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HARMONIC ANALYSIS ON TREES

1. Trees. I will present the standard geometric language on trees. This is mostly a flash course in CAT(-1) geometry. 2. Tree lattices. I will discuss tree lattices and their relationship with graph theory and symbolic dynamics. 3. The Ikehara trace formula. This formula draws a link between the eigenvalues of the discrete Laplace operator and the lengths of closed geodesics on a finite graph. It is a discrete analogue of the Selberg trace formula. I will state it precisely and prove it. 4. Thermodynamic formalism and unitary representations. Give a tree lattices, I will exhibit relationships between the thermodynamic formalism on the associated subshift and the theory of its unitary representations.