COURSES (Academic year 2020/2021)

*Click on the ► sign to see the content of a course.*

**First Year (M1)**

**First semester**

## Differential geometry

Differentiable manifolds. Vector fields and flow-box theorem. Differential forms and Stokes' theorem. Tensors and vector bundles. Riemannian manifolds and connections. Geometry of gauge fields.

## Fourier analysis

Fourier series for periodic functions of a real variable, Riemann–Lebesgue lemma, Dirichlet's conditions
and Parseval's theorem.
Convolution of two complex-valued functions defined on *R ^{d}*. Approximation to the identity.
Fourier transform on

*R=L*and Fourier inversion theorem. Plancherel theorem and Fourier transform on

^{1}*R=L*. Application to the resolution of some partial differential equations: Schrödinger equations, wave equations and heat equations.

^{2}## Ordinary differential equations

General existence and uniqueness theorems. Linear differential systems. Floquet theorem. Nonlinear autonomous systems. Phase portraits in two dimensions. Stability theorems. Elementary bifurcations in one dimension.

## Quantum Physics

1. Introduction:

1.1 Observables in classical mechanics

1.2. Finite dimensional model of quantum mechanics

2. Basic principles of quantum mechanics:

2.1 States and observables in quantum mechanics

2.2 Quantum entanglement

2.3 Heisenberg uncertainty principle

2.4 Coordinate and momentum representations

2.6 Quantum dynamics: Schrödinger and Heisenberg pictures

2.5 Schrödinger equation

2.6 Classical limit

3. Quantum mechanics in one dimension:

3.1 Harmonic oscillator. Creation and annihilation operators

3.2 Scattering problem in one dimension

4. Quantum mechanics in 3D:

4.1 Free particle

4.2 Rotation group and angular momentum

4.3 Hydrogen atom

4.4 Spin

5. Multi-particle quantum systems, introduction

**Second semester**

## Groups and representations

Notion of a group representation. Development of the structure theory for complex representations of finite groups: Theorems of Maschke and Schur. Tensor products and duality. Character theory. Induced representations. Some outlook beyond finite groups as time permits.

## Mathematical methods of classical mechanics

Lagrangian and Hamiltonian formalisms. Hamiltonian systems on symplectic manifolds. Variational principle and Hamilton-Jacobi equations. Poisson manifolds. Symmetries and momentum map.

## Numerical methods for Physics

Interpolation and/or Linear systems. Numerical integration (classical rules, Gaussian quadrature rules). Fourier approximation. Numerical methods for solving ODE & PDE.

## Partial differential equations

Distributions on ^{n}

**Second Year (M2)**

**First semester**

## Mathematical methods of quantum physics

Mathematical framework of quantum mechanics. Field theory: Symmetries, Noether theorem. Lorentz invariance. Representations of the Lorentz group. Klein-Gordon equation. Free bosons. Spinors. Dirac field. Quantum fermions. Introduction to Quantum Integrable Systems: lattice and field theories.

## Riemann geometry and integrable systems

The course will be an introduction to Riemann surfaces and to some of their relations with integrable systems. We will start with an introduction to classical integrable systems: review of classical Hamiltonian mechanics, Poisson manifolds, Liouville-Arnold theorem. Later we will focus on Riemann surfaces, with the (ideal) aim of constructing solutions to the KP hierarchy via the Baker-Akhiezer functions. Topics will include: definition of Riemann surface and basic examples, plane algebraic curves, hyperelliptic curves, holomorphic coverings, fundamental group, Riemann-Hurwitz theorem, homology groups, abelian differentials, meromorphic functions, Abel theorem, Riemann-Roch theorem.

## Lie groups and Lie algebras

1. Lie algebras: Basic definitions. Ideals and Lie subalgebras. Lie theorems. Real and complex forms. Universal enveloping algebra. Poincaré-Birkhoff-Witt theorem. Campbell-Hausdorff formula. 2. Structure of Lie algebras: Solvable, nilpotent and semisimple Lie algebras. Killing form. Lie and Engel theorems. Cartan criterion. Jordan decomposition. 3. Semisimple Lie algebras: Cartan subalgebra. Root system. Dynkin diagram. Classification of simple Lie algebras. Finite dimensional representations of sl(2).

**Second semester**

## Quantum Field Theory

## Quantum groups

The course is an introduction to the elements of Hopf algebra theory. The aim is to understand the modern examples of such objects associated to the term “quantum group”. In particular, these can be quantized enveloping algebras of Lie algebras or certain deformations of the function algebras on matrix groups. Topics: Bialgebras and Hopf algebras. Classical examples from group and Lie algebra theory. Representations : modules and comodules ; tensor categories. Braided tensor categories ; (Co)quasitriangular Hopf algebras, R-matrices and the Drinfeld double. Examples of quantum enveloping algebras and deformed matrix groups.

## Mathematical methods of gravitation

Gravity as spacetime curvature. A first case study: spherical collapse. Gravitational collapse in General Relativity: theorems, conjectures and initial value problem. Black hole region and Horizons. The Kerr solution. Particle motion in a black hole background. Physics of black holes: spin bounds, no-hair properties, gravitational radiation. Classical and quantum fields in the presence of a black hole.

*The students should prepare a master's thesis under the supervision
of a researcher of the institute. Every year, several master's thesis proposals
are available, covering a broad range of research topics.*