Cohomological methods in integral geometry

积分几何中的上同调方法

• 批准号: 520350299
• 负责人: Professor Dr. Andreas Bernig
• 金额: --
• 依托单位: Schwerpunkt Analysis und Numerik
• 依托单位国家: 德国
• 项目类别: Research Grants
• 资助国家: 德国
• 项目状态: 未结题
• 来源: https://gepris.dfg.de/gepris/projekt/520350299?language=en
• 关键词: Cohomological; methods; integral; geometry

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.

Lefschetz包是一组语句，出现在非常不同的数学环境中，如多面体理论、组合学、代数几何。它起源于紧的Kähler流形的上同调理论。Lefschetz包已经找到了许多应用，例如McMullen的g猜想的证明，组合学中的Erdös-Moser和Dowling-Wilson猜想，以及凸几何中的Alexandrov-Fenchel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一个（主要是推测的）Lefschetz包。赋值是有限维向量空间中紧凸体空间上的有限加性测度，或者更一般地说，是光滑流形的一类正则子集上的有限加性测度。用于估值的Lefschetz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亚历山德罗夫-芬切尔型几何不等式中具有深远的应用。在项目的第一部分，我们将尝试证明这些猜想的一些重要的特殊情况，并建立新的几何不等式。在第二部分，我们将研究黎曼流形和Kähler流形的赋值，特别是实流形和复流形的赋值，以及它们与上同调、概率舒伯特微积分和积分几何的联系。