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。 该提案涉及四个一般领域的问题（见下文）。这些进展应能阐明长期存在的问题，并开辟新的研究方向。一个重要的焦点是计算方面的解决方案的奇异性，反映了哲学在我们最近的文章中开发的desingularization不变量可以与自然的几何信息一起使用，以计算当地的正规形式的奇异性。主要内容有：（1）除极小奇点外的去奇异化。我们的目标是找到模型的双有理等价类具有温和的奇异性，必须承认在自然情况下。(For例如，为了同时解决曲线族的奇点，必须允许具有法向交叉奇点的特殊纤维。(2)希尔伯特-塞缪尔函数的微积分，一个基本的去奇异化不变量，在次解析集的分析中也起着重要的作用。(3)度量的去奇异化，着眼于奇异簇上微分算子的微积分。(4)从奇异集扩展可微函数，保持重要的几何性质（发展最近的贡献惠特尼扩展问题）。

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. The proposal addresses problems in four general areas (below). Progress should shed light on long-standing conjectures and open new directions of research. An important focus is computational aspects of resolution of singularities, reflecting a philosophy developed in our recent articles that the desingularization invariant can be used together with natural geometric information to compute local normal forms of singularities. The main subjects are: (1) Desingularization except for minimal singularities. The goal is to find models of birational equivalence classes with mild singularities that have to be admitted in natural situations. (For example, to simultaneously resolve the singularities of a family of curves, one has to allow special fibres with normal crossings singularities.) (2) Calculus of the Hilbert-Samuel function, a fundamental desingularization invariant that plays an important role also in analysis on subanalytic sets. (3) Desingularization of a metric, with a view towards a calculus of differential operators on singular varieties. (4) Extension of differentiable functions from singular sets, preserving important geometric properties (developing recent contributions to the Whitney extension problem).