Cohomological methods in integral geometry

Lefschetz包是一组出现在非常不同的数学环境中的语句，例如多面体理论、组合学、代数几何。它起源于紧致Kähler流形的上同调理论。Lefschetz包有许多应用，如证明麦克马伦g猜想、组合数学中的S-莫泽猜想和道林-威尔逊猜想，以及凸几何中的亚历山大-芬切尔不等式。最近的一个发展是在凸体上的连续平移不变赋值理论中出现了(主要是猜测的)Lefschetz包。赋值是有限维向量空间中紧凸体空间上的有限可加测度，或者更一般地说，是光滑流形的某类正则子集上的有限可加测度。Lefschetz估值包包含混合硬Lefschetz定理的一个版本以及混合Hodge-Riemann关系。一个完整的证明将在积分几何和Alexandrov-Fichel型几何不等式中有广泛的应用。在项目的第一部分，我们将尝试证明这些猜想的一些重要特例，并建立新的几何不等式。在第二部分中，我们将研究黎曼流形和Kähler流形上的赋值，特别是实和复Grassmann流形上的赋值，以及它们与上同调、概率Schubert演算和积分几何的联系。

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.