It is a major insight of the preceding two centuries that one can use a geometrical language to study the solutions of a given set of polynomial equations with coefficients in an arbitrary field, such as the complex numbers or even in an arbitrary ring, such as the integers. The corresponding objects, called algebraic varieties, are extremely rich and mysterious due to their dual nature, geometric and arithmetic.

The driving force of the project is the use of the recent and powerful theory of motivic A1-homotopy introduced by Voevodsky to produce new, and study classical, invariants of algebraic varieties of both geometric and arithmetic nature.

The expected applications have a very wide range: advances in the understanding of Voevodsky’s theory, producing new knowledge in affine algebraic geometry, extending previously known computations of invariants for families of algebraic varieties, and improvement of our arithmetical knowledge of certain kinds of algebraic varieties.

A one year study group based on Fabien Morel's book *$\mathbb{A}^1$-algebraic topology over a field* held weekly at the IMB starting September 2019.

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