpiの超越性 ( Hilbertの証明 ) p進数 ( 初心者向けの解説 )
”26 Perfectoid algebraic geometry as an example of anabelomorphy” というのがあって、”Perfectoid”をちょっと調べてみようということです
Inter-universal geometry と ABC予想 43 https://rio2016.5ch.net/test/read.cgi/math/1577401302/299 299 自分:132人目の素数さん[] 投稿日:2020/03/28(土) 18:23:32.21 ID:MRwZqC/h [1/3] メモ https://arxiv.org/pdf/2003.01890.pdf On Mochizuki’s idea of Anabelomorphy and its applications Kirti Joshi 20200305 (抜粋) P61 26 Perfectoid algebraic geometry as an example of anabelomorphy A detailed treatment of assertions of this section will be provided in [DJ] where we establish many results in parallel with classical anabelian geometry. In particular this suggests that the filtered absolute Galois group of a perfectoid field of characteristic zero has non-trivial outer automorphisms which does not respect the ring structure of K. This is the perfectoid analog of the fact that the absolute Galois group GK of a p-adic field K has autormorphisms which do not preserve the ring structure of K. Now let me explain that the main theorem of [Sch12b] provides the perfectoid analog of anabelomorphy (in all dimensions).
In some sense Scholze’s proof of the weight monodromy conjecture does precisely this: Scholze replaces the original hypersurface by a (perfectoid) nabelomorphic hypersurface for which the conjecture can be established by other means. <References> [DJ] Taylor Dupuy and Kirti Joshi. Perfectoid anbelomorphy. 0036現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2020/03/29(日) 17:16:53.55ID:PhmwLbdrhttps://www.ms.u-tokyo.ac.jp/~mieda/workshop201305.html p可除群とそのモジュライ空間に関する最近の進展
参考文献 P. Scholze and J. Weinstein, Moduli of p-divisible groups, arXiv:1211.6357 P. Scholze, Perfectoid spaces, Publ. Math. de l'IHES 116 (2012), no. 1, 245--313. Also available here L. Fargues and J.-M. Fontaine, Courbes et fibres vectoriels en theorie de Hodge p-adique, preprint L. Fargues and J.-M. Fontaine, Vector bundles on curves and p-adic Hodge theory, preprint J. Weinstein, Semistable models for modular curves of arbitrary level, arXiv:1010.4241v2 M. Rapoport and Th. Zink, Period spaces for p-divisible groups, Annals of Mathematics Studies, vol. 141, Princeton University Press, Princeton, NJ, 1996. 0037現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2020/03/29(日) 17:21:12.00ID:PhmwLbdr メモ https://repository.kulib.kyoto-u.ac.jp/dspace/bitstream/2433/243673/1/B64-17.pdf RIMS Kokyuroku Bessatsu B64 (2017), 219?253 Perfectoid空間論の基礎 (Foundations for theory of perfectoid spaces) By 津嶋貴弘 (Takahiro TSUSHIMA)*
https://en.wikipedia.org/wiki/Random_close_pack Random close pack (抜粋) Random close packing (RCP) is an empirical parameter used to characterize the maximum volume fraction of solid objects obtained when they are packed randomly. For example, when a solid container is filled with grain, shaking the container will reduce the volume taken up by the objects, thus allowing more grain to be added to the container. In other words, shaking increases the density of packed objects. But shaking cannot increase the density indefinitely, a limit is reached, and if this is reached without obvious packing into a regular crystal lattice, this is the empirical random close-packed density.
