Project description

Classical algebraic topology pursues the classification of topological manifolds and spaces with the aid of algebraic invariants. Especially interesting ones are the representable algebraic invariants, also known as cohomology theories, whose representing object (known as a spectrum) may be seen as a topological space endowed with a particularly nice type of addition law. The collection of all spectra forms the stable homotopy category and this viewpoint has led to quite amazing progress in the original classification problem. In contrast to algebraic topology, algebraic geometry deals with the considerably more inflexible algebraic manifolds, given locally as the solutions of polynomial equations. In the 1990s, Fabien Morel and Vladimir Voevodsky extended the topological approach via generalised cohomology theories to the setting of algebraic geometry, initiating a great deal of new research in this direction. This research furnished a framework for the construction and study of many very interesting cohomology theories on algebraic manifolds, now known as motivic stable homotopy category over a field or more generally over a base-scheme. For the study of these algebraic invariants, a technical tool, the use of filtrations, is essential. In the topological setting, the use of many different filtrations, or towers, have been used and played off against each other in order to gain a better theoretic understanding of basic objects of study, as well as for making concrete computations. The motivic stable homotopy category has given rise to many such filtrations, some direct generalisations of the topological ones, others quite new, whose study in the last decade has brought astounding results. The goal of this project is to study several of these motivic filtrations and to use their properties to achieve concrete results. Especially important examples for study are Voevodsky’s slice filtration and the filtrations by connectivity. These filtrations and their relation with one another should be helpful in understanding algebraic K-theory, cooperations for mod p motivic cohomology, homotopy sheaves of the sphere-spectrum, as well as motivic orientations, generalising the classical Todd genus.

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