Description: We will introduce the problems and methods on eigenfunctions of the Laplacian.Contents: Review of the Laplacian and the d'Alembertian, the Hadamard parametrix, the sharp Weyl formula, stationary phase and microlocal analysis, improved spectral asymptotics and periodic geodesics, classical and quantum ergodicity.Prerequisite：Real analysis, functional analysisReference：C.D. Sogge, ...
AbstractI will show the emergence of type III_1 von Neumann algebras from large N quantum mechanical models. These are effective models constructed from gauge theories on a circle. Their partition functions are matrix models that are explicitly solved in the large N limit. The spectral densities underlying the algebras are computed in closed form, and expressed in terms of the eigenvalue densit...
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