Flag vectors of polytopes and a personal experience in experimental mathematics

Gil Kalai, Hebrew University of Jerusalem and Yale University

For a d-dimensional convex polytope P, their flag-numbers counts the number of chains of faces of prescribed dimensions. There are altogether 2^d flag numbers, which together form the flag-vector. The flag numbers satisfy various linear equalities derived from the Euler-Poincare theorem. A remarkable theorem of Bayer and Billera determines the dimension of their affine hulls as one less the dth Fibonacci number. Some mysterious linear equalities between flag numbers of polytopes and their duals are related to "mirror symmetry". Linear inequalities between flag numbers and their combinatorial consequences are very interesting and in some cases could only be explored and proved by an automatic system. After describing the theoretical background concerning flag numbers I will tell about my experience in the direction of automating proofs and experimentation.