31640 - Advanced Mathematical Methods of Physics

Learning outcomes

At the end of the course, the student has acquired some mathematical tools necessary for the study of modern theoretical physics. The students learns some critical techniques, both analytical and geonetrical, essential to the theoretical physicist.

Course contents

SPECTRAL THEORY IN HILBERT SPACES (Prof. Ortolani)
Review on Hilbert spaces: separability, subspaces, orthogonality. Projection operators; continuous, closed, compact  linear operators. Symmetric and unitary operators. Normal operators. Spectral mesure. Integration. Spectral theorem  for unitary and self-adjoint operators.

DIFFERENTIAL GEOMETRY (Prof.ssa Ercolessi) Differentiable manifolds: definition and examples; maps; topological properties: homotopy groups. Tangent space: vector fields; Lie brackets; one-parameter groups. Tensors: covectors; cotangent space, tensors. Differential forms: p-forms; exterior algebra; exterior differential; volume form and orientability. Integration: integration of differential forms; Stokes theorem; remarks on homology and cohomology groups. Riemannian geometry: metric; geodesics; covariant derivative; orthogonal group; Hodge-star operator. Applications: gradient, laplacian, divergence and rotor operators; classical theorems; Maxwell equations. Symplectic geometry: symplectic form; Darboux theorem; Poisson brackets; canonical transformations; hamiltonian fields. Applications: lagrangian and hamiltonian mechanics; first integrals; examples of physical systems; notes on quantum mechanics. Fibre bundles: definitions and examples; sections; covering spaces; vector bundles; principle bundles.

Readings/Bibliography

M. Reed, B. Simon: Methods of modern mathematical physics.

N. I. Akhiezer, I. M. Glazman: Theory of linear operators in Hilbert space.

M. S. Birman, M. Z. Solomjak: Spectral theory of self-adjoint operators in Hilbert space.

C.J. Isham, Modern differential Geometry for Physicists.

Y. Talpaert, Differential Geometry with applications to mechanics and physics

Teaching methods

Lectures in class

Assessment methods

Oral exam

Teaching tools

Lecture notes at AMS Campus

Links to further information

http://www-th.bo.infn.it/activities/na41/

Office hours

See the website of Elisa Ercolessi

See the website of Fabio Ortolani