Representation of Galois groups and descent in algebraic K-theory

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DMS-0104162Gunnar CarlssonThis project includes several different directions of research. Carlsson will study a homotopy theoretic model he has constructed for the algebraic K-theory spectrum of a field, which is built out of the representation theory of the absolute Galois group of the field. He will try to prove that the model is indeed equivalent to the K-theory of the field, as well as to work out the consequences of this result for the Quillen-Lichtenbaum conjectures and the relationship between this result and the Bloch-Kato conjecture. He will also investigate potential applications of algebraic topology in high dimensional data analysis. He expects to improve software which he and V. De Silva have developed for homology computation, with the goal of identifying features such as singular points as well as global topology for data sets of dimension greater than three. He also plans to study homotopy theoretic issues which arise in computational questions, relating to sampling subcomplexes from large simplicial complexes. Kiem plans to compute intersection numbers for singular moduli spaces of curves. This is an important problem since the intersection numbers in this case are related to well-known geometric invariants, such as the Casson invariant.One of the great themes in mathematics over the last two centuries is the strong relationship between arithmetic and geometry. The study of this theme was initiated by Abel and Galois in the early 19th century, and in this century its further development has resulted in our obtaining very precise information concerning sets of integer or rational solutions to systems of equations. The goal of this project is to explore another manifestation of this theme, in the form of the so-called algebraic K-theory of fields. Algebraic K-theory is a geometric construction attached to arithmetic objects, called fields. Fields are arithmetic objects in which one can add, multiply, and divide. Algebraic K-theory attaches geometric objects to these fields, and in such a way that already well understood invariants of fields can be extracted easily. Algebraic K-theory also contains many less well understood invariants as well, and the goal of understanding these invariants has been one toward which topologists have been striving since the early 1970's, when Quillen defined higher algebraic K-theory. Two important conjectures have been formulated concerning algebraic K-theory, the Quillen-Lichtenbaum and Bloch-Kato conjectures. They would relate the algebraic K-theory of a field to properties of its "absolute Galois groups", an object which incorporates all possible symmetries of sets of solutions of sets of equations over the field in question. This project aims to understand very clearly how algebraic K-theory is built out of information of this large group of symmetries, and to use this understanding to approach the two central conjectures mentioned above.