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。 该提案涉及四个一般领域的问题(见下文)。这一进展应该会揭示长期存在的猜想，并开辟新的研究方向。一个重要的焦点是奇点分解的计算方面，反映了我们在最近的文章中发展的一种哲学，即去奇点不变量可以与自然几何信息一起用于计算奇点的局部范式。主要内容有：(1)除极小奇点外的去奇异。我们的目标是找到在自然情况下必须承认的具有轻微奇点的种族等价类的模型。(2)Hilbert-Samuel函数的微积分，这是一个基本的去奇异不变量，它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，以求奇异簇上的微分算子演算。(4)奇异集可微函数的延拓，保持重要的几何性质(发展惠特尼延拓问题的最新贡献)。

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).