[過去ﾛｸﾞ] 現代数学の系譜11 ガロア理論を読む8 (779ﾚｽ)
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574(2): 2014/06/07(土)21:40 AAS
>>569-571
http://www.maths.ed.ac.uk/~aar/papers/exoticsmooth.pdf
Exotic Smoothness and Physics Differential Topology and Spacetime Models 2007
Exoticな時空が、物理学にどう影響するのか
そこを詳しく書いている
名著だと思う
575: 2014/06/08(日)05:33 AAS
>>574
著者
http://www.researchgate.net/profile/Torsten_Asselmeyer-Maluga
Torsten Asselmeyer-Maluga PhD Researcher
German Aerospace Center (DLR)
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.
http://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.
578(3): 2014/06/14(土)05:47 AAS
>>574 補足
http://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.
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