Dimensions, universal spaces, and continua

普遍对象的概念在数学中非常重要。它出现在代数、分析、几何和拓扑学中。一个对象X对于某类对象是泛在的，这意味着什么？一个标准要求是X是族本身的元素。其他要求可能有所不同。尽管如此，一般的想法是X本身必须包含来自族的每个对象的副本，或者来自族的每个对象可以通过某种标准程序从X获得。我研究宇宙物体的各种维度。本文的主要目的之一是研究在上同调维数的情况下是否存在泛紧。为了解决这个问题，我打算应用扩展维数的方法，特别是进一步发展我所介绍的准有限复形理论。我的程序的另一部分是关于无限维拓扑的。无穷维空间在数学的许多应用中是重要的。例如，希尔伯特空间是量子力学的基本组成部分。我研究各种各样的无限维空间。我希望找到这些类的“好”特征，以更好地理解空间的结构。我的建议的第三部分专门讨论连续统的跨度。连续体的最简单的例子是度量图，它可以被可视化为道路网络。这种图的跨度可以描述为两个旅行者在同时遍历图时彼此之间所能保持的最大距离。我将研究跨度的属性，例如跨度类型之间的关系。我还想知道跨度为零的连续统是否与线段相似。我的计划的最后一部分是致力于交换和非交换拓扑之间的各种类似物的研究。通过研究函数代数的性质及其与底层空间性质的联系，我希望能发现C*-代数的新性质。

The concept of a universal object is very important in mathematics. It appears in algebra, analysis, geometry, and topology. What does it mean for an object X to be universal for a certain family of objects? One standard requirement is that X is an element of the family itself. Other requirements may vary. Nevertheless, the general idea is that X must contain in itself a copy of each object from the family, or each object from the family can be obtained from X by means of a certain standard procedure. I study universal objects for various versions of dimensions. One of my main goals is to find out whether a universal compactum exists in the case of cohomological dimension. To solve this problem, I intend to apply methods of extension dimension, and, in particular, further develop the theory of quasi-finite complexes that I have introduced. Another part of my program is concerned with infinite-dimensional topology. Infinite-dimensional spaces are important in many applications of mathematics. For example, Hilbert space is an essential ingredient of quantum mechanics. I study various classes of infinite-dimensional spaces. I wish to find "nice" characterizations for these classes to better understand the structure of spaces. The third part of my proposal is devoted to spans of continua. The simplest example of a continuum is a metric graph, which can be visualized as a network of roads. The span of such graph can be described as the maximal distance two travellers can keep between each other while simultaneously traversing the graph. I am going to investigate properties of span, such as relations between types of spans. I also want to find out whether continua with span zero are similar to a line segment. The concluding part of my program is devoted to the study of various analogs between commutative and non-commutative topology. By studying properties of algebras of functions and their connection to the properties of underlying spaces I hope to discover new properties of C*-algebras.