1 Introduction: description of the problem and some results on polynomials of one variable as heating. 2 Extensions body : algebraicity , algebraically closed , Lemma primitive element. 3 Minimal polynomial , combined elements. 4 Finite : Frobenius automorphisms , extensions of finite fields. 5 Group theory I: basic results , order of an element , Lagrange's theorem . 6 Galois : Lemma Artin , Galois groups , Galois . 7 Group Theory II: solvable groups , non solvability of the symmetric group Sn for n greater than or equal to 5. 8 Cyclotomy I: General cyclotomic extension, Kummer theory 9 Theorems of solvability of Galois : test solvability theorem, Galois degree p 10 Reduction mod p : calculating Galois groups of polynomials with integer coefficients by reduction modulo p 11 Supplements : cyclotomy Q ( through the reduction modulo p) and other applications
由来 この言葉はリンカーンのオリジナルではない[要出典]。今知られている一番古いものはジョン・ウィクリフ（1320年頃 - 1384年）が聖書を英訳した著作の序言に "This Bible is for the government of the people, by the people, and for the people"（「この聖書は人民の、人民による、人民のための統治に資するものである」）とあるのに始まる[要出典]。 その言葉を引用したウェブスター（1782年 - 1852年、雄弁家、政治家）、さらにそれを引用したパーカー（1810年 - 1860年、牧師、雄弁家、黒人解放運動家）と順次引用され、リンカーンの引用に至る[要出典]。
従来にない数学と物理の概念について公開しています。ただし，これらの概念は世の中でまだ認められたものではないのでこのような考え方もあるということを提言しているものです。 （The complex vector and the second imaginary number by Katsuhiro Manabe) Author:眞鍋克裕 1954年生まれ 東京都出身 東京工業大学理学部物理学科卒 職業：国家公務員
By classical results due to Nikulin, Mukai, Xiao and Kondo in the 1980's and 90's, the finite symplectic automorphism groups of K3 surfaces are always subgroups of the Mathieu group M24. This is a simple sporadic group of order 244823040. However, also by results due to Mukai, each such automorphism group has at most 960 elements and thus is by orders of magnitude smaller than M24. On the other hand, according to a recent observation by Eguchi, Ooguri and Tachikawa, the elliptic genus of K3 surfaces seems to contain a mysterious footprint of an action of the entire group M24: If one decomposes the elliptic genus into irreducible characters of the N=4 superconformal algebra, which is natural in view of superconformal field theories (SCFTs) associated to K3, then the coefficients of the so-called non-BPS characters coincide with the dimensions of representations of M24.
In joint work with Dr. Anne Taormina, first results of which are presented in Anne Taormina, Katrin Wendland, The overarching finite symmetry group of Kummer surfaces in the Mathieu group M24; JHEP 1308:152 (2013); arXiv:1107.3834 [hep-th]
we develop techniques which eventually should overcome the above-mentioned "order of magnitude problem": For Kummer surfaces which carry the Kahler class that is induced by their underlying complex torus, we find methods that improve the classical techniques due to Mukai and Kondo, and we give a construction that allows us to combine the finite symplectic symmetry groups of several Kummer surfaces to a larger group. Thereby, we generate the so-called overarching finite symmetry group of Kummer surfaces, a group of order 40320, thus already mitigating the "order of magnitude problem".
In 4 dimensions it is not known whether there are any exotic smooth structures on the 4-sphere. The statement that they do not exist is known as the "smooth Poincare conjecture", and is discussed by Michael Freedman, Robert Gompf, and Scott Morrison et al. (2010) who say that it is believed to be false.
Some candidates for exotic 4-spheres are given by Gluck twists (Gluck 1962). These are constructed by cutting out a tubular neighborhood of a 2-sphere S in S4 and gluing it back in using a diffeomorphism of its boundary S2×S1. The result is always homeomorphic to S4. But in most cases it is unknown whether or not the result is diffeomorphic to S4. (If the 2-sphere is unknotted, or given by spinning a knot in the 3-sphere, then the Gluck twist is known to be diffeomorphic to S4, but there are plenty of other ways to knot a 2-sphere in S4.)
