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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. In this talk, Karen Vogtmann of Warwick University will describe these spaces and show why you might expect them to have holes. She will then explain why these spaces are of particular interest to mathematicians studying group theory and low-dimensional topology.