AEI-DFG: Combinatorial Polytope Theory

• Applicants: Professor Dr. Raman Sanyal; Professor Dr. Christian Stump
• Subject Area: Mathematics
• Term: since 2024
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 541393733

Combinatorial Polytope Theory is the study of the two-way fruitful interactions between polytopes and combinatorics. The first research line on "Combinatorics of Polytopes" studies combinatorial properties of general polytopes. We will study face numbers of important families of polytopes that are neither simple nor simplicial and fall outside the scope of the g-theorem. Moreover, we will explore novel polytope constructions inspired by the study of the behavior of linear programming, and the connection of max-slope polytopes with deformed permutahedra. The research topics in this project are motivated by recent breakthroughs in combinatorial polytope theory, and are interconnected by transversal subjects including deformed permutahedra, virtual polytopes, linear programming, oriented matroids, and projections/sections of polytopes. The second research line on "Polytopes of Combinatorics" concerns the study of families of polytopes arising from combinatorics and algebra, as the classical permutahedra and associahedra. Motivated by different perspectives on the associahedron, we will follow three specific research directions: the (semi)lattice congruences of the weak order and its various generalizations, the recent poset associahedra and their interpretation as nested complexes of face lattices of polytopes, and the geometric properties of the deformation cones of permutahedra.

• DFG Programme: Research Grants
• International Connection: Spain
• Partner Organisation: Agencia Estatal de Investigación
• Cooperation Partners: Professor Arnau Padrol; Dr. Vincent Pilaud