Large Structures Seminar: Nicolai Vorobjov

There are many ways to generalize the familiar concept of a continuous univariate monotone function $R \to R$ to cover functions in many variables or maps $R^n \to R^m$. I will suggest a number of equivalent definitions which imply that graphs of monotone maps are topologically regular cells. Since the property of being a monotone cell is usually easier to verify than regularity as such, it gives a convenient tool to prove regularity of certain sets of particular interest. The definition of regularity resembles the standard definition of convexity, and it is possible to prove the analogy of Helly’s theorem for graphs of monotone maps.