Semi-monotone sets and triangulation of definable families

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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.