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The conjectural theory of mixed motives would be a universal cohomology theory in arithmetic algebraic geometry. The monograph describes the approach to motives via their well-defined realizations. This includes a review of several known cohomology theories. A new absolute cohomology is introduced and studied.
The book assumes knowledge of the standard cohomological techniques in algebraic geometry as well as K-theory. So the monograph is primarily intended for researchers. Advanced graduate students can use it as a guide to the literature.

This book casts the theory of periods of algebraic varieties in the natural setting of Madhav Nori's abelian category of mixed motives. It develops Nori's approach to mixed motives from scratch, thereby filling an important gap in the literature, and then explains the connection of mixed motives to periods, including a detailed account of the theory of period numbers in the sense of Kontsevich-Zagier and their structural properties.
Period numbers are central to number theory and algebraic geometry, and also play an important role in other fields such as mathematical physics. There are...