A Proof of Grünbaum’s Lower Bound Conjecture for polytopes, lattices, and strongly regular pseudomanifolds

Abstract:

In 1967, Grünbaum conjectured that any $d$-dimensional polytope with $d+s\leq 2d$ vertices has at least $${\varphi }_k(d+s,d)=(\genfrac{}{}{0ex}{}{d+1}{k+1})+(\genfrac{}{}{0ex}{}{d}{k+1})-(\genfrac{}{}{0ex}{}{d+1-s}{k+1})$$ $k$-faces. In the talk, we will prove this conjecture and discuss equality cases. We will then extend our results to lattices with diamond property (the inequality part) and to strongly regular normal pseudomanifolds (the equality part). We will also talk about recent results on $d$-dimensional polytopes with $2d+1$ or $2d+2$ vertices.

Note: There will be no pre-seminar this week—the main talk will start at 3:30. This talk will be online only.

Join Zoom Meeting: https://washington.zoom.us/j/91547335974
Meeting ID: 915 4733 5974