In this talk I will introduce a notion of monotonicity for definable maps in an
o-minimal structure (for example, semi-algebraic maps). This generalizes the usual notion of monotonicity for functions of several variables as well as for vector functions mapping $\mathbb{R}$ to $\\mathbb{R}^n$. The graphs of such maps, which we call monotone cells, while not necessarily convex, share some properties of convex sets. For example, they satisfy a version of Helly's theorem. The main result is that any monotone cell $C$ is topolgically regular (in other words the pair $(\overline{C},C)$ is definably
homemorphic to the pair $([0,1]^{\dim(C)}, (0,1)^{\dim(C)})$. Since, the property of being a monotone cell is easier to verify than regularity it gives a convenient tool to prove regularity of certain semi-algebraic sets. I will mention some applications in the context of triangulations of monotone families, as well as some conjecture applications having to do with the theory of total positivity of matrices. (Joint work with A. Gabrielov and N. Vorobjov.)
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