Semi-monotone sets and triangulation of definable families

Andrei Gabrielov和Sugata Basu建议将可在o-极小结构中定义的集合上的一些基本几何运算推广到这类集合族的单参数可定义。特别地，给定可定义紧K的一个单参数单调(递增)可定义子集族，目标是构造K的一个可定义三角剖分，使得在三角剖分的每个单形内，该族等价于由Lex单调布尔函数分类的其中一个“标准”族。其他可以扩展到可定义族的经典几何结构包括柱面胞格分解和惠特尼分层。这种几何结构的存在将使人们能够研究一个可定义族的“精细结构”，并计算它的拓扑不变量，如零同调、交同调以及它的Hausdorff极限的同伦型。单调单参数可定义族也可以看作是可定义函数的次水平集族，因此所提出的研究可以看作是可定义函数奇点的拓扑分解。这项研究的最初动机来自Gabrielov和Vorobjov发展的同伦等价可定义紧集族逼近可定义集的理论。可定义族的三角剖分将为证明该理论的主要猜想提供一个重要的工具，所提出的研究将极大地增强我们对可定义在o-极小结构中的集合的几何、拓扑和组合学的理解，以及对此类集合族的理解。它为可定义函数奇点的求解提供了一种新的途径。即使对于所有o-极小结构的最基本(也是最重要的应用)的实半代数集，预期的结果也是新的。这项拟议的研究将对纯数学和应用数学的几个不同领域产生影响。首先，它将介绍实代数和o-极小几何领域的基本新工具，这些工具将对研究任意o-极小结构中可定义集的几何和拓扑性质，包括本文中奇点的拓扑分解产生直接影响。它还将潜在地对目前在代数几何和拓扑组合学中活跃的某些领域产生影响。最后，半单调集和单调映射的理论很有可能在离散和计算几何的非常活跃的领域(围绕持久同调理论)以及在控制理论和动力系统中得到应用。最近出现了一个这样的应用程序，即“环形立方体”(Toric Cube)。这些与系统发育中的边积集有关的半代数集是单调映射图的闭包，因此它们是拓扑闭球。在更高的层面上，拟议的研究将把最初在o-极小几何背景下开发的思想和技术应用于当前几个其他领域的重要问题-特别是代数几何、离散和计算几何以及控制理论。

Andrei Gabrielov and Saugata Basu propose to extend some fundamental geometric operations on the sets definable in an o-minimal structure to one-parametric definable families of such sets. In particular, given a one-parametric monotone (increasing) definable family of subsets of a definable compact K, the goal is to construct a definable triangulation of K such that, inside each simplex of the triangulation, the family is equivalent to one of the "standard" families, classified by lex-monotone Boolean functions. Other classical geometric constructions that can be extended to definable families include cylindrical cell decomposition and Whitney stratification. Existence of such geometric constructions would allow one to investigate the "fine structure" of a definable family, and to compute its topological invariants, such as vanishing homology, intersection homology and the homotopy type of its Hausdorff limit. A monotone one-parametric definable family can be alternatively viewed as the family of sub-level sets of a definable function, so the proposed research can be viewed as a topological resolution of singularities of definable functions. The original motivation for the proposed research comes from the theory of approximation of definable sets by homotopy equivalent definable families of compact sets developed by Gabrielov and Vorobjov. Triangulation of a definable family would provide a crucial tool for the proof of the main conjecture of that theory.The proposed research would substantially enhance our understanding of geometry, topology and combinatorics of the sets definable in an o-minimal structure, and of the families of such sets. It suggests a new approach to the resolution of singularities of definable functions. The expected results would be new even for real semi-algebraic sets, the most basic (and the most important in applications) of all o-minimal structures. The proposed research will have impact in several different areas of pure and applied mathematics. Firstly, it will introduce fundamental new tools in the areas of real algebraic and o-minimal geometry, which will have direct impact in the the study of geometric and topological properties of definable sets in arbitrary o-minimal structure, including topological resolution of singularities in this context. It will also potentially have impact in certain areas of currently active interest in algebraic geometry and topological combinatorics. Finally, it is very likely the theory of semi-monotone sets and monotone maps will find applications in the extremely active areas of discrete and computational geometry (around the theory of persistent homology), as well as in control theory and dynamical systems. One such application to "toric cubes" emerged recently. These semi-algebraic sets which are related to edge-product sets in phylogenetics are closures of graphs of monotone maps, thus they are topologically closed balls. At a higher level, the proposed research will bring ideas and techniques developed originally in the context of o-minimal geometry, to currently important problems in several other areas - in particular, algebraic geometry, discrete and computational geometry and control theory.