The multiplicative structure on polynomial continuous valuations

Theory of valuations on manifolds, IV. New properties of the multiplicative structure

This is the fourth part in the series of articles math.MG/0503397, math.MG/0503399, math.MG/0509512 where the theory of valuations on manifolds is developed. In this part it is shown that the filtration on valuations introduced in math.MG/0503399 is compatible with the product. Then it is proved that the Euler-Verdier involution on smooth valuations introduced in math.MG/0503399 is an automorphism of the algebra of valuations. Then an integration functional on valuations with compact support is introduced, and a property of selfduality of valuations is proved. Next a space of generalized valua…

Theory of valuations on manifolds, III. Multiplicative structure in the general case

This article is the third part of the series of articles where the theory of valuations on manifolds is constructed. In math.MG/0503399 the notion of a smooth valuation on a manifold was introduced. The goal of this article is to put a canonical multiplicative structure on the space of smooth valuations on general manifolds, thus extending some of the affine constructions from math.MG/0301148, math.MG/0503397.

平面多边形上的多项式估值

Polynomial valuations on plane polygons

本文从朴素一致的观点出发，通过描述多边形上所有简单估值并刻画平移不变性的影响，得到所有多项式简单估值的描述，这是平移不变理论的新推广。

Hard Lefschetz theorem for valuations and related questions of integral geometry

We study the properties of the multiplicative structure on valuations on convex sets. We prove a new version of the hard Lefschetz theorem for even translation invariant continuous valuations, and discuss related problems of integral geometry. Then we formulate a conjectural analogue of this result for odd valuations.

凸函数上的多项式赋值及其极大扩张

Polynomial valuations on convex functions and their maximal extensions

该研究探讨凸函数不同锥上的多项式赋值扩张问题，证明其可归结为赋值支集的几何阻碍，借助齐次分解与分布支集性质推导结论，并给出最高次赋值的显式积分表示。

凸集上赋值的非阿基米德类比的 Hodge-Riemann 关系

Towards Hodge-Riemann relations for non-Archimedean analogs of valuations on convex sets

本文在非阿基米德类比空间中引入乘积与卷积两种乘法结构，提出混合硬 Lefschetz 定理和 Hodge-Riemann 关系的猜想，并证明乘积的度 1 与卷积的余度 1 的非混合 Hodge-Riemann 关系。