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

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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. ***