Experiments and computer simulations have shown that the most compact way to pack hard perfect spheres randomly gives a maximum volume fraction of about 64%, i.e., approximately 64% of the volume of a container is occupied by the spheres. It seems as if because it is not possible to precisely define 'random' in this sense it is not possible to give an exact value.[1] The random close packing value is significantly below the maximum possible close-packing of (equal sized) hard spheres into a regular crystalline arrangements, which is 74.04% -- both the face-centred cubic (fcc) and hexagonal close packed (hcp) crystal lattices have maximum densities equal to this upper limit. 0042132人目の素数さん2020/03/30(月) 16:24:37.22ID:SsupeAn8 おっちゃんです。 新型コロナもはやく収まらないモノかね。 0043132人目の素数さん2020/03/30(月) 16:25:39.83ID:SsupeAn8 それじゃ、おっちゃんもう寝る。 0044現代数学の系譜 雑談 ◆e.a0E5TtKE 2020/03/30(月) 18:39:25.99ID:zICzxEKY おっちゃん、どうも、スレ主です。 同意です。おやすみなさい(^^; 0045酒浸り2020/03/30(月) 21:51:31.26ID:Y+NgZsAC 間違って踏んで仕舞った。未だに何故、 Surreal(1-0.999…)=0 & Game(1-0.999…)=ε≠0 に成るか 理由が分からない。Gameに順序性と演算規則性を補完してSurrealが構築されるならば 益々以て上記式のεはSurrealではないGameにしか成り得ない筈なのに、ε自体はSurrealだ!
分からない人に言うなら、これは実数と超実数。 Real(1-a)=0 & Hypereal(1-0.999…)=a≠0 ならば 此の a はRealではないHyperealにしか成り得ない。
厳密な定義 「超実数#超実体」も参照 X はチホノフ空間(英語版)(T3?-空間とも)とし、C(X) で X 上定義される実数値連続函数全体の成す線型環を表す。C(X) の素イデアル P に対し、剰余線型環 A := C(X)/P は、定義により環として整域を成す実線型環で、全順序付けられていると考えることができる。 A の商体 F が準超実体 (super-real field) であるとは、F が真に実数体 ? を含む?ゆえに F は ? に順序同型 (order isomorphic) でない?ときに言う。
https://en.wikipedia.org/wiki/Hahn_series Hahn series (抜粋) In mathematics, Hahn series (sometimes also known as Hahn?Mal'cev?Neumann series) are a type of formal infinite series. They are a generalization of Puiseux series (themselves a generalization of formal power series) and were first introduced by Hans Hahn in 1907[1] (and then further generalized by Anatoly Maltsev and Bernhard Neumann to a non-commutative setting). They allow for arbitrary exponents of the indeterminate so long as the set supporting them forms a well-ordered subset of the value group (typically {Q} or {R} ). Hahn series were first introduced, as groups, in the course of the proof of the Hahn embedding theorem and then studied by him as fields in his approach to Hilbert's seventeenth problem.
Contents 1 Formulation 2 Properties 2.1 Properties of the valued field 2.2 Algebraic properties 3 Summable families 3.1 Summable families 3.2 Evaluating analytic functions 4 Hahn?Witt series 0059現代数学の系譜 雑談 ◆e.a0E5TtKE 2020/03/31(火) 11:48:04.23ID:YIE+6BeO>>58 関連
英語のページが、実に充実しているね https://ja.wikipedia.org/wiki/%E5%BD%A2%E5%BC%8F%E7%9A%84%E5%86%AA%E7%B4%9A%E6%95%B0 形式的冪級数 https://en.wikipedia.org/wiki/Formal_power_series Formal power series (抜粋) Contents 1 Introduction 2 The ring of formal power series 2.1 Definition of the formal power series ring 2.1.1 Ring structure 2.1.2 Topological structure 2.1.3 Alternative topologies 2.2 Universal property 3 Operations on formal power series 3.1 Power series raised to powers 3.2 Inverting series 3.3 Dividing series 3.4 Extracting coefficients 3.5 Composition of series 3.5.1 Example 3.6 Composition inverse 3.7 Formal differentiation of series 4 Properties 4.1 Algebraic properties of the formal power series ring 4.2 Topological properties of the formal power series ring 4.3 Weierstrass preparation 5 Applications 6 Interpreting formal power series as functions 7 Generalizations 7.1 Formal Laurent series 7.1.1 Formal residue 7.2 The Lagrange inversion formula 7.3 Power series in several variables 7.3.1 Topology 7.3.2 Operations 7.3.3 Universal property 7.4 Non-commuting variables 7.5 On a semiring 7.6 Replacing the index set by an ordered abelian group 8 Examples and related topics 0060現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2020/03/31(火) 20:55:54.25ID:zp6RcyFj>>46 > 11 ゲーム
https://en.wikipedia.org/wiki/Combinatorial_game_theory Combinatorial game theory (抜粋) Combinatorial game theory (CGT) is a branch of mathematics and theoretical computer science that typically studies sequential games with perfect information.
