- Arnau Padrol (IMJ-PRG, Université Pierre et Marie Curie Paris 6)
This project studies combinatorial problems on (possibly high dimensional) convex polytopes
and polyhedral subdivisions. These arise naturally in several research areas in mathematics and
computer science, both theoretical and applied. Although many of these problems are strongly
related, they are traditionally formulated and approached from completely different points of
view. We want to unveil and exploit such connections by sharing problems, techniques and
perspectives from our different complementary backgrounds.
The range of topics we want to study covers different aspects of the field. They include doing a
combinatorial analysis of geometric constructions as well as understanding geometric constraints
imposed by combinatorial structures. We want to study enumeration problems for combinatorial
types of polytopes, with a view to revealing typical properties of random combinatorial instances.
We also want to explore applications and extensions of the theory of generalized Gale transforms
for the operations of projection and section. We expect these advances to be relevant in the study
of extended formulations, a very active topic in the combinatorial optimization community, but
also in the context of shadow and stabbing problems, and in relation to the convex dimension of
graphs. Finally, we want to investigate realizability problems on tropical polytopes and oriented
matroids. These are motivated by applications of tropical linear programming as well as by the
search of polyhedral realizations of combinatorial structures.