Dimensions, universal spaces, and continua

宇宙物体的概念在数学中是非常重要的。它出现在代数、分析、几何和拓扑学中。对于某一类对象来说，对象X是通用的意味着什么？一个标准要求是X是族本身的一个元素。其他要求可能会有所不同。然而，一般的想法是，X本身必须包含来自该家族的每个对象的副本，或者来自该家族的每个对象可以通过某种标准程序从X获得。我研究各种不同维度版本的宇宙对象。我的主要目标之一是找出在上同调维度的情况下是否存在泛紧关系。为了解决这个问题，我打算应用可拓维的方法，特别是进一步发展我所介绍的拟有限复形理论。我的程序的另一部分是关于无限维拓扑的。无限维空间在数学的许多应用中都很重要。例如，希尔伯特空间是量子力学的重要组成部分。我研究各种类型的无限维空间。我希望为这些类找到“好”的特征，以便更好地理解空间的结构。我的提案的第三部分致力于连续体的跨度。连续体最简单的例子是公制图，它可以被可视化为一个道路网络。这种图的跨度可以被描述为两个旅行者在同时遍历该图时彼此之间所能保持的最大距离。我将研究跨度的性质，例如跨度类型之间的关系。我还想知道跨度为零的连续体是否类似于直线段。我的程序的最后部分致力于研究交换拓扑和非交换拓扑之间的各种类比。通过研究函数代数的性质及其与基础空间性质的关系，我希望发现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.