讲座内容：

We will survey recent developments in the positivity of the matroid Kazhdan–Lusztig–Stanley theory. Elias, Proudfoot, and Wakefield introduced the matroid Kazhdan–Lusztig polynomial and posed two conjectures: nonnegativity and log-concavity. While the nonnegativity conjecture has been proved, log-concavity remains a central open problem. This talk reviews current progress on some invariants: the (inverse) Kazhdan–Lusztig polynomials and the (inverse) Z-polynomials. We will outline the latest results and open questions concerning their unimodality and log-concavity. Finally, we will present our recent contributions as follows: log-concavity of inverse Kazhdan–Lusztig polynomials for paving matroids;log-concavity of inverse Z-polynomials for sparse paving matroids; unimodality of Kazhdan–Lusztig polynomials for sparse paving matroids.This is joint work with Alice L. L. Gao, Yun Li, Philip B. Zhang, and Zuo-ru Zhang.