Colloquium by Karen Vogtmann

Many phenomena in mathematics and science can be modeled by drawing finite networks whose nodes are connected by edges with specified lengths. After imposing various natural constraints, the set of all such models forms a geometric space, with one point for each possible model. Although some properties of these spaces are easy to verify (such as the fact that they are connected), deeper aspects (such as whether they have “holes” of various dimensions) remain quite mysterious and are the focus of much current mathematical activity.

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