Back-to-School Seminar: A Tour of h-numbers

The theory of \$h\$--numbers is an incredibly rich one. In this talk, we will begin by highlighting various viewpoints that one can use to understand the \$h\$--numbers of simplicial complexes. From each new perspective, we will gain new insights, and through this journey, we will traverse the lands of commutative algebra, topology, convex geometry, and combinatorics. After seeing how each of these tools applies to simplicial complexes, we will mention how these classical techniques can be extended to the flag \$h\$--numbers of Eulerian posets. In doing so, we will illustrate a recursive decomposition that one can use to compute the \$cd\$--index of a polytope (among many other spaces). This decomposition is the main result of joint work with Felipe Caster and José Samper.