History In the 1960s, Elwyn R. Berlekamp, John H. Conway and Richard K. Guy jointly introduced the theory of a partisan game, in which the requirement that a play available to one player be available to both is relaxed. Their results were published in their book Winning Ways for your Mathematical Plays in 1982. However, the first work published on the subject was Conway's 1976 book On Numbers and Games, also known as ONAG, which introduced the concept of surreal numbers and the generalization to games. On Numbers and Games was also a fruit of the collaboration between Berlekamp, Conway, and Guy. 0062132人目の素数さん2020/04/01(水) 04:24:25.41ID:+nGXqagc>>60 長年って具体的に何年?。 コピペ作業始めてから?。 0063現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2020/04/01(水) 07:26:13.42ID:RqQA8SNl>>62 良い質問ですね ガロアスレ 1 http://uni.2ch.net/test/read.cgi/math/1328016756/1- 初代スレ 1 名前:名無しさん[] 投稿日:2012/01/31(火) 22:32:36.78 ID:LTM9xtnu [1/10]
https://ja.wikipedia.org/wiki/%E3%83%95%E3%82%A3%E3%83%BC%E3%83%AB%E3%82%BA%E8%B3%9E フィールズ賞 (抜粋) 1990年(京都) ヴォーン・ジョーンズ(Vaughan F. R. Jones, 1952年 - ) ニュージーランド 「 for his discovery of an unexpected link between the mathematical study of knots ? a field that dates back to the 19th century ? and statistical mechanics, a form of mathematics used to study complex systems with large numbers of components. エドワード・ウィッテン(Edward Witten, 1951年 - )アメリカ合衆国の旗 アメリカ合衆国 「 proof in 1981 of the positive energy theorem in general relativity
P20 (iii) は不分岐な v では Weil 予想(Deligne の定理),一般には未解決のウェイト・モノドロミー予想の帰結であり,(iv) は (i) と同様の制限下でL 進エター ルコホモロジーの構成の帰結である.
これらの予想は主に代数幾何学における重さの哲学を反映するものであるから,代数幾何学を通して証明され るものが多いが,保型表現の解析的理論がもっとも強力に定性的な結果をもたらすものとしては,有限性がある. 代数的な Π, R の導手を,すべての有限素点における Πv,WD(Rv) の導手 pmv の積 ?vpmv で定めると,これは有 限積で OF のイデアルとなり,Π と R が対応すれば互いの導手は等しい.Π の導手は Hecke 指標のモジュラス・ 保型形式のレベルにあたるものである. 0093現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2020/04/02(木) 23:01:01.92ID:kD9YEDnI 以上、”ウェイト・モノドロミー予想”とは? について、調べた むずいww(^^; 0094現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2020/04/03(金) 00:16:16.17ID:DyKRdYgChttps://en.wikipedia.org/wiki/Perfectoid_space Perfectoid space (抜粋) In mathematics, perfectoid spaces are adic spaces of special kind, which occur in the study of problems of "mixed characteristic", such as local fields of characteristic zero which have residue fields of characteristic prime p.
A perfectoid field is a complete topological field K whose topology is induced by a nondiscrete valuation of rank 1, such that the Frobenius endomorphism Φ is surjective on K°/p where K° denotes the ring of power-bounded elements.
