Geometry and analysis on singular spaces

奇点描述了数学及其应用的许多分支中的空间和函数的不规则性。我们最近的研究和这个提议的总体目标是更好地理解二次几何、分析和次解析几何中的奇点。这些主题跨越了广阔的区域。这些问题取决于所考虑的函数类别(代数函数、解析函数、可微函数、形式函数)。我们以前的工作为奇异空间开发了同时适用于各种类别的“微积分”技术，从而发现了奇点的代数、几何、组合和分析方面之间的联系(例如，可微函数在次解析几何中的惊人作用)。我们的构造性分解奇点的算法开辟了一个以前无法进入的区域。

Singularities describe irregularities of spaces and of functions in many branches of mathematics and its applications. The general goal of our recent research and this proposal is to better understand singularities in birational geometry, analysis and subanalytic geometry. These subjects span a vast area. The questions depend on the classes of functions considered (algebraic, analytic, differentiable, formal). Our previous work has developed "differential calculus" techniques for singular spaces, that apply simultaneously to the various classes, leading to the discovery of links among algebraic, geometric, combinatorial and analytic aspects of singularities (for example, the surprising role of differentiable functions in subanalytic geometry). Our algorithm for constructive resolution of singularities has opened up a previously inaccessible area.