Mathematical Sciences: Homotopy Limit Problems and AlgebraicK-Theory

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Carlsson plans to study the stable structure of a delooping of Quillen's algebraic K-theory, with the hope of producing a stable splitting of a natural filtration of it. He also plans to study the algebraic K-theory of certain non-singular curves, which are used to describe the failure of the so-called profinite descent map. Both these projects, if successful, would be important progress in the resolution of the Lichtenbaum-Quillen conjectures for algebraic K-theory. This work comes under the general heading of exploring and perfecting algebraic techniques for treating qualitative features of spaces, e.g. connectedness, knottedness, stability, and so forth. Such features can be every bit as significant for applications as are the usual features examined by methods of classical analysis.