Abstract

An interesting question surfaced (pun intended) in the 1800’s: What is the largest number of regions into which one can divide a given surface so that every two regions share a segment of frontier boundary? (This question is related to, but not the same as the generalization of the four color conjecture to all surfaces) The answer was found in 1968 after a long and winding historical road.

The first part of this talk will consist of a discussion of some stretches of this fascinating road, profusely illustrated by surfaces crocheted* by the speaker with maps having the maximal number of regions (with each pair of regions sharing a segment of boundary). Some of these explicit higher genus surface maps were recently determined by undergraduate students Yanbing Gu, Connor Steward and Ajmain Yamin.

In the second part of this talk we will discuss patterns related to curves on surfaces: Two closed curves on a surface that can be deformed one into the other (sliding, stretching, contracting is allowed, breaking is forbidden) are called equivalent. Each of the members of this class crosses itself a certain number of times. The smallest number of these crossing numbers is the self-intersection number of the class. We will discuss relations between the length and the number of crossings of closed curves on a surface. There are also dramatic statistical patterns which are surprising or even unexpected by experts.

* Note for the uninitiated: crochet is a fiber craft similar to knitting

About the Speaker

Prof. Moira Chas is an Associate Professor of Mathematics at Stony Brook University, New York.