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 Master  LMD
Master Mathematics
Résumé
The Master in Mathematics offers a highlevel twoyear training in pure or applied mathematics for careers in research, teaching, insurance and finance. Read moreAccéder aux sections de la fiche
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Details
Introduction
The master addresses a broad variety of topics, all of which are fascinating, and it also offers a great scientific overview to its students. It relies on a recognized research team, the AGM Research Center in Mathematics (UMR CNRS 8088), which develops many collaborations with academic and industrial labs.
The Master in Mathematics benefits from a program of excellence scholarships, which provides a 7 300 euros annual support to the selected secondyear applicants.
Objectives
Places
Person in charge of the academic program
Heads of the master
Philippe Gravejat (Master 2)
philippe.gravejat@cyu.fr
Michal Wrochna (Master 1)
michal.wrochna@cyu.fr
Administration
Marie Chef
marie.chef@cyu.fr
+ 33 (0)1 34 25 65 61
Partnership
Research center
Institutions
Admission
Prerequisite
Prerequisites training
The second year is open to candidates holding the first year of a master's degree in mathematics or applied mathematics.
Application
Conditions