Perfectoid spaces may be used to (and were invented in order to) compare mixed characteristic situations with purely finite characteristic ones. Technical tools for making this precise are the tilting equivalence and the almost purity theorem. The notions were introduced in 2012 by Peter Scholze.[1]
Contents 1 Tilting equivalence 1.1 Almost purity theorem 2 See also 0095現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2020/04/03(金) 00:43:32.96ID:DyKRdYgC <ウェイト・ モノドロミー予想>
1.伊藤哲史先生>>87-88 「Langlands 対応などへの応用上は, 残された混標数の場合が重要であると考えら れる. しかし, この場合は, 様々な部分的な結果はあるものの, 一般には未解決である」 2.Perfectoid space >>94 「In mathematics, perfectoid spaces are adic spaces of special kind, which occur in the study of problems of "mixed characteristic"」 で、"mixed characteristic"混標数の性質の良い空間を作って そこで、ウェイト・ モノドロミー予想を部分解決したってことかな?(>>31) 3.「ウェイト・モノドロミー予想(weight-monodromy conjecture)とは,Deligneにより1970年の国際数学者会議において提出された予想である([D1]).」 「これは,完備離散付値体上の固有かつ滑らかな代数多様体のl進コホモロジーに定義されたモノドロミー・フィルトレーションの重み(weight)が純であるという予想として定式化されており,」 「"Deligneによるモノドロミー・フィルトレーションの純性予想"とも呼ばれている.」 か。さっぱり分からんが、下記 Kirti Joshi先生のPDFとの関連はついたかな(^^
(参考) https://arxiv.org/pdf/2003.01890.pdf On Mochizuki’s idea of Anabelomorphy and its applications Kirti Joshi 20200305 (抜粋) P61 26 Perfectoid algebraic geometry as an example of anabelomorphy A detailed treatment of assertions of this section will be provided in [DJ] where we establish many results in parallel with classical anabelian geometry. In particular this suggests that the filtered absolute Galois group of a perfectoid field of characteristic zero has non-trivial outer automorphisms which does not respect the ring structure of K.
つづき This is the perfectoid analog of the fact that the absolute Galois group GK of a p-adic field K has autormorphisms which do not preserve the ring structure of K. Now let me explain that the main theorem of [Sch12b] provides the perfectoid analog of anabelomorphy (in all dimensions). Suppose that K is a complete perfectoid field of characteristic zero. Let X/K be a perfectoid variety over K, which I assume to be reasonable, to avoid inane pathologies. Let π1(X/K) be its ´etale site. Let Xb/Kb be its tilt. Then the main theorem of [Sch12b] asserts that Theorem 26.1. The tilting functor provides an equivalence of categories π1(X/K) → π1(Xb/Kb). If L is any untilt of Kb and Y/L is any perfectoid variety with tilt Yb/Lb =〜 Xb/Kb. Then one has π1(X/K) =〜 π1(Y/L) and in particular X/K and Y/L are perfectoid anabelomorphs of each other. In particular one says that X/K and Y/L are anabelomorphic perfectoid varieties over anabelomorphic perfectoid fields K ←→ L. Thus one can envisage proving theorems about X/K by picking an anabelomorphic variety in the anabelomorphism class which is better adapted to the properties (of X/K) which one wishes to study. In some sense Scholze’s proof of the weight monodromy conjecture does precisely this: Scholze replaces the original hypersurface by a (perfectoid) anabelomorphic hypersurface for which the conjecture can be established by other means. (引用終り) 以上 0097現代数学の系譜 雑談 ◆e.a0E5TtKE 2020/04/03(金) 13:42:18.87ID:eln2Kr6c メモ貼る http://imetrics.co.jp/academy/Topos-presentation.pdf iMetrics Academy Press AI 時代の数学 (層・圏論・そしてトポスへの道のり) 2019 SPRING 2019. 6. 21 数学とは言語 Author: Sage Kusafusa 草房誠二郎 Production:iMetrics.co.jp (Japanese/ ENGLISH) http://imetrics.co.jp/academy/Topos-presentation-print.pdf