Towards a concrete theory of cohomology: a fundamental concept in geometry

In geometry an important tool is linearisation, whereby hard, nonlinear problems are converted into easier,linear ones. One of the most useful linearisation techniques, and one of the triumphs of 20th century mathematics, is cohomology theory. In algebraic geometry, we have not one but infinitely many cohomologytheories. It is believed that we can unify them all into a single universal cohomology theory, the theory of motives. In the 1990s, there was important abstract progress in developing this theory, but for applications it is vital to have a concrete approach. This project will develop big de Rham-Witt cohomology, a concrete cohomology theory unifying many existing ones. It is thus one step closer to a concrete theory of motives.