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建议的总目标是更好地理解双有理几何，分析和亚解析几何中的奇点。这些主题涉及的范围很广。这些问题取决于类的功能考虑（代数，分析，微分，正式）。我们以前的工作已经开发了“微分”技术的奇异空间，同时适用于各种类，导致发现之间的联系代数，几何，组合和分析方面的奇点（例如，令人惊讶的作用，可微函数在次解析几何）。我们的算法建设性的解决方案的奇点开辟了一个以前无法进入的领域。

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.