Akbulut (2009) showed that a certain family of candidates for 4-dimensional exotic spheres constructed by Cappell and Shaneson are in fact standard.
plus.maths.org/content/richard-elwes Submitted by mf344 on January 12, 2011 Exotic spheres, or why 4-dimensional space is a crazy place by Richard Elwes 抜粋 The weird world of four dimensions So, is the smooth Poincare conjecture true? Most mathematicians lean towards the view that it is probably false, and that 4-dimensional exotic spheres are likely to exist. The reason is that 4-dimensional space is already known to be a very weird place, where all sorts of surprising things happen. A prime example is the discovery in 1983 of a completely new type of shape in 4-dimensions, one which is completely unsmoothable.
As discussed above, a square is not a smooth shape because of its sharp corners. But it can be smoothed. That is to say, it is topologically identical to a shape which is smooth, namely the circle. In 1983, however, Simon Donaldson discovered a new class of 4-dimensional manifolds which are unsmoothable: they are so full of essential kinks and sharp edges that there is no way of ironing them all out.
Beyond this, it is not only spheres which come in exotic versions. It is now known that 4-dimensional space itself (or R4) comes in a variety of flavours. There is the usual flat space, but alongside it are the exotic R4s. Each of these is topologically identical to ordinary space, but not differentially so. Amazingly, as Clifford Taubes showed in 1987, there are actually infinitely many of these alternative realities. In this respect, the fourth dimension really is an infinitely stranger place than every other domain: for all other dimensions n, there is only ever one version of Rn. Perhaps after all, the fourth dimension is the right mathematical setting for the weird worlds of science fiction writers' imaginations.
Exotic sphere home page on the home page of Andrew Ranicki. Assorted source material relating to exotic spheres. www.maths.ed.ac.uk/~aar/exotic.htm
www.maths.ed.ac.uk/~aar/exotic.htm Exotic spheres An exotic sphere is an n-dimensional differentiable manifold which is homeomorphic but not diffeomorphic to the standard n-sphere Sn. The articles on exotic spheres on the Wikipedia and the Manifold Atlas Project. On manifolds homeomorphic to the 7-sphere, by J.Milnor, Ann. of Math. (2) 64, 399--405 (1956) www.maths.ed.ac.uk/~aar/papers/exotic.pdf Hedrick Lectures on Differential Topology by J. Milnor (1965) 略 The structure set by A.Ranicki, Chapter 13 of Algebraic and Geometric Surgery, Oxford (2002) Exotic spheres and curvature by M.Joachim and D.J.Wraith, Bull. A.M.S. 45, 595--616 (2008) A minimal Brieskorn 5-sphere in the Gromoll-Meyer sphere and its applications. by C.Duran and T.Puttmann, Michigan Math. J. 56, 419--451 (2008) On the work of Michel Kervaire in surgery and knot theory by A.Ranicki, Slides of lecture given at Kervaire memorial symposium, Geneva, 10-12 February, 2009. Addendum Exotic spheres and the Kervaire invariant (8 May 2009) An introduction to exotic spheres and singularities by A.Ranicki, Slides of lecture given in Edinburgh, 4 May 2012 Dusa McDuff and Jack Milnor (Somewhere in Scotland, 2011)
>The statement that they do not exist is known as the "smooth Poincare conjecture", and is discussed by Michael Freedman, Robert Gompf, and Scott Morrison et al. (2010) who say that it is believed to be false.
arxiv.org/abs/0906.5177 Freedman, Michael; Gompf, Robert; Morrison, Scott; Walker, Kevin (2010), "Man and machine thinking about the smooth 4-dimensional Poincare conjecture", Quantum Topology 1 (2): 171?208, arXiv:0906.5177
で、コンピュータパワーで、結び目理論で計算したらしい 5.3 Results Computing the two-variable polynomial for K2 took approximately 4 weeks on a dual core AMD Opteron 285 with 32 gb of RAM. At this point, we haven’t been able to do the calculation for K3 . With the current version of the program, after about two weeks the program runs out of memory and aborts.
