>>574 補足 www.maths.ed.ac.uk/~aar/papers/exoticsmooth.pdf Exotic Smoothness and Physics Differential Topology and Spacetime Models 2007 より 11.2.4 Geometric structures on %manifolds and exotic differential structures
To summarize, we hope to have provided support for the conjecture: Conjecture: The differential structures on a simply-connected compact, 4-manifold M are determined by the homotopy classes [M, BGl(T)+] and by the algebraic K-theoy K3(T) where T is the hyperfinite II1 factor C*-algebra. The classes in K3(T) are given by the geometric structure and/or a codamension-1 foliation of a homology 3-sphere in M determining the Akbulut cork of M.
From the physical point of view, this conjecture is very interesting because it connects the abstract theory of differential structures with well-known structures in physics like operator algebras or bundle theory. Perhaps such speculations may provide a geometrization of quantum mechanics or more. We close this section, and book, which these highly conjectural remarks.
579 名前：１３２人目の素数さん [2014/06/14(土) 05:55:46.39 ]
>>578 補足 >Conjecture: The differential structures on a simply-connected compact,
Epilogue Under the light of the various examples seen in this book, it seems reasonable to conjecture that, if a topological 4-manifold admits a smooth structure at all, then it might admit infinitely many. While gauge theory was how the door was opened on those vast unexplored realms, it might not be how these will be charted. We have seen that there are whole realms where the Seiberg-Witten invariants cannot help us. For example, the theory is blind on 4-manifolds that admit metrics of positive scalar curvature, on homology 4-spheres (which in particular leaves the smooth 4-dimensional Poincare conjecture with no solution in sight), on all manifolds with hi even, and in general on 4-manifolds that are far from complex. More, gauge theory offers only negative results (as in "two manifolds are not diffeomorphic"). Indeed, the field of 4-manifolds lacks enough techniques for obtaining affirmative results (as in "two manifolds are diffeomorphic"). Looking back, the only affirmative results we encountered came either from ad hoc constructions, from Kirby calculus, or from complex geometry. The field also lacks techniques for building enough examples, which might one day be organized into any sort of classification scheme. We are lost in an ever-growing jungle. Hence the final conclusion of this volume can only be that We know that we don't know. This only makes it all the more exciting ...
>>580 >We have seen that there are whole realms where the Seiberg-Witten invariants cannot help us. >For example, the theory is blind on 4-manifolds that admit metrics of positive scalar curvature, >on homology 4-spheres (which in particular leaves the smooth 4-dimensional Poincare conjecture with no solution in sight), >Hence the final conclusion of this volume can only be that >We know that we don't know. >This only makes it all the more exciting ...
>>566 関連PDF 新しい www.mathematik.uni-marburg.de/~agricola/material/Turin2014.pdf Dirac operators in Riemannian geometry Thomas Friedrich Torino February 2014 www.euro-math-soc.eu/node/4177 The European Mathematical Society School "CARNIVAL DIFFERENTIAL GEOMETRY" Start: Feb 24 2014 - 12:00 Short description of the event: The School is mainly targeted at PhD students and young researchers working in the field of differential geometry. The aim is to expose participants to current research and point to applications. At the heart of the school are two mini-courses, consisting of four 90' lectures each, that will start from the basics and reach the latest results. Lecturers: Ilka Agricola (Marburg): "Non integrable geometries, torsion, and holonomy" Thomas Friedrich (HU Berlin): "Dirac operators in Riemannian geometry"
588 名前：１３２人目の素数さん [2014/06/21(土) 22:35:26.55 ]
これが面白そう
arxiv.org/abs/1301.3628. On the origin of inflation by using exotic smoothness T Asselmeyer-Maluga 著 - ‎2013 - ‎引用元 3 - ‎2013/01/16
www.hindawi.com/journals/ahep/2014/867460/abs/ Inflation and Topological Phase Transition Driven by Exotic Smoothness Torsten Asselmeyer-Maluga1 and Jerzy Król2 Published 19 March 2014
589 名前：１３２人目の素数さん [2014/06/21(土) 22:41:27.25 ]
>>588 >On the origin of inflation by using exotic smoothness
ほい prac.us.edu.pl/~us2009/ XXXIII International Conference of Theoretical Physics MATTER TO THE DEEPEST: Recent Developments in Physics of Fundamental Interactions, USTROŃ'09
>>593 補足 上記Krol氏のpdfのP10/19に ”In the AdS/CFT correspondence the exotic 4-structures can cause the additional susy breaking (important for approaching the realistic QCD).”とあって ”一方、AdS/CFT対応から、突破口・・みたいなことを夢想しないでもない ”（ >>586）と関連しているねと やっぱ、exotic 4-structuresの突破口は、物理からやってくるのか
I will only address the question to know why R4 admits some non-standard, "exotic", differentiable structure. The question to know why there are infinitely many (and even uncountably many) examples simply requires an extension of the same kind of techniques.
