From Riemann to Poincaré, topological methods pervaded the development of geometry, algebra and arithmetic. Homology and homotopy, sheaf theory and spectral sequences, derived and model categories have shaped a wealth of invariants applied to a wide variety of problems, crowned by Grothendieck’s refoundation of algebraic geometry. In this parentage, Voevodsky’s motivic theory has burst out in the nineties led by the proof of the Bloch-Kato conjecture. After Morel and Voevodsky’s foundation of motivic homotopy, the theory has strongly evolved around its close connection with algebraic topology, abutting to important foundational results such as cohomological orientation theory (motivic cohomology, algebraic cobordism), and the discovery of the quadratic nature of the $\mathbb{A}^1$-homotopical invariants: Morel’s computations of stable homotopy sheaves of spheres, Panin and Walter’s generalized orientations leading to quadratic enumerative geometry à la Levine-Kass-Wickelgren.

The aim of the project is to extend and apply $\mathbb{A}^1$-homotopical methods in three main complementary directions, a unifying moto being the role of characteristic classes, especially that of the fundamental class of the diagonal:

• The study of non necessarily proper algebraic varieties through $\mathbb{A}^1$-homotopical methods, with the aim to establish a theory of "$\mathbb{A}^1$-homotopy at infinity", whose cornerstone would be the understanding of the fundamental class of the diagonal of open algebraic varieties. Our aim is to develop methods of computations from different perspectives, which could also give an original approach for computing these fundamental classes. We also hope to develop this line of thoughts to initiate an $\mathbb{A}^1$-homotopical study of singularities, knots and links following Mumford, Milnor, and others. A long-term motivation is to get unstable $\mathbb{A}^1$-homotopical invariants at infinity, and formulate an accurate $\mathbb{A}^1$-homotopical analogue of the Poincaré conjecture.
• Applications of quadratic invariants from $\mathbb{A}^1$-homotopy theory to arithmetic problems, such as the development and the study of a quadratic Riemann-Roch formula, based on the quadratic invariants of $\mathbb{A}^1$-homotopy such as Hermitian K-theory, and Chow-Witt groups. We intend to develop in this vein the recent notion of formal ternary laws, an $\mathbb{A}^1$-homotopical analogue of formal group laws, fundamental in algebraic topology. We also propose paths to apply quadratic invariants of $\mathbb{A}^1$-homotopy to problems of arithmetic flavor, in particular $L$-functions and special values.
• Developing new decomposition theorems for relative motives. This is indeed related to the decomposition of the fundamental class of the diagonal, this time in the proper case. Some of our recent results give new cases where the relative Chow-Künneth conjecture can be proved. We hope to be able to extend this result to more arithmetical cases, by working on number fields or even on rings of integers. We also hope to develop the theory of relative Nori motives, and its links with Voevodsky's theory.