Ma 157b: Riemannian Geometry II (Spectral geometry of Riemannian manifolds)

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Instructor: Alexander Gorodnik
Office: 172 Sloan
Phone: 395-4325
E-mail: gorodnik "at" caltech "dot" edu
Office Hours: TBA and by appointment

Homework Problems

Course Description:

• Math 157b is continuation of Math 157a. This year we concentrate on the spectral geometry of Riemannian manifolds.
• The Laplacian is the most fundamental differential operator defined on Riemannian manifolds which describes propagation of heat. Its spectrum also contains vast information about the topology of the manifold, which is manifested by the Hodge theory.
• We prove spectral theorem for the Laplace operator, introduce the heat kernel, and establish the asymptotics of the eigenvalues (Weyl law).
• The first nonzero eigenvalue of the Laplacian is of fundamental importance. In this direction, we discuss the Cheeger inequality and results around Selberg 1/4 conjecture, which leads to a construction of expander graphs.
• There are many profound connections between the spectral theory and geometry. In particular, we discuss the Selberg trace formula, which relates the eigenvalues of the Laplacian and the lengths of closed geodesics on surfaces, and prove the prime geodesic theorem.
• Statistical properties of the eigenvalues and the eigenfunctions of the Laplace operator are the subject of quantum chaos theory. The main scheme here is that the chaotic properties of the geodesic flow should have quantum counterparts reflected in asymptotic distribution of eigenvalues/eigenfunctions.
• The central question of the inverse spectral theory is "whether one can hear the shape of the drum". Namely, knowing the spectrum of the Laplacian, can we determine the Riemannian manifold up to isometry.

Prerequisites: Math 157a

Grading Policy: Homework problems will be collected.

References:

• Buser, Geometry and spectra of compact Riemann surfaces, Progress in Mathematics, 106. Birkhauser Boston, Inc., Boston, MA, 1992.
• Craioveanu, Puta, Rassias, Old and new aspects in spectral geometry. Mathematics and its Applications, 534. Kluwer Academic Publishers, Dordrecht, 2001.
• Rosenberg, The Laplacian on a Riemannian manifold. An introduction to analysis on manifolds. London Mathematical Society Student Texts, 31. Cambridge University Press, Cambridge, 1997.
• Sakai, Riemannian Geometry, AMS, 1996