Коллоквиум. Spaces of finite graphs

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Many phenomena in mathematics and science can be modeled by drawing finite graphs whose nodes are connected by edges with specified lengths. The set of all such models forms a geometric space, with one point for each possible model. Although some properties of this space are easy to verify, deeper aspects (such as their cohomology) remain quite mysterious and are the focus of much current mathematical activity. In this talk I will describe these spaces and explain why they are of particular interest to mathematicians studying group theory and low-dimensional topology. At the end I hope to explain some recently discovered connections with number theory.