Abstracts
for the Seminar

Fall 2016

**Speaker: **Ernest Chong, Nanyang Technological University

**Title: **Beyond the $g$-theorem

**Time:** 2:30 PM, Tuesday, December 6, 2016

**Place:** Malott 206

**Abstract:** The $g$-theorem is a momentous result in geometric combinatorics that characterizes the $f$-vectors of simplicial polytopes $\Delta$. In all its known proofs, a suitable graded algebra $R$ is considered, whose h-vector equals the h-vector of $\Delta$, and the difficult part is to show that a certain Artinian reduction of $R$ satisfies the (weak) Lefschetz property. More recently, there has been much interest in the $g$-conjecture, which asserts that the $g$-theorem can be extended to all simplicial homology spheres. In this talk, we introduce the "stress algebra", whose existence is proven using ideas from rigidity theory. We show that the stress algebra is an Artinian Gorenstein graded algebra that can be associated to any simplicial homology sphere, and we explain its relevance to the $g$-conjecture. This is ongoing joint work with Tiong Seng Tay.

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