There are several ways to construct examples of exotic R4. All use some deep results of topological nature due to Freedman and some deep results of differentiable nature due to Donaldson. I don't know if the results of Freedman have any physical interpretation. The Yang-Mills theory appears in Donaldson's results, on the differentiable side (one needs a differentiable structure to write partial differential equations).
Here is a sketch of one of the standard construction. （以下略）
1 Answer Milnor gives an example of two homeomorphic smooth manifolds whose tangent bundles are not isomorphic as vector bundles, see his ICM-1962 address, Corollary 1. I think, this was the first such example.
597 名前：１３２人目の素数さん [2014/06/28(土) 11:41:13.88 ]
>>593 補足 Gerbes en.wikipedia.org/wiki/Gerbe In mathematics, a gerbe is a construct in homological algebra and topology. Gerbes were introduced by Jean Giraud (Giraud 1971) following ideas of Alexandre Grothendieck as a tool for non-commutative cohomology in degree 2. They can be seen as a generalization of principal bundles to the setting of 2-categories. Gerbes provide a convenient, if highly abstract, language for dealing with many types of deformation questions especially in modern algebraic geometry. In addition, special cases of gerbes have been used more recently in differential topology and differential geometry to give alternative descriptions to certain cohomology classes and additional structures attached to them.
"Gerbe" is a French (and archaic English) word that literally means wheat sheaf.
”Riemann Hypothesis solved through physics-math in new cosmological model ” 検索でヒットしたが、半分ジョークです。 the Netherlands, independent cosmologist. 世の中いろんな考えがあると
vixra.org/why vixra.org/pdf/1308.0034v1.pdf Riemann Hypothesis solved through physics-math in new cosmological model: the Double Torus Hypothesis. Author: Dan Visser, Almere, the Netherlands, independent cosmologist. Date: July 23 2013
600 名前：１３２人目の素数さん [2014/06/29(日) 08:58:32.10 ]
>>598 アラン・コンヌによる非可換幾何学とRiemannで検索したら下記ヒット 修士論文らしいが、証明が一つもない(>>591関連) 文系の論文かとおもうくらい 日本では通らないように思うが、なにかのご参考に www.math.northwestern.edu/~jcutrone/Work/J.%20Cutrone%20Master%27s%20Thesis.pdf On Riemann’s 1859 paper “Über die Anzahl der Primzahlen unter einer gegebenen Grösse”and Its Consequences by Joseph W. Cutrone A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science Department of Mathematics New York University September 2005
数論 (代数的整数論) での両側剰余類の話が出てきた. 私自身は使ったことないが, Connes の数論での相転移論文にも出てきたことを思い出した. Hecke algebras, type III factors and phase transitions with spontaneous symmetry breaking in number theory www.alainconnes.org/docs/bostconnesscan.pdf という論文だが, 学生時代は学生時代できちんと読もうとして訳が分からず挫折した経緯があり, 結局あまり内容を把握していない. 時々 Twitter でネタにするので, この機会に軽く眺めてみようと思い, 自分用メモとして残しておく.
あと, 関係する話として新井先生の Infinite dimensional analysis and analytic number theory eprints3.math.sci.hokudai.ac.jp/637/ という話もある. 両方とも量子統計と数論の関係がテーマで, 分配関数が Riemann の ζ になる, という話. 新井先生の論文の方は直接的に Fock 空間と第 2 量子化作用素の話をしていて, 数学的にはこちらの方が簡単で読みやすい. ただ, 基本的には全く違う話なので両方読み比べた方が楽しいだろう.
www.alainconnes.org/docs/imufinal.pdf NONCOMMUTATIVE GEOMETRY AND THE RIEMANN ZETA FUNCTION Alain Connesの P12 ”The C algebra closure of HC is Morita equivalent (cf. M. Laca) to the crossed product C algebra,”
Morita？ 検索すると下記。あまり知られていないが、森田紀一さんすごいね
en.wikipedia.org/wiki/Morita_equivalence Morita equivalence In abstract algebra, Morita equivalence is a relationship defined between rings that preserves many ring-theoretic properties. It is named after Japanese mathematician Kiiti Morita who defined equivalence and a similar notion of duality in 1958.
en.wikipedia.org/wiki/Kiiti_Morita www.ams.org/notices/199706/morita.pdf Arhangelskii, A.V.; Goodearl, K.R.; Huisgen-Zimmermann, B. (June–July 1997), "Kiiti Morita 1915-1995" (PDF), Notices of the American Mathematical Society (Providence, RI: American Mathematical Society) 44 (6): 680–684
