Hadwiger famously conjectured that every $K_h$-minor-free graph is properly $(h-1)$-colourable. This talk will present the following improper analogue of Hadwiger’s Conjecture: for fixed $h$, every $K_h$-minor-free graph is $(h-1)$-colourable with monochromatic components of bounded size. The number of colours is best possible regardless of the size of monochromatic components. This solves an open problem of Edwards, Kang, Kim, Oum and Seymour [SIAM J. Disc. Math. 2015], and concludes a line of research initiated in 2007. Similarly, for fixed $t\geqslant s$, we show that every $K_{s,t}$-minor-free graph is $(s+1)$-colourable with monochromatic components of bounded size. The number of colours is best possible, solving an open problem of van de Heuvel and Wood [J. London Math. Soc. 2018]. We actually prove a single theorem from which both of the above results are immediate corollaries. For an excluded apex minor, the result is strengthened  as follows: for fixed $t \geqslant s \geqslant 3$, and for any fixed apex graph $X$, every $K_{s,t}$-subgraph-free $X$-minor-free graph is $(s+1)$-colourable with monochromatic components of bounded size. The number of colours is again best possible. This is joint work with Vida Dujmović, Louis Esperet and Pat Morin [arXiv:2306.06224].

Zeta and L-functions have been classically associated, first to schemes of finite type over Z, and then to the l-adic cohomology of smooth projective varieties over a global field. In the latter definition, independence from l and actual existence remain partly conjectural in characteristic 0. In these lectures, I will explain how to associate unconditionally an L-function to objects in triangulated categories of motives. For the motive of a smooth projective variety this definition differs from the one above in general, but only up to a finite number of Euler factors. This might be useful to tackle the Beilinson conjectures.

Zeta and L-functions have been classically associated, first to schemes of finite type over Z, and then to the l-adic cohomology of smooth projective varieties over a global field. In the latter definition, independence from l and actual existence remain partly conjectural in characteristic 0. In these lectures, I will explain how to associate unconditionally an L-function to objects in triangulated categories of motives. For the motive of a smooth projective variety this definition differs from the one above in general, but only up to a finite number of Euler factors. This might be useful to tackle the Beilinson conjectures.

Zeta and L-functions have been classically associated, first to schemes of finite type over Z, and then to the l-adic cohomology of smooth projective varieties over a global field. In the latter definition, independence from l and actual existence remain partly conjectural in characteristic 0. In these lectures, I will explain how to associate unconditionally an L-function to objects in triangulated categories of motives. For the motive of a smooth projective variety this definition differs from the one above in general, but only up to a finite number of Euler factors. This might be useful to tackle the Beilinson conjectures.