Hard Lefschetz theorem and Hodge-Riemann-relations for unitarily invariant valuations

In the early 2000s, Alesker introduced the algebra of smooth translation invariant valuations, which turns out to satisfy a variety of properties analogous to those of the cohomology of a compact Kähler manifold, e. g. Poincaré duality, Hard Lefschetz theorem or Hodge-Riemann-relations. In 2020, Kotrbatý conjectured a mixed version of hard Lefschetz theorem and Hodge-Riemann-relations for smooth valuations, that have recently been proven in full generality by Bernig, Kotrbatý and Wannerer. Their proof not only holds for the corresponding theorems but further provides an algebraic criterion to check for different relations of a similar kind. This method is presented for the unitarily invariant case in the talk.