Early inflationary models Inflation was proposed in January 1980, by Alan Guth as a mechanism for resolving these problems.[41][42] At the same time, Starobinsky argued that quantum corrections to gravity would replace the initial singularity of the universe with an exponentially expanding deSitter phase.[43] In October 1980, Demosthenes Kazanas suggested that exponential expansion could eliminate the particle horizon and perhaps solve the horizon problem,[44] while Sato suggested that an exponential expansion could eliminate domain walls (another kind of exotic relic).[45] In 1981 Einhorn and Sato[46] published a model similar to Guth's and showed that it would resolve the puzzle of the magnetic monopole abundance in Grand Unified Theories.
1981年にアラン・ハーヴェイ・グースとほぼ同時期に、インフレーション宇宙論を提唱した。 この理論の最初の論文投稿者は佐藤であるが[4][5]、グースは1980年1月に佐藤と同様のインフレーションモデルをスタンフォード大学のセミナーで発表している[6]。 また、Alexei Starobinskyも1979年に同様のモデルについてのアイデアを示し[7]、1980年に論文を発表している[8]。なお、“インフレーション”という言葉を最初に用いたのはグースである[5]。 (注：佐藤の論文は、”Recieved 1980 September 9;in original form 1980 February 21”、Alan Guthは”Recieved 11 August 1980”)
K3曲面って面白いね ja.wikipedia.org/wiki/K3%E6%9B%B2%E9%9D%A2#CITEREFBrown2007 K3曲面は、複素トーラスとともに 2次元のカラビ・ヤウ多様体である。ほとんどの複素K3曲面は代数的ではない。このことは、K3曲面を多項式により定義される曲面として射影空間へ埋め込むことができないことを意味する。 Andre Weil (1958) は、これらに 3人の代数幾何学者の名前、エルンスト・クンマー(Ernst Kummer)、エーリッヒ・ケーラー（英語版）(Erich Kahler)、小平邦彦(Kunihiko Kodaira)にちなむと同時に、 （当時は未踏の山であった）カシミールの山であるK2にちなみK3曲面と名付けた。 “ Dans la seconde partie de mon rapport, il s'agit des varietes kahleriennes dites K3, ainsi nommees en l'honneur de Kummer, Kahler, Kodaira et de la belle montagne K2 au Cachemire ” ?Andre Weil (1958, p.546)の「K3曲面」という名前の理由について引用
クヌースによるアルゴリズムに関する著作 The Art of Computer Programming のシリーズはプログラミングに携わるものの間ではあまりにも有名[3]。 アルゴリズム解析と呼ばれる分野を開拓し、計算理論の発展に多大な貢献をしている。その過程で漸近記法で計算量を表すことを一般化させた。
理論計算機科学への貢献とは別に、コンピュータによる組版システム TeX とフォント設計システム METAFONT の開発者でもあり、Computer Modern という書体ファミリも開発した。
具体的な個人名が記されていないね en.wikipedia.org/wiki/Microsoft_Research The research division of Microsoft was created in 1991 and employs computer scientists, physicists, engineers, and mathematicians, including Turing Award winners, Fields Medal winners, MacArthur Fellows, and Dijkstra Prize winners. These 1,100 scientists and engineers collaborate with academic, government, and industry researchers to advance the state of the art of computing, and solve difficult world problems through technological innovation.
en.wikipedia.org/wiki/Banff_International_Research_Station The Banff International Research Station (BIRS) for Mathematical Innovation and Discovery was established in 2003.[1] It provides an independent research institute for the mathematical sciences in North America, a counterpart to the Mathematical Research Institute of Oberwolfach in Europe.[2] The research station, commonly known by its acronym, "BIRS", hosts over 2000 international scientists each year to undertake research collaboration in the mathematical sciences.[3]
In physics and mathematics, a sequence of n numbers can also be understood as a location in n-dimensional space. When n = 7, the set of all such locations is called 7-dimensional Euclidean space. Seven-dimensional elliptical and hyperbolic spaces are also studied, with constant positive and negative curvature.
