本项目研究几何测度论中受到广泛关注并且相互关联的两个问题：.Plateau问题与切结构分析。Plateau问题是推动几何测度论理论兴起的根本问题，主要目标是对一大类具有某种极小性质的几何对象（如肥皂膜）进行刻划。对于余维大于一，或空间维数高于三维时的情形，人们对 Plateau问题解的奇点附近结构的认识非常有限，很多基本问题未获解决。切结构分析是几何测度论中研究奇点的一个基本思想，意在将光滑对象中切空间的思想引入远为复杂的非光滑对象，亦即通过集合上支撑的测度在无穷小处的渐近极限来研究奇点周围的渐近性质。本项目将重点探讨以下问题：Plateau问题中的奇点分类，奇点附近切结构的唯一性，以及通过切结构来对集合进行参数化。

We propose to investigate two interrelated problems that are of main interest in geometric measure theory: Plateau’s problem and tangent structure analysis. Plateau's problem is one of the fundamental problem in geometric measure theory, which aims at understanding the existence and regularity of a large class of geometric objects that admit some minimal property, such as soap films. So far, in case of dimension bigger than 3 or codimension larger than 1, the structure around the singularities of the solutions of Plateau’s problem is far from clear, and many fundamental problems are unsolved. Tangent structure analysis is a main tool of studying singularities in geometric measure theory. The basic idea is to introduce the idea of tangent space (for smooth objects) to much more complicated non smooth objects: that is, we study the infinitesimal behavior of a set around singularities by investigating the infinitesimal limits of measures supported on the set. Due to the above motivation, this project will discuss in turn the following questions: the classification of singularities in Plateau’s problem, the uniqueness of the tangent structure near singularities, and the parameterization of a set around a singular point by the tangent object.