https://upsilon.cc/~zack/research/hdr/

La logique tensorielle est une logique primitive du tenseur et de la négation, dont l'objectif est de circonscrire les ingrédients élémentaires du raisonnement logique, et de les étudier au moyen des outils de l'algèbre contemporaine.

La logique est aussi conçue pour fonder la sémantique des jeux en théorie des types, et pour l'articuler de manière précise et harmonieuse avec la logique linéaire et la théorie des continuations dans les langages de programmation.

https://www.irif.fr/~mellies/habilitation.html

The four-color theorem states that every simple planar graph admits a homomorphism to $K_4$. In many proposed extensions or reformulations of this theorem $K_4$ is regarded as the complete graph on four vertices. In this work we consider $K_4$ as the Cayley graph “$\mathbb Z_2^2, {01,10,11}$”. Main observation, which is hidden behind the fact that 2 is a very small number, is that {01,10} is a basis of $\mathbb Z_2^2$ and that 11=01+10.

The generalization of this view is the Cayley graph $\mathbb Z_2^k, {e_1,e_2, \cdots, e_k, J}$ which is isomorphic to the projective cube of dimension $k$ also known as the folded cube.

Thus we consider the problem of mapping planar graphs into projective cubes, and show that this question is related to several other notions of coloring such as the edge-chromatic number of planar multi-graphs, circular chromatic number and the fractional chromatic number.

Finally, after providing a test to decide if a graph $B$ of odd-girth $2k+1$ admits a homomorphism from any graph of tree-width at most $t$ and odd-girth at least $2k+1$, we show that every 3-tree of odd-girth at least $2k+1$ admits a homomorphism to the projective cube of dimension $2k$.