Geometry and analysis on singular spaces

点击翻译按钮获取中文摘要

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).