The main goal of this research project was to extend some fundamental geometric operations on the "tame sets" (such as semialgebraic sets defined by polynomial equations and inequalities) to one-parametric monotone (increasing) families of such sets. One such fundamental property is triangulation: partition of a set into "standard" subsets called simplices. There is one kind of a simplex in each dimension (a point, a line segment, a triangle, etc.). A triangulation conjecture formulated as one of the outcomes of this project states that for each monotone family of tame compact sets inside a compact K, there is a triangulation of K such that restriction of the family to each simplex is equivalent to a "standard family" from an explicitly described list. There are 2^n+1 standard families in dimension n (5 two-dimensional families, 9 three-dimensional, etc.). As a result of this project, the conjecture has been proved for two-dimensional monotone families, and for monotone families in R^3.

One of the most important applications of monotone families is approximation of the general (tame) sets by monotone families of compact sets. Such approximation allows one to obtain good upper bounds on the topological complexity of the general tame sets. Triangulation of monotone families implies invariance of such approximations under projections to subspaces, greatly increasing applicability of these approximations. As an outcome of this project, we obtained improved lower bounds on the computational complexity of an algebraic network, in terms of the topological complexity of the semialgebraic set associated with such a network.

A novel technical tool, the theory of semi-monotone sets and monotone maps was developed in this project. A graph of a monotone map is a multivariate generalization of a monotone (either increasing or decreasing or constant) univariate function. It can be also considered as a generalization of a convex set. In particular, such a graph is a topologically regular cell. This theory already found an application in a problem related to genomics. A generalization of Helly's theorem in convex geometry has also been proved for families of  semi-monotone sets as well as graphs of monotone maps. 

Classification of tame two-dimensional singularities with respect to bi-Lipschitz equivalence (preserving the distances between points up to multiplication by a bounded constant) was suggested as an outcome of this project. A new Lipschitz invariant, the "width function," has been discovered.

Classification of spherical quadrilaterals with at least one integer angle, and of spherical rectangles, was obtained as a result of this project. Relations between spherical quadrilaterals and real solutions of the Heun equations (second order linear differential equations with four regular singular points) and between circular quadrilaterals and real solutions of the Painleve VI equations (most general nonlinear equations without movable singular points) were established.

Amongst, other accomplishments, investigations into the quantitative aspects of semi-algebraic sets have led to proofs of more refined bounds on the topological complexities of semi-algebraic sets, which in turn has proved important in applications to incidence geometry -- including a new multi-level polynomial partitioning theorem that has led to new bounds on points-hypersurfaces incidences in higher dimensions.  A new o-minimal version of  the basic Szemeredi-Trotter theorem in incidence geometry has been proved. Topological complexity of semi-algebraic sets invariant under the action of finite reflection groups has been studied and new results on the equivariant topology of such symmetric semi-algebraic sets has been obtained. 

Last Modified: 09/28/2017
Modified by: Andrei Gabrielov