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软件包是一组在非常不同的数学环境中出现的陈述，例如多层理论，组合学，代数几何。它起源于紧凑型kähler歧管的共同体学理论。 Lefschetz软件包已经找到了许多应用程序，例如McMullen的G-conjocture的证明，Erdös-Moser和Dowling-Wilson在Compinatorics中的猜想，以及Convex几何形状中的Alexandrov-Fenchel不平等。最近的发展是在凸体上连续翻译不变估值理论中的（主要是猜想的）lefschetz包的外观。在有限维矢量空间中或更一般地，在平滑歧管的某些常规子集上，一个值最终是在紧凑型凸体空间上的一个添加度量。用于估值的Lefschetz软件包包含混合硬Lefschetz定理的版本以及混合的Hodge-Riemann关系。完整的证明将在整体几何形状和Alexandrov-Fenchel型几何不等式中具有较远的应用。在项目的第一部分中，我们将尝试证明这些猜想的一些重要案例并确定新的几何不平等现象。在第二部分中，我们将研究Riemann和Kähler歧管的价值，尤其是关于真实和复杂的Grassmann歧管，及其与共同体的联系，概率Schubert calculus和整体几何形状。