Master Mathematics

I. Introduction to differential equations
II. Basics of analysis and linear algebra
1. Basics of topology
2. Basics of linear algebra
3. Basics of differential calculus
III. Cauchy–Lipshitz theorem
1. Fixed point theorem and applications
2. Cauchy-Lipschitz theorem
IV. Linear differential equations
1. Equations with constant coefficients
2. Resolvent and perturbation theory
3. Equations with periodic coefficients
V. Nonlinear differential equations
1. Lifetime of solutions
2. Perturbation theory
3. Flows and vector fields
4. Stability analysis

Algebra (5 ECTS)

Differential geometry (5 ECTS)

I. Advanced differential calculus
1. Basics of differential calculus in Rⁿ
2. Differential calculus in Banach spaces
3. Immersions, submersions, constant rank theorem
II. Submanifolds
1. Equivalent definitions of submanifolds
2. Diffeomorphisms
3. Tangent space and tangent bundle
4. Vector fields and integral curves
III. Surfaces
1. Second fundamental form
2. Elements of Riemannian geometry

Continuous-time stochastic processes (5 ECTS)

Statistics (5 ECTS)

1. Statistical models
2. Basic notions of pointwise estimation: bias, quadratic risk, convergence
3. Construction of estimators: method of moments and maximum likelihood estimation
4. Optimal estimation: sufficient statistics, complete statistics, Lehmann-Scheffé theorem, Fisher information, Cramér-Rao inequality. Exponential families.
5. Confidence regions
6. Statistical tests

The goal of this lecture is to review quickly algebraic notions already covered in the bachelor's degree: especially linear algebra and a bit of algebraic structures.

1. Vector spaces
2. Linear maps
3. Eigenvectors and eigenvalues
4. Reduction of endomorphisms
5. Bilinear and quadratic forms
6. Orthogonality
7. Algebraic structures

Finite element methods (4 ECTS)

Modelling (4 ECTS)

This lecture deals with the main themes of numerical modelling. It is aimed at various student profiles: those wishing to deepen their knowledge in numerical analysis, or improve their programming. It will alternate between theoretical sessions (convergence of algorithms,...) and numerical work in Python with Jupyter Lab. The lecture will cover many classical algorithms of numerical analysis, related to mathematical analysis in the broad sense, and will contain precise applications in its exercises. The goal is to obtain strong bases in programming, fluency in coding, and an understanding of the mathematical results at the heart of the algorithms.

1. Graphics
2. Linear systems
a. Direct methods
b. Iterative methods
c. Principal component analysis
3. Numerical integration
a. Newton-Cotes formulas
b. Monte-Carlo method
4. Function approximation
a. Lagrange interpolation
b. Fourier transform
5. Nonlinear systems
a. Iterative methods in dimension one
b. Newton-Raphson algorithm
6. Optimization
a. Least squares
b. Descent algorithms
c. Constraints and Lagrange multipliers
7. Ordinary differential equations
a. Explicit and implicit Euler schemes
b. High order schemes
c. Numerical illustration of the properties of solutions
8. Partial differential equations
a. Introduction to finite differences
b. Poisson equations, transport equations, and heat equations in dimension one

Dynamics of parabolic equations

This lecture will deal with the study of evolution equations modelling the phenomena of advection, diffusion and reaction. These partial differential equations appear for example in fluid mechanics, in population dynamics or in combustion. The goal is to understand how the solutions behave asymptotically, either in large time or near the singularities that they could form in finite time. A universal phenomenon will then be observed: resolution into self-similar solutions. The lecture will begin with classical examples of equations for which representation formulas have been discovered, then it will describe the modern analytical framework, which allows to study the solutions in the absence of such formulas (tools of perturbative analysis, harmonic and spectral analysis).

