Convex valuation theory and integral geometry

凸几何是一个经典而又迅速发展的数学分支，它以各种形式出现在各种各样的数学学科中，从代数和辛几何，到概率论和组合学，再到算法。积分几何在十九世纪以几种形式出现。它通过研究整体不变量(如全长)而不是局部不变量(如曲率)来区别于微分几何。*我的研究涉及赋值理论。赋值理论始于Dehn对Hilbert关于三维空间中多面体体积定义的第三个问题的解决。在最简单的设置下，估值是凸体上的连续变化的有限可加度量。凸体的许多积分几何不变量，如体积和表面积，实际上都是赋值的。今天，赋值理论是凸几何、积分几何、随机几何和几何概率的核心。赋值理论中的一个自然问题是找出在一组对称下不变的所有赋值。例如，体积和表面积在实体的平移和旋转过程中保持不变。当对称的各向同性群是紧凑的时，例如旋转群，在过去15年中开发的大型工具箱就可以很容易地用于分类问题。我主要对对称的非紧群感兴趣，在那里必须开发新的方法。*著名的等周不等式有一个深远的推广，即亚历山大-芬切尔不等式，它涉及一种特殊类型的估值，称为混合体积。因此，寻找其他涉及赋值的几何不等式是很自然的。*积分几何中的另一个核心角色是Radon变换，它名义上是用它在直线上的积分来替换函数，并形成了层析成像领域的基础，用于医学成像、地震学等。赋值理论提供了Radon变换的广泛推广，我计划探索这一点。**

Convex geometry is a classical, yet rapidly developing branch of mathematics, appearing in various guises in a great variety of mathematical disciplines, ranging from algebraic and symplectic geometry, through probability theory and combinatorics, and on to algorithmics. Integral geometry appeared in the nineteenth century in several flavors. It distinguishes itself from differential geometry by studying global invariants (such as total length), rather than local invariants (such as curvature).*** ***My research concerns valuation theory. Valuation theory started with Dehn's solution of Hilbert's third problem on the definition of volume of polytopes in 3-dimensional space. In the simplest setting, valuations are continuously varying, finitely-additive measures on convex bodies. Many of the integral-geometric invariants of convex bodies, such as volume and surface area, are in fact valuations. Today, valuation theory is central in convex geometry, integral geometry, stochastic geometry and geometric probability.*********A natural problem in valuation theory is to find all valuations that are invariant under a group of symmetries. For example, volume and surface area remain unchanged during translations and rotations of a body. When the isotropy group of symmetries is compact, such as the group of rotations, a large toolbox, developed over the past 15 years, is readily available for the classification problem. I am interested primarily in non-compact groups of symmetries, where new approaches must be developed. *********The famous isoperimetric inequality has a far reaching generalization, the Alexandrov-Fenchel inequality, which concerns valuations of a particular kind known as mixed volumes. It is therefore natural to look for other geometric inequalities involving valuations.*********Another central player in integral geometry is the Radon transform, which nominally replaces a function by its integrals over lines, and forms the basis of the field of tomography, used in medical imaging, seismography etc. Valuation theory offers a vast generalization of the Radon transform, which I plan to explore.**