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

9504789 Carlsson 该项目将在代数 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，通常是所涉及流形的基本群的群环。 K 理论对于研究给定同伦类型内的同胚分类非常有用。 第二个将使用这些几何问题中使用的类似技术来研究场的代数 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。 拓扑学关注在变形下不会改变的空间属性（曲线、曲面和更高维的类似物）。 非正式地，人们会想到拉伸或收缩部分或全部空间。 例如，如果将大写字母“A”视为一个空格，则可以用不同的字体打印它，尽管结果会发生变化（字符的大小、倾斜度或字符内水平线的高度），但它不会在拓扑上发生变化，因为一个图形可以拉伸或弯曲到另一个图形中。 人类能够从视觉上识别出这些图形在拓扑上相同的事实，因为我们可以轻松阅读以任何一种字体编写的文本。 这种等价概念被称为“同胚”。 还有一种更强的等价概念，称为“同伦等价”，例如，它不仅允许弧被拉伸，而且可以被压缩为点。 因此，大写字母“H”与大写字母“X”同伦等价但不同胚，因为可以通过将水平线压缩到一点来从“H”获得“X”。 然而，大写“O”并不与“I”同伦，因为“O”包含循环，而“I”则不包含。 同伦等价通常是比同态更容易确定的关系。 该项目关注的是对于某个“同伦类型”家族，对已经同伦等价的同态空间进行分类的问题。 令人惊讶的是，虽然这个问题看起来完全是几何问题，但所需的技术大量使用抽象代数。 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. ***