Homotopy Theoretic Investigations in Higher K-theory, High-dimensional Data Analysis, and High Dimensional Manifold Theory

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DMS-0406992Gunnar E. CarlssonThis project concerns the descent problem in higher algebraic K-theory. There has been much recent progress in identifying the initial term of a spectral sequence converging to the K-groups of a field, notably by V. Voevodsky and A. Suslin. This project is an attempt to prove that the algebraic K-theory spectrum, not just its homotopy groups, can be described purely in terms of the absolute Galois group of the field in question, and the algebraic K-theory spectrum of an algebraically closed field. The key ingredients are the equivariant K-theory spectrum attached to complex representations of the absolute Galois group, which can be regarded as a highly structured ring spectrum, and a derived version of a completion construction at an augmentation ideal. The model has already been developed, and what remains is to prove the agreement of the model with algebraic K-theory of the field in general. The validity of these results should give striking connections between, on the one hand, the derived completion of the representation ring of the absolute Galois group, and on the other the Milnor K-groups of the field, which are groups defined directly from the arithmetic of the field. This project seeks to connect the behavior of the arithmetic in a field (an algebraic object which generalizes our usual notions of real, complex, and rational numbers) and the behavior of so-called complex representations of a group of symmetries of a larger field containing all possible solutions to algebraic equations of the original field. These representations contain a great deal of geometric information, and drawing this kind of connection (between geometric and arithmetic information) has been a longstanding theme in mathematics. The relationship between such seemingly distinct kinds of ideas has been responsible for many of the recent striking developments in arithmetic and algebraic geometry, including Delinge's proof of the so-called Weil conjectures as well as the recent work on the Geometric Langlands program. The present project represents another facet of this circle of ideas.