Semi-monotone sets and triangulation of definable families

Andrei Gabrielov和Saugata Basu提出将一些基本的几何运算推广到o-极小结构中可定义的集合上，并推广到此类集合的单参数可定义族。特别是，给定一个可定义紧致K的子集的单参数单调（递增）可定义族，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，该族等价于一个“标准”族，由lex-monotone布尔函数分类。其他经典的几何结构，可以扩展到可定义的家庭包括圆柱形细胞分解和惠特尼分层。这样的几何构造的存在将允许人们研究一个可定义族的“精细结构”，并计算其拓扑不变量，如消失同调、相交同调及其Hausdorff极限的同伦类型。单调单参数可定义函数族可以看作是可定义函数的子水平集族，因此本文的研究可以看作是可定义函数奇异性的一种拓扑解析.提出的研究的最初动机来自于Gabrielov和Vorobjov开发的同伦等价可定义紧集族对可定义集的逼近理论。可定义集合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，所提出的研究将大大提高我们对可定义集合族的几何、拓扑和组合学的理解。它提出了一种新的方法来解决可定义函数的奇异性。预期的结果将是新的，即使是真实的半代数集，最基本的（和最重要的应用）的所有o-最小结构。拟议的研究将在纯数学和应用数学的几个不同领域产生影响。首先，它将介绍真实的代数和o-极小几何领域的基本新工具，这将直接影响到任意o-极小结构中可定义集的几何和拓扑性质的研究，包括奇点的拓扑分解。它也将有可能在某些领域的影响，目前积极关注的代数几何和拓扑组合。最后，半单调集和单调映射的理论很可能会在离散和计算几何（围绕持久同调理论）以及控制理论和动力系统的非常活跃的领域中找到应用。 最近出现了一个这样的应用程序“复曲面立方体”。这些半代数集与遗传学中的边积集有关，它们是单调映射图的闭包，因而是拓扑闭球。在更高的层次上，拟议的研究将带来的想法和技术开发的背景下，O-最小的几何，目前重要的问题，在其他几个领域-特别是代数几何，离散和计算几何和控制理论。

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.