Resolution of Singularities and its applications. Analysis on and geometry of singular spaces.

奇点在许多数学分支中表达形式的不规则性，其方式类似于它在日常语言中的含义，并且是大多数数学及其应用中的基本研究对象。形式的重要特征往往集中在奇点上。我的研究目的是找到奇点的几何学和分析中所编码的信息之间的联系。这使我发现了奇点的代数、解析和几何方面之间的基本联系，特别是在解决惠特尼、托姆和平中--几何和代数奇点理论的创始人--提出的长期存在的问题时。这也使我将经典的Sobolev-Nirenberg和Bernstein-马尔可夫不等式推广到奇异环境，并发现了可以在其上进行经典局部分析的“驯服”次分析集。

Singularities express irregularities of form in many branches of mathematics in a way similar to what it means in an everyday language and are a basic object of study in most of the mathematics and its applications. The important features of form are often concentrated at singularities. The objective of my research is to find links between the information encoded in the geometry of and the analysis on singular objects.This led me to a discovery of fundamental links between algebraic, analytic and geometric aspects of singularities, particularly in solutions of longstanding problems posed by Whitney, Thom and Hironaka - the originators of the singularity theories in geometry and algebra. This also led me to an extension of classical Sobolev-Nirenberg and Bernstein-Markov inequalities to a singular setting, and to a discovery of `tame' subanalytic sets on which one can do classical local analysis.