Geometry and analysis on singular spaces

数学中的奇点描述空间和函数的不规则性，就像日常语言中的奇点意味着物体或行为的不规则性一样。因此，奇点是大多数数学学科及其应用中的基本思想。物体的重要特征往往集中在奇点上。我的研究目标是使用几个数学领域的强大工具--代数、分析和几何--更好地理解奇点。数学中的许多重要问题取决于所考虑的空间或函数的类别(代数、解析、可微等)。我试图采取一种普遍的方法，为奇异空间开发一种同时适用于各种类别的“微积分”。这导致了奇点的代数、组合和分析方面之间意想不到的联系的发现，并解决了奇点理论的先驱惠特尼、托姆和平中的问题。我的相对初级的奇点分解算法(投影到光滑或非奇异空间的奇异对象上)有助于揭开奇异空间的隐藏特征。它打开了一个以前无法触及的领域，并提供了一些技术来解决不同领域的基本问题，如逻辑和分析。我正在研究的问题在物理和信息科学中具有潜在的重要意义。例如，奇点分解算法的有效实现是计算机代数中一个活跃的研究领域。

Singularities in mathematics describe irregularities of spaces and functions, in the same way that a singularity in everyday language means an irregularity of an object or of behaviour. Singularity is thus a basic idea in most mathematical disciplines and their applications. The important features of an object are often concentrated at the singularities. The goal of my research is to better understand singularities using the powerful tools of several areas of mathematics -- algebra, analysis and geometry.Many important questions in mathematics depend on the class of spaces or functions considered (algebraic, analytic, differentiable,...). I have tried to take a universal approach, developing a "differential calculus" for singular spaces, that applies simultaneously to the various classes. This has led to the discovery of unexpected links among algebraic, combinatorial and analytic aspects of singularities, and to the solution of problems of Whitney, Thom and Hironaka -- the pioneers of singularity theory.My relatively elementary algorithm for resolution of singularities (projection onto a singular object of a space that is smooth, or nonsingular) helps to unravel the hidden features of singular spaces. It opens a previously inaccessible area, and provides techniques to attack fundamental problems in diverse areas like logic and analysis.The problems that I am working on are potentially important in the physical and information sciences. Efficient implementation of algorithms for resolution of singularities, for example, is an active area of research in computer algebra.