Motivic Category Theory and Rational Motivic Gamma-spaces

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The primary goal of motivic homotopy theory is to study algebraic varieties from a homotopy theoretic viewpoint. Many of the basic ideas and techniques in this subject originate in algebraic topology. The theory was invented by Morel and Voevodsky in the 90s and it is now one of the most active areas in algebraic geometry and algebraic topology, bringing together researchers in topology, algebra and representation theory. Motivic homotopy theory led to such striking applications as the solution of the Milnor conjecture and the Bloch- Kato conjecture, in algebraic geometry. Besides these quite spectacular applications, the fact that one can use the ideas and techniques of homotopy theory to solve problems in algebraic geometry has attracted mathematicians from both fields and has led to a wealth of new constructions and applications.This project will investigate categorical aspects of various triangulated categories of motives and rational motivic Gamma-spaces. The main purpose of this project is to advance our understanding of enriched motivic category theory, rational connected motivic spectra and motivic infinite loop spaces. Important applications for a range of associated cohomology theories of algebraic varieties and for the classical stable homotopy theory are expected as well.