In a seminal paper Kalai (1983) extended the notion of a tree to higher dimensions. Formally, an n-vertex d-dimensional hypertee is a Q-acyclic simplicial complex with a full (d-1) dimensional skeleton and {n-1 \choose d} d-dimensional faces. We will use instead an equivalent intuitive definition that relies only on elementary linear algebra. In this talk I will try to give a flavor of these exciting concepts. I will discuss several of the many open problems that arise here and describe some of our new discoveries.

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