We study the cohomology ring of the Bott--Samelson varieties, which are useful in studying the topology of compact Lie groups and constructing resolutions of Schubert varieties. We compute an explicit presentation of this ring via Soergel's result, which implies that it is a combinatorial invariant. We prove that it is a Koszul algebra and we provide a quadratic (reduced) Gröbner basis. Furthermore, we prove that it satisfies the whole Kähler package, including the Poincaré duality, the hard Lefschetz theorem, and the Hodge--Riemann bilinear relations. I will also discuss an open question on the isomorphism problem on Bott—Samelson varieties. This is the product of a project of the PACE 2024 program (a research experience for undergraduates) at Peking University, joint with Lin Sun (Fudan University), Shihao Wang (Tsinghua University) and Haoyu Zhu (Sun Yat-sen University).
