Algebraic Aspects of Dimension

最直观的几何概念之一是维数。对于一个非数学家来说，猜测一个特定几何对象的维度是很容易的，但是要以一种正式的、严格的方式定义维度是相当困难的。维数的第一个数学定义出现在线性代数中。根据这个定义，我们很容易理解直线、平面和三维空间的尺寸分别是1、2和3。自从拓扑学作为分析学的一个分支形成以来，数学家们就试图用非代数的方式来定义维数。最直观的定义就是所谓空间X的小归纳维数ind(X)。本质上，ind(X)最多n意味着边界维数最多为（n-1）的开集U构成了X的一组基。扩展维数理论是空间不被自然数参数化的一种广义维数理论。相反，它们被CW配合物参数化。在这个理论中，dim(X)最大等于K意味着K是X的绝对扩展量。特别地，dim(X)最大等于n维球面S(n)当且仅当X的覆盖维数最大等于n时，dim(X)=K意味着K对于所有L是最小的，使得dim(X)最大等于L。事实证明，扩展维数同时包含覆盖维数和上同维数。它是丰富的几何和代数之间的相互作用，这对首席研究员来说，是所有数学的基石。在有限维紧实的情况下，我们有一个相关的代数对象，叫做布克斯坦代数。有一个扩展维的对偶理论，它处理连续波复形，在可数连续波复形的情况下，有一个相关的代数对象，称为对偶伯克斯坦代数。这个项目涉及“维度”的研究。研究不同维度对象的需要不仅仅是数学。随着时间的推移，我们对我们生活的基本物体——宇宙——的维度的感知经历了重大变化。在牛顿力学中，宇宙被认为是三维的；爱因斯坦又增加了一个维度（时间）。目前，理论物理学家们正在思考各种旨在统一引力和量子力学的宇宙模型；在一些模型中，宇宙的维度是10，而在另一些模型中，宇宙的维度是26。

One of the most intuitive geometric concepts is that of dimension. While it is very easy for a non-mathematician to guess the dimension of a particular geometric object, it is fairly difficult to define the dimension in a formal, rigorous way. The first mathematical definition of dimension arises in linear algebra. From that definition one easily understands that the dimensions of the line, the plane, and the 3-space are respectively 1,2, and 3. Ever since the formation of topology as an offshoot of analysis, mathematicians attempted to define dimension in a non-algebraic way. The most intuitive such definition is the so-called small inductive dimension ind(X) of a space X. Essentially, ind(X) at most n means that open sets U with boundary of dimension at most (n-1) form a basis of X. Extension dimension theory is a general theory of dimension in which spaces are not parametrized by natural numbers. Instead, they are parametrized by CW complexes. In this theory dim(X) being at most K means that K is an absolute extensor of X. In particular, dim(X) is at most the n-dimensional sphere S(n) if and only if the covering dimension of X is at most n. dim(X)=K means that K is minimal with respect to all L such that dim(X) is at most L. It turns out that extension dimension encompasses both the covering dimension and the cohomological dimension. It is rich in interplay between geometry and algebra which, to the Principal Investigator, is the cornerstone of all mathematics. In the case of finite-dimensional compacta one has an associated algebraic object called the Bockstein algebra. There is a dual theory to extension dimension which deals with CW complexes and in the case of countable CW complexes there is an associated algebraic object called the dual Bockstein algebra. This project involves the study of 'dimension.' The need to study objects of various dimensions is not unique to mathematics alone. The perception of the dimension of the basic object we live in, the Universe, has undergone signi ficant changes over time. In Newtonian mechanics, the Universe is assumed to be 3-dimensional; Einstein added one more dimension (time). Currently theoretical physicists ponder various models of the Universe aimed at unifying gravity and quantum mechanics; in some models the Universe is of dimension 10 while in others its dimension is 26.