Abstract seven-dimensional space occurs frequently in mathematics, and is a perfectly legitimate construct. Whether or not the real universe in which we live is somehow seven-dimensional (or indeed higher) is a topic that is debated and explored in several branches of physics, including astrophysics and particle physics, but it does not matter for mathematics.
Formally, seven-dimensional Euclidean space is generated by considering all real 7-tuples as 7-vectors in this space. As such it has the properties of all Euclidian spaces, so it is linear, has a metric and a full set of vector operations. In particular the dot product between two 7-vectors is readily defined, and can be used to calculate the metric. 7 × 7 matrices can be used to describe transformations such as rotations which keep the origin fixed.
A distinctive property is that a cross product can be defined only in three or seven dimensions (see seven-dimensional cross product). This is due to the existence of quaternions and octonions.
>>425 補足 Octonion＝８元数 これが結構応用があるという en.wikipedia.org/wiki/Octonion In mathematics, the octonions are a normed division algebra over the real numbers, usually represented by the capital letter O, using boldface O or blackboard bold \mathbb O. There are only four such algebras, the other three being the real numbers R, the complex numbers C, and the quaternions H. The octonions are the largest such algebra, with eight dimensions, double the number of the quaternions from which they are an extension. They are noncommutative and nonassociative, but satisfy a weaker form of associativity, namely they are alternative.
Octonions are not as well known as the quaternions and complex numbers, which are much more widely studied and used. Despite this, they have some interesting properties and are related to a number of exceptional structures in mathematics, among them the exceptional Lie groups. Additionally, octonions have applications in fields such as string theory, special relativity, and quantum logic.
The octonions were invented in 1843 by John T. Graves, inspired by his friend William Hamilton's discovery of quaternions. Graves called his discovery octaves. They were discovered independently by Arthur Cayley[1] and are sometimes referred to as Cayley numbers or the Cayley algebra.
Abstract: The octonions are the largest of the four normed division algebras. While somewhat neglected due to their nonassociativity, they stand at the crossroads of many interesting fields of mathematics. Here we describe them and their relation to Clifford algebras and spinors, Bott periodicity, projective and Lorentzian geometry, Jordan algebras, and the exceptional Lie groups. We also touch upon their applications in quantum logic, special relativity and supersymmetry.
www.math.msu.edu/~akbulut/vita/pub.html Publications: 84. Gluck twisting 4-manifolds with odd intersection form (with K.Yasui) (to appear Math Research Letters)
>>447 参考 en.wikipedia.org/wiki/Selman_Akbulut Selman Akbulut (born 1949) is a Turkish mathematician and a Professor at Michigan State University. His research is in topology.
Career In 1975 he earned his Ph.D. from the University of California, Berkeley as a student of Robion Kirby. In topology, he has worked on handlebody theory, low-dimensional manifolds, symplectic topology, G2 manifolds. In the topology of real-algebraic sets, he and Henry C. King proved that every compact piecewise-linear manifold is a real-algebraic set; they discovered new topological invariants of real-algebraic sets.
He has developed 4-dimensional handlebody techniques, settling conjectures and solving problems about 4-manifolds, such as Zeeman conjecture,[1] Harer-Kas-Kirby conjecture, Scharlemann problem,[2] and Cappell-Shaneson problems.[3][4][5] He constructed an exotic compact 4-manifold (with boundary)[6] from which he discovered "Akbulut corks".[7]
His most recent results concern the 4-dimensional smooth Poincare conjecture.[8] He has supervised 10 Ph.D students as of 2011. He has more than 80 papers and 2 books published, and several books edited.
