Cohomological methods in integral geometry

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The Lefschetz package is a set of statements that appear in very different mathematical contexts such as polytope theory, combinatorics, algebraic geometry. It has its origin in the cohomology theory of compact Kähler manifolds. The Lefschetz package has found numerous applications, such as proofs of McMullen's g-conjecture, the Erdös-Moser and the Dowling-Wilson conjectures in combinatorics, and the Alexandrov-Fenchel inequality in convex geometry. A recent development is the appearance of a (mainly conjectural) Lefschetz package in the theory of continuous translation invariant valuations on convex bodies. A valuation is a finitely additive measure on the space of compact convex bodies in a finite-dimensional vector space, or more generally on some class of regular subsets of a smooth manifold. The Lefschetz package for valuations contains a version of the mixed hard Lefschetz theorem as well as mixed Hodge-Riemann relations. A full proof would have far reaching applications in integral geometry and in Alexandrov-Fenchel type geometric inequalities. In the first part of the project we will try to prove some important special cases of these conjectures and establish new geometric inequalities. In the second part, we will study valuations on Riemannian and Kähler manifolds, in particular on real and complex Grassmann manifolds, and their connection to cohomology, probabilistic Schubert calculus and integral geometry.