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

小行星9504789 这个项目将在代数K理论中追求两个方向。 第一次尝试解决问题，在高维几何拓扑研究的代数K-理论的群环，一般群环的基本群体的流形。 K-理论在研究给定同伦类型的同胚分类中是有用的。 第二个将使用类似的技术中使用的这些几何问题来研究代数K-理论领域，特别是下降问题领域的拓扑循环绝对伽罗瓦群。 拓扑学关注的是空间（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。 非正式地说，人们会想到拉伸或收缩部分或全部空间。 例如，如果一个人认为大写字母“A”是一个空格，他可以用不同的字体打印它，虽然结果会改变（大小，以及字符的倾斜度，或字符内水平线的高度），但它不会改变拓扑，因为一个数字可以拉伸或弯曲到另一个。 人类能够从视觉上识别出这些图形在拓扑结构上是相同的，因为我们很容易阅读用任何一种字体书写的文本。 这种等价的概念被称为“同胚”。有一个更强的等价概念被称为“同伦等价”，例如，它不仅允许弧被拉伸，而且可以被压缩成点。 因此大写字母“H”是同伦等价的，但不同胚于大写字母“X”，因为人们可以通过将水平线压缩成一个点来从“H”得到“X”。 然而，大写字母“O”不是同伦等价于“I”，因为“O”包含一个循环，而“I”不包含。 同伦等价通常是比同胚更容易确定的关系。 本文研究的问题是，对于某类同伦型，如何将同伦空间分类到同伦等价的同胚空间。令人惊讶的是，虽然这个问题看起来完全是几何问题，但所需的技术使用了大量抽象代数。 代数和几何之间的这种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. ***