1. Representation formulas for linear equations
2. Local resolution of quasilinear transport equations
3. Classical solutions of the Burgers equation
a. Cauchy problem
b. Self-similar resolution of singularities
4. Heat equation in Lebesgue spaces
a. Semi-group decay estimates
b. Long time behavior
5. Viscous Burgers equation
a. Cauchy problem
b. Convergence towards blow-up profiles or solitary waves
6. Weak solutions of the Burgers equation
a. Evanescent viscosity limit
b. Convergence to blow-up profile
7. Local Cauchy problem for semilinear parabolic equations
8. Formation of singularities for the Keller-Segel system
a. Virial formulas
b. Backward self-similar profiles
9. Stability analysis
a. Renormalization and modulation
b. Spectral theory
c. Nonlinear stability

Regularity for partial differential equations: elliptic equations, homogenization and fluid mechanics

The question of whether solutions of Partial Differential Equations (PDEs) are regular or not is central in the field. One of the most famous open problems is that of the global existence of smooth solutions to the Navier-Stokes equations in fluid mechanics, or the finite-time breakdown of regularity (Millenium problem of the Clay's institute). Another famous problem is Hilbert's 19^th problem on the regularity of minimizers of certain functionals in the calculus of variations. This problem was solved in three independent works by De Giorgi, Nash and Moser in the late fifties.

The purpose of these lectures is to give some fundamental tools for the analysis of the regularity of PDEs of elliptic or parabolic type. The material presented in the course is well-known to the PDE community since (at least) the eighties. However some results (De Giorgi-Nash-Moser theorem, epsilon-regularity for the Navier-Stokes equation, uniform estimates in homogenization) are still inspiring new mathematical developments today.

Outline of the lecture:
1. Constant or smooth coefficient elliptic equations (Caccioppoli's inequality, perturbative methods)
2. Crash course in Harmonic Analysis (Calderon-Zymund decomposition, analysis of singular integral operators)
3. Regularity for elliptic equations with bounded measurable coefficients (non perturbative methods of Moser and De Giorgi)
4. Improved regularity in homogenization (compactness methods, quantitative approach, Liouville-type theorems)
5. Epsilon-regularity for the Navier-Stokes equations (compactness proof by Lin)

References:
- Evans, Partial Differential Equations.
- Giaquinta and Martinazzi, An Introduction to the Regularity Theory for Elliptic Systems, Harmonic Maps and Minimal Graphs.
- Seregin, Lecture Notes on Regularity Theory for the Navier Stokes Equations.
- Lemarié-Rieusset, The Navier-Stokes Problem in the 21st Century.

The goal of this lecture is to review quickly algebraic notions already covered in the bachelor's degree: especially linear algebra and a bit of algebraic structures.

1. Vector spaces
2. Linear maps
3. Eigenvectors and eigenvalues
4. Reduction of endomorphisms
5. Bilinear and quadratic forms
6. Orthogonality
7. Algebraic structures

This lecture deals with the main themes of numerical modelling. It is aimed at various student profiles: those wishing to deepen their knowledge in numerical analysis, or improve their programming. It will alternate between theoretical sessions (convergence of algorithms,...) and numerical work in Python with Jupyter Lab. The lecture will cover many classical algorithms of numerical analysis, related to mathematical analysis in the broad sense, and will contain precise applications in its exercises. The goal is to obtain strong bases in programming, fluency in coding, and an understanding of the mathematical results at the heart of the algorithms.

1. Graphics
2. Linear systems
a. Direct methods
b. Iterative methods
c. Principal component analysis
3. Numerical integration
a. Newton-Cotes formulas
b. Monte-Carlo method
4. Function approximation
a. Lagrange interpolation
b. Fourier transform
5. Nonlinear systems
a. Iterative methods in dimension one
b. Newton-Raphson algorithm
6. Optimization
a. Least squares
b. Descent algorithms
c. Constraints and Lagrange multipliers
7. Ordinary differential equations
a. Explicit and implicit Euler schemes
b. High order schemes
c. Numerical illustration of the properties of solutions
8. Partial differential equations
a. Introduction to finite differences
b. Poisson equations, transport equations, and heat equations in dimension one

Statistical learning (4 ECTS)

Financial risk management (4 ECTS)

Time series methods (4 ECTS)

Numerical methods in finance (4 ECTS)