PS 101 Spectral Geometry

The Laplace-Beltrani operator associated to a Riemannian manifold. The heat kernel for a closed Riemannian manifold. The spectrum of a closed Riemannian manifold. Examples of explicit spectra. Characterizing eigenvalues of the Laplace-Beltrami operator. Asymptotic expansion of Minakshisundaram-Pleijel type and consequences: Weyl’s asymp-totic formula, geometric properties that can be derived from the spectrum, etc. Estimates for eigenvalues through geometric data. Isospectral closed Riemannian manifolds: Sunada’s theorem and generalizations.

PS 102 Lie Groups and Applications

Lie groups. Lie algebras. Exponential map. Ad(Ad*) action. Action of Lie groups on smooth manifolds. Symplectic reduction. Applications in hydrodynamics, plasma physics and rigid body motion. Stability of the equilibrium states via the Energy-Casimir method.

PS 103 Dynamical Systems

Differentiable manifolds. Fiber bundles. Connections. Lagrangian and Hamiltonian dynamical systems. Birkhoffian formalism. Conservation laws. Applications in physics, biology and economics.

PS 104 Riemannian Geometry

Fundamental concepts in Riemannian geometry: Riemannian measure, curvature, irreducible decomposition of the space of curvature tensors, geodesics, Jacobi fields. Riemannian covering. Riemannian submersion. Global concepts in Riemannian geometry : variation formulas and Jacobi fields, Morse index theorem, the Myers theorem, cut locus, Ambrose’s theorem. Comparison theorems and applications : spaces of constant curvature, comparison theorems for Jacobi fields and applications, Toponogov’s comparison theorem, symmetric spaces.

PS 105 Fibre Bundles

The categories of fibre bundles (Steenrod), principal bundles and vector bundles; sections in fibre bundles. Fields of geometric objects and connections on differential manifolds. Yang-Mills and gauge theories.

PS 106 Geometrical Techniques in Gauge Theory

Meaning of gauge nivariance. Nonrelativistic particles with spin. Yang-Mills particles. The canonical formalism and quantization. Pseudoclassical description. Local and global Lagrangians.

PS 107 Geometric Models in Relativity Theory

The symmetry in special relativity. The classical theory of relativistic free fields. Local Minkowski manifolds. The General relativity. The theory of quantum fields. Static geometric models.