非阿基米德几何,从何而来??
(2019-08-11 19:09:43)非阿基米德几何,从何而来??
在几何学中,不存在无穷小量,被人们称为阿基米德原理,但是,1899年,希尔伯特在《几何基础》中指出:几何学含有无穷小量不会导致逻辑矛盾,预言了非阿基米德几何学的合理性。
袁萌
附件1:
Several approaches to non-archimedean geometry
Brian Conrad1
Introduction Let k be a non-archimedean eld: a eld that is complete with respect to a specied nontrivial non-archimedean absolute value |·|. There is a classical theory of k-analytic manifolds (often used in the theory of algebraic groups with k a local eld), and it rests upon versions of the inverse and implicit function theorems that can be proved for convergent power series over k by adapting the traditional proofs over R and C. Serre’s Harvard lectures [S] on Lie groups and Lie algebras develop this point of view, for example. However, these kinds of spaces have limited geometric interest because they are totally disconnected. For global geometric applications (such as uniformization questions, as rst arose in Tate’s study of elliptic curves with split multiplicative reduction over a non-archimedean eld), it is desirable to have a much richer theory, one in which there is a meaningful way to say that the closed unit ball is “connected”. More generally, we want a satisfactory theory of coherent sheaves (and hence a theory of “analytic continuation”). Such a theory was rst introduced by Tate in the early 1960’s, and then systematically developed (building on Tate’s remarkable results) by a number of mathematicians. Though it was initially a subject of specialized interest, in recent years the importance and power of Tate’s theory of rigid-analytic spaces (and its variants, due especially to the work of Raynaud, Berkovich, and Huber) has become ever more apparent. To name but a few striking applications, the proof of the local Langlands conjecture for GLn by Harris–Taylor uses ´etale cohomology on non-archimedean analytic spaces (in the sense of Be
ural maps
Tn →
Tn+n0 andT n0 →
Tn+n0 onto the rst n and last n0 variables, it makes sense
to let J,J0 ⊆ Tn+n0 be the
ideals generated by I and I0 respectively. Consider the k-anoid
algebra Tn+n0/(J + J0). There are
evident k-algebra maps ι : A →
Tn+n0/(J +J0) and ι0 : A0 →
Tn+n0/(J +J0). Prove that this pair
ofmapsisuniversalinthefo
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附件2:
百度一下“无穷小微积分”,访问该网站,下载“Elementary Calculus”,查找第二章46页Figure 2.1.2,此时,在你的眼前就是非阿基米德几何的导数示意图了。
在无穷小微积分教材中到处都是非阿基米德几何示意图,因为,无穷小就是一种非阿基米德量。
坦率地,学习微积分,,离不开希尔伯特5组几何公理。
其实,超实平面几何就是一种非阿基米德几何。
告别传统(阿基米德几何),迎向未来(非阿基米德几何)。
袁萌