"On the g-conjecture"
Abstract:In 1970 McMullen conjectured a complete characterization of the possible face-vectors of boundary complexes of simplicial polytopes. These numerical conditions were proved in 1980, necessity by Stanley and sufficiency by Billera and Lee, known as the g-theorem.
The proof of necessity shows that a hard-Lefschetz decomposition holds for an appropriate ring associated with the polytope. A major open problem, known as the g-conjecture, is to extend these numerical and algebraic assertions to the larger family of simplicial sphere, and beyond.
This problem illustrates interesting relations (some are only conjectured) between combinatorics, commutative algebra, algebraic topology and geometry. We will describe these relations and indicate recent developments on the g-conjecture.
Some of the new results are joint work with Eric Babson, some are joint work with Martina Kubitzke.