Abstract:
In geometric, algebraic, and topological combinatorics, we often study the unimodality of combinatorial generating polynomials, which in turn follows when the polynomial is (real) stable, a property often deduced via the theory of interlacing polynomials. Motivated by a conjecture of Brenti and Welker on the real-rootedness of the -polynomial of the barycentric subdivision of the boundary complex of a convex polytope, we introduce a framework for proving real-rootedness of h-polynomials for subdivisions of polytopal complexes by relating interlacing polynomials to shellability via the existence of so-called stable shellings. We show that any shellable cubical, or simplicial, complex admitting a stable shelling has barycentric and edgewise (when well-defined) subdivisions with real-rooted h-polynomials. Such shellings are shown to exist for well-studied families of cubical polytopes, giving a positive answer to the conjecture of Brenti and Welker in these cases. The framework of stable shellings is also applied to answer to a conjecture of Mohammadi and Welker on edgewise subdivisions in the case of shellable simplicial complexes,
Note: This talk begins with a pre-seminar (aimed at graduate students) at 3:30–4:00. The main talk starts at 4:10.
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Meeting ID: 915 4733 5974