He was a visiting scholar several times at the Institute for Advanced Study (in 1975-76, 1980?81, 2002, and 2005).[9]
望月理論の応用 ”Teichmuller space below the Planck length, then it would, with Mochizuki’s discovery of a deep connection between the Teichmuller theory and number theory [3], give number theory an important place in the fundamental laws of nature.”という
vixra.org/abs/1301.0146 Teichmuller Space Interpretation of Quantum Mechanics Authors: Friedwardt Winterberg Submission history [v1] 2013-01-23 18:36:44 抜粋 1. Introduction
In this regard it is remarkable that a deep connection between the Teichmuller theory [2] and number theory has most recently been discovered by S. Mochizuki in his groundbreaking work “Inter-universal Teichmuller Theory,” which is an arithmetic version of the Teichmuller theory for number fields with an elliptic curve [3].
Conclusion Schrodinger said: “I would not call the entanglement one, but rather the characteristic trait of quantum mechanics, the one that enforces its entire departure from classical lines of thought.” If it should find its rational explanation in a conjectured Teichmuller space below the Planck length, then it would, with Mochizuki’s discovery of a deep connection between the Teichmuller theory and number theory [3], give number theory an important place in the fundamental laws of nature.
>>471 補足 en.wikipedia.org/wiki/AdS/CMT In theoretical physics, AdS/CMT correspondence is the program to apply string theory to condensed matter theory using the AdS/CFT correspondence. Over the decades, experimental condensed matter physicists have discovered a number of exotic states of matter, including superconductors, superfluids and Bose?Einstein condensates. These states are described using the formalism of quantum field theory, but some phenomena are difficult to explain using standard field theoretic techniques. Some condensed matter theorists hope that the AdS/CFT correspondence will make it possible to describe these systems in the language of string theory and learn more about their behavior.[1]
References Merali, Zeeya (2011). "Collaborative physics: string theory finds a bench mate". Nature 478 (7369): 302?304. Bibcode:2011Natur.478..302M. doi:10.1038/478302a. PMID 22012369. dx.doi.org/10.1038%2F478302a Sachdev, Subir (2013). "Strange and stringy". Scientific American 308 (44): 44. Bibcode:2012SciAm.308a..44S. doi:10.1038/scientificamerican0113-44. dx.doi.org/10.1038%2Fscientificamerican0113-44
この中に www.nilesjohnson.net/seven-manifolds.html An animation of exotic 7-spheres Video from a presenation by Niles Johnson at the Second Abel conference in honor of John Milnor.
mas.lvc.edu/~lyons/pubs/ David W. Lyons' Publications and Presentations Selected Expository Work [1] David W. Lyons. An elementary introduction to the Hopf fibration. Mathematics Magazine, 76(2):87-98, 2003. [ journal | e-print ]
In mathematics, Seiberg?Witten invariants are invariants of compact smooth 4-manifolds introduced by Witten (1994), using the Seiberg?Witten theory studied by Seiberg and Witten (1994a, 1994b) during their investigations of Seiberg?Witten gauge theory.
Seiberg?Witten invariants are similar to Donaldson invariants and can be used to prove similar (but sometimes slightly stronger) results about smooth 4-manifolds. They are technically much easier to work with than Donaldson invariants; for example, the moduli spaces of solutions of the Seiberg?Witten equations tend to be compact, so one avoids the hard problems involved in compactifying the moduli spaces in Donaldson theory.
For detailed descriptions of Seiberg?Witten invariants see (Donaldson 1996), (Moore 2001), (Morgan 1996), (Nicolaescu 2000), (Scorpan 2005, Chapter 10). For the relation to symplectic manifolds and Gromov?Witten invariants see (Taubes 2000). For the early history see (Jackson 1995).
This is the home page for the Polymath8 project, which has two components:
Polymath8a, "Bounded gaps between primes", was a project to improve the bound H=H_1 on the least gap between consecutive primes that was attained infinitely often, by developing the techniques of Zhang. This project concluded with a bound of H = 4,680.
Polymath8b, "Bounded intervals with many primes", is an ongoing project to improve the value of H_1 further, as well as H_m (the least gap between primes with m-1 primes between them that is attained infinitely often), by combining the Polymath8a results with the techniques of Maynard.
Contents 1 World records 1.1 Current records 1.2 Timeline of bounds 2 Polymath threads 3 Writeup 4 Code and data 4.1 Tuples applet 5 Errata 6 Other relevant blog posts 7 MathOverflow 8 Wikipedia and other references 9 Recent papers and notes 10 Media 11 Bibliography
つづき levitopher.wordpress.com/2013/05/08/exotic-smoothness-iii-existence/ May 8, 2013 by cduston Exotic Smoothness III: Existence Dimension 4 So the problem is that decomposition techniques generally fail in dimension 4, due to the added complexity but failure of the Whitney disk trick. Now, the topological version of the h-cobordism theorem works; meaning that two manifolds that are homotopic in dimension four are also homeomorphic. Of course, that doesn’t help us very much because we are at least in the category of continuity; want we want is the difference between continuous and smooth. 略 By a complete classification of these forms (done by Freedman and Donaldson), you can do things like try and decompose the manifold while preserving the intersection form. This leads to some contradictions, the most interesting of which leads to the existence of exotic R^4. These would be smooth 4-manifolds which are homeomorphic to our usual R^4, but which are not diffeomorphic to the usual R^4. Things are even worse (or better!) ? there are infinitely-many exotic R^4!
So the situation is this; in terms of exotic smoothness, dimension 4 is special. This presents a major motivation for studying exotic smoothness in the context of physics. We have already discussed that since exotic smooth structures are not smoothly equivalent, we would not expect any results which relied on calculus (like physics!) to be the same on both of them. Of course, this would not matter if we were studying the physics of space alone ? since it is 3-dimensional, there is no exotic smoothness. But as soon as we move to the dimension in which all our fundamental theories are based, exotic smoothness suddenly becomes non-trivial.
This is either a very significant observation, or it is not! The next post will discuss how we might try to study exotic smoothness in physics, from both model-building and observational standpoints.
So far, I have introduced some of the basic notions of smooth manifolds, what exotic smoothness is, and (very superficially!) how we know it exists. In this post I will talk about how one can go about constructing a physical model which includes exotic smooth structures, and what kinds of behavior we can expect. “What problems can exotic smoothness solve?” might be a summary for this post, but as we will see, there is more conjecture then problem solving.
Large and Small Exotic R^4 略
Dark Matter 略
The Brans Conjecture Localized exotic smoothness can mimic an additional source for the gravitational field. Of course, this conjecture is quite vague, but what Brans had in mind was exactly a solution to the dark matter problem. 略
Normal Matter
I think it’s fair to say the Brans conjecture has not been proven yet ? specifically, there is not currently a model of dark matter which can be compared to (and thus verified by) observations. However, there has certainly been work done which suggests that exotic smoothness can mimic mass in more limited ways. For instance, Torsten Asselmeyer-Maluga (you will see his name come up frequently in connection with this topic ? he has been diligently working on getting very interesting results for over a decade now) has shown that the intersection of some special surfaces in 4-manifolds (which represent points of which a homeomorphism f:M →M' fails to be a diffeomorphism) can create non-zero curvature terms (1997). In other words, the failure of two 4-manifolds to be diffeomorphic at points can mimic mass terms. This can be extended (see here and here), so that it appears that this result is quite general, and can be used to construct matter with a variety of internal symmetries.
Thus, in semiclassical gravity there are at least some instances when the Bran Conjecture is certainly true.
Well, this was long post but I wanted to give the current state of model-building based on exotic smooth structures. I think I will stop here; much of my other work is related to this topic, but this is enough to know in terms of exotic smooth structure.
