Mathematical Sciences: Algebraic K-Theory of Group Rings and Fields

9504789 Carlsson这个项目将在代数K-理论中追求两个方向。第一个尝试通过研究群环的代数K-理论来解决高维几何拓扑中的问题，通常是所涉及的流形的基本群的群环。K-理论在研究给定同伦型内的同胚分类时是有用的。第二部分将使用类似于这些几何问题中使用的技巧来研究场的代数K-理论，特别是具有拓扑循环绝对伽罗瓦群的场的下降问题。拓扑学本身涉及空间(曲线、曲面和高维类似物)的性质，这些性质在变形下不会改变。非正式地，人们会想到伸展或收缩部分或全部空间。例如，如果把大写字母“A”看作一个空格，可以用不同的字体打印它，虽然结果会改变(字符的大小、倾斜或字符内水平线的高度)，但它不会在拓扑上改变，因为一个图形可以拉伸或弯曲成另一个图形。人类能够从视觉上识别这些图形在拓扑结构上相同的事实，因为我们能够很容易地阅读以任何一种字体书写的文本。这种等价性的概念被称为“同胚”。还有一种更强的等价性概念，称为“同伦等价”，例如，它不仅允许圆弧被拉伸，而且可以被压缩成点。因此，大写字母“H”与大写字母“X”同伦等价，但不是同胚的，因为人们可以通过将水平线压缩到一点来从“H”获得“X”。然而，大写的“O”并不等同于“I”，因为“O”包含一个循环，而“I”不包含。同伦等价通常比同胚关系更容易确定。这个问题涉及到对于某一族的“同伦型”，上至同伦等价的同态空间的分类问题。令人惊讶的是，尽管这个问题看起来完全是几何问题，但所需的技术使用了高度抽象的代数。代数和几何之间的相互作用是当今数学中最令人兴奋的领域之一。***

9504789 Carlsson This project will pursue two directions within algebraic K-theory. The first attempts to resolve problems within high dimensional geometric topology by studying the algebraic K-theory of group rings, generally the group rings of fundamental groups of the manifolds involved. K-theory turns out to be useful in studying the homeomorphism classification within a given homotopy type. The second will use an analogue of the techniques used in these geometric problems to study the algebraic K-theory of fields, specifically the descent problem for fields with topologically cyclic absolute Galois group. Topology concerns itself with properties of spaces (curves, surfaces, and higher dimensional analogues) which do not change under deformations. Informally, one thinks of stretching or shrinking part or all of the space. For instance, if one thinks of the capital letter "A" as a space, one could print it with different fonts, and although the results would change (the size, as well as the slant of the character, or the height of the horizontal line within the character), it would not change topologically, since the one figure could be stretched or bent into the other. Humans are able to recognize visually the fact that these figures are topologically the same, since we are easily able to read text written in either font. This notion of equivalence is referred to as "homeomorphism." There is an even stronger notion of equivalence referred to as "homotopy equivalence," which, for instance, allows arcs not only to be stretched but to be compressed into points. Thus the capital letter "H" is homotopy equivalent but not homeomorphic to the capital letter "X," since one can obtain "X" from "H" by compressing the horizontal line to a point. However, capital "O" is not homotopy equivalent to "I," since the "O" contains a loop while the "I" does not. Homotopy equivalence is typically an easier relation to determine than homeomorphism. This proje ct concerns itself with the problem of classifying up to homeomorphism spaces which are already homotopy equivalent, for a certain family of "homotopy types." Surprisingly, although this problem seems entirely geometric, the techniques required use heavily abstract algebra. This interplay between algebra and geometry is one of the most exciting areas in mathematics today. ***