Reconstructing simplicial polytopes from partial information

A polytope is the convex hull of finitely many points. A polytope is simplicial if all of its faces are simplices. What partial information about a simplicial polytope P is enough to uniquely determine P (up to certain equivalences)? We will discuss two versions of this question and report on recent progress on two conjectures of Gil Kalai. Our answers involve a mixture of tools from combinatorics, discrete geometry, and commutative algebra. No prior knowledge on polytopes will be assumed.