A fundamental tool in Random Matrix Theory: the Cauchy-Stieltjes transform - Part 1

• Mireille Capitaine (IMT)

We will introduce the so-called Cauchy-Stieltjes transform of a probability measure on the real line and present some of its fundamental properties. Then, we will explain how this analytic tool can be used to establish asymptotic spectral properties of random Hermitian matrices when the dimension goes to infinity.

Bézout's theorem and applications - Part 1

• Enrica Floris (IMT)

Bézout's theorem for plane curves states that two irreducible curves in the complex projective plane of degree d and e meet in exactly de points counted with multiplicity. In these lectures we will explain the statement of the theorem, give some applications and generalizations and give some elements of the proof.

An Introduction to Viscosity Solutions for Hamilton–Jacobi Equations - Part 1

• Jessica Guerand (IMAG)

This mini-course provides an introduction to viscosity solutions, a weak notion of solution introduced by Crandall and Lions. This framework is particularly well suited for Hamilton–Jacobi equations, where classical solutions may fail to exist. We will focus on first-order Hamilton–Jacobi equations, presenting the main ideas, illustrating them with examples and applications such as traffic flow modeling and optimal control, and discussing the well-posedness of these equations.

A fundamental tool in Random Matrix Theory: the Cauchy-Stieltjes transform - Part 2

• Mireille Capitaine (IMT)

We will introduce the so-called Cauchy-Stieltjes transform of a probability measure on the real line and present some of its fundamental properties. Then, we will explain how this analytic tool can be used to establish asymptotic spectral properties of random Hermitian matrices when the dimension goes to infinity.

Bézout's theorem and applications - Part 2

• Enrica Floris (IMT)

Bézout's theorem for plane curves states that two irreducible curves in the complex projective plane of degree d and e meet in exactly de points counted with multiplicity. In these lectures we will explain the statement of the theorem, give some applications and generalizations and give some elements of the proof.

An Introduction to Viscosity Solutions for Hamilton–Jacobi Equations - Part 2

• Jessica Guerand (IMAG)

This mini-course provides an introduction to viscosity solutions, a weak notion of solution introduced by Crandall and Lions. This framework is particularly well suited for Hamilton–Jacobi equations, where classical solutions may fail to exist. We will focus on first-order Hamilton–Jacobi equations, presenting the main ideas, illustrating them with examples and applications such as traffic flow modeling and optimal control, and discussing the well-posedness of these equations.