有名なJohn Milnorが、”The Poincare Conjecture 99 Years Later”を書いて、そのときPerelman の論文が投稿されて、半信半疑だと書いていた・・ www.math.sunysb.edu/~jack/PREPRINTS/poiproof.pdf The Poincare Conjecture 99 Years Later: A Progress Report John Milnor, Stony Brook University, February 2003
抜粋 Three months ago, Grisha Perelman in St. Petersburg posted a preprint describing a way to resolve some of the major stumbling blocks in the Hamilton program and suggesting a path toward a solution of the full Elliptization Conjecture. The initial response of experts to this claim has been carefully guarded optimism, although, in view of the long history of false proofs in this area, no one will be convinced until all of the details have been carefully explained and veri￣ed. Perelman is planning to visit the United States in April, at which time his arguments will no doubt be subjected to detailed scrutiny.
ユーザープログラムを配信・利用できるCRANネットワーク機能 世界中のRユーザが開発したRプログラム（ライブラリ）（これを「パッケージ」と呼ぶ）がCRAN (The Comprehensive R Archive Network) と呼ばれるネットワークで配信されており、 それらをR環境単独でオンラインでダウンロード・インストール・アップグレードと一連の管理が可能である。
４次元で、下記が、なかなか絵が充実しているね これが置いてある、Andrew Ranicki’s Homepageでも”wild ”という言葉が良くヒットする。”wild ”すきみたいだね。すぎちゃんの系統かね www.maths.ed.ac.uk/~aar/papers/scorpan.pdf Scorpan, A. (2005), The wild world of 4-manifolds, Providence, R.I.: American Mathematical Society, ISBN 0-8218-3749-4
the following are rough draft versions of a text-to-be on applied algebraic topology, all in pdf. enjoy! the bibliographic entries are not yet added, and some of the cross-references and pictures are muddled...sorry!
下記がよくまとまっている en.wikipedia.org/wiki/4-manifold#cite_note-2 （抜粋） 4-manifold From Wikipedia, the free encyclopedia Contents 1 Topological 4-manifolds 2 Smooth 4-manifolds 3 Special phenomena in 4-dimensions 4 Failure of the Whitney trick in dimension 4
Smooth 4-manifolds For manifolds of dimension at most 6, any piecewise linear (PL) structure can be smoothed in an essentially unique way,[1] so in particular the theory of 4 dimensional PL manifolds is much the same as the theory of 4 dimensional smooth manifolds. A major open problem in the theory of smooth 4-manifolds is to classify the simply connected compact ones. As the topological ones are known, this breaks up into two parts:
1. Which topological manifolds are smoothable? 2. Classify the different smooth structures on a smoothable manifold.
There is an almost complete answer to the first problem of which simply connected compact 4-manifolds have smooth structures.
>>535 > 1. Which topological manifolds are smoothable? >There is an almost complete answer to the first problem of which simply connected compact 4-manifolds have smooth structures.
>>541 ども 4次元だとキャッソンハンドルに同相らしい "Casson's idea was to iterate this construction an infinite number of times, in the hope that the problems about double points will somehow disappear in the infinite limit." ということなので、”disappear in the infinite limit."が結論だな
en.wikipedia.org/wiki/Casson_handle In 4-dimensional topology, a branch of mathematics, a Casson handle is a 4-dimensional topological 2-handle constructed by an infinite procedure. They are named for Andrew Casson, who introduced them in about 1973. They were originally called "flexible handles" by Casson himself, and Michael Freedman (1982) introduced the name "Casson handle" by which they are known today. In that work he showed that Casson handles are topological 2-handles, and used this to classify simply connected compact topological 4-manifolds.
en.wikipedia.org/wiki/N-sphere#Spherical_coordinates An n-sphere is the surface or boundary of an (n + 1)-dimensional ball, and is an n-dimensional manifold. For n ≥ 2, the n-spheres are the simply connected n-dimensional manifolds of constant, positive curvature. The n-spheres admit several other topological descriptions: for example, they can be constructed by gluing two n-dimensional Euclidean spaces together, by identifying the boundary of an n-cube with a point, or (inductively) by forming the suspension of an (n － 1)-sphere.
4-sphere Equivalent to the quaternionic projective line, HP1. SO(5)/SO(4).
en.wikipedia.org/wiki/Quaternionic_projective_line In mathematics, quaternionic projective space is an extension of the ideas of real projective space and complex projective space, to the case where coordinates lie in the ring of quaternions H. Quaternionic projective space of dimension n is usually denoted by
\mathbb{HP}^n
and is a closed manifold of (real) dimension 4n. It is a homogeneous space for a Lie group action, in more than one way.
Projective line From the topological point of view the quaternionic projective line is the 4-sphere, and in fact these are diffeomorphic manifolds. The fibration mentioned previously is from the 7-sphere, and is an example of a Hopf fibration.
As mentioned above, in dimensions smaller than 4, there is only one differential structure for each topological manifold. That was proved by Johann Radon for dimension 1 and 2, and by Edwin E. Moise in dimension 3.[3] By using obstruction theory, Robion Kirby and Laurent Siebenmann [4] were able to show that the number of PL structures for compact topological manifolds of dimension greater than 4 is finite. John Milnor, Michel Kervaire, and Morris Hirsch proved that the number of smooth structures on a compact PL manifold is finite and agrees with the number of differential structures on the sphere for the same dimension (see the book Asselmeyer-Maluga, Brans chapter 7) By combining these results, the number of smooth structures on a compact topological manifold of dimension not equal to 4 is finite.
Dimension 4 is more complicated. For compact manifolds, results depend on the complexity of the manifold as measured by the second Betti number b_2. For large Betti numbers b_2>18 in a simply connected 4-manifold, one can use a surgery along a knot or link to produce a new differential structure. With the help of this procedure one can produce countably infinite many differential structures. But even for simple spaces like S^4, {\mathbb C}P^2,... one doesn't know the construction of other differential structures. For non-compact 4-manifolds there are many examples like {\mathbb R}^4,S^3\times {\mathbb R},M^4\setminus\{*\},... having uncountably many differential structures.
Chapter 11 From Differential Structures to Operator Algebras and Geometric Structures This chapter surveys some of the interesting interplay of exotic smoothness with other areas of mathematics and physics. In the first section we consider the “change” of a differential structure on a given TOP manifold to a differential structure on a second manifold homeomorphic but not diffeomorphic to the first one. Harvey and Lawson introduced the notion of singular bundle maps and connections to study this problem. This leads to speculations that such a process could give rise to singular string-like sources to the Einstein equations of General Relativity, including torsion. The next section deals with formal properties of a connection change and its relation to cyclic cohomology, providing a relationship between Casson handles and Ocneanus string algebra. This approach motivates introduction of the hyperfinite II1 factor C* algebra T leading to the conjecture that the differential structures are classified by the homotopy classes [M, BGl(T)+]. This conjecture may have some significance for the the 4-dimensional, smooth Poincark conjecture. The last section introduces a conjecture relating differential structures on 4-manifolds and geometric structures of homology 3-spheres naturally embedded in them.
About The differential or smoothness structure of a topological manifold (if it exists) can be non-unique. In all dimension except 4 there are only a finite number of different (i.e. non-diffeomorphic) smoothness structures. But dimension 4 is exceptional. Here there are an infinite number of different smoothness structures, countable infinite for most compact and uncountable many for many non-compact 4-manifolds. But what is the physical meaning of this fact, that is my main research program.
en.wikipedia.org/wiki/Carl_H._Brans Carl Henry Brans (born December 13, 1935) is an American mathematical physicist best known for his research into the theoretical underpinnings of gravitation elucidated in his most widely publicized work, the Brans–Dicke theory.
Recently Brans began study of developments in differential topology concerning the existence of exotic (non-standard) global differential structures and their possible applications to physics. This work includes looking at the exotic 7-sphere of Milnor as an exotic Yang-Mills bundle, and most especially the infinity of exotic differential structure on Euclidean four space (exotic R4) as alternative models for space-time in general relativity. Much of this work has been done in collaboration with Torsten Asselmeyer-Maluga of Berlin. In particular, they made the proposal that exotic smoothness structures can be resolve some of the problems in cosmology like dark matter or dark energy. Together they published a book, Exotic Smoothness and Physics World Scientific Press, 2007.
>>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
