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
