Mikolaj Fraczyk: Arithmetic Quantum Uniquie Ergodocity

Let $X$ be a a finite volume hyperbolic surface and let $\phi_i$ be an enumeration of the normalized eigenvectors of the Laplacian operator acting on $L^2(X)$. The Quantum Unique Ergodicity conjecture predicts that the mass of $\phi^2_i$ equidistributes over $X$ as i tends to infinity. In this talk I will explain the proof of ( non-unique) quantum ergodicity using the geodesic flow on $X$ and introduce some of the techniques used by Lindenstrauss in his proof of QUE for compact arithmetic hyperbolic surfaces.