Abstract: This talk aims to advertise a pattern/phenomenon that has emerged in many different mathematical areas during the past decades but is not currently well-understood. I will begin with a broad overview of the Kahler packages (Poincare duality, Hard Lefschetz, and Hodge-Riemann relations) that appear in topology, geometry, algebra, and combinatorics, from the classical work of Poincaré, Lefschetz, Hodge, to the recent work of last year's Fields medalist June Huh, in a down-to-earth way. Then I will discuss two new Kahler packages we discovered that are equivariant and might have no geometric origin. The equivariant log-concavity conjectures of the cohomology of configuration spaces and flag manifolds hints at our discoveries. Partly based on joint work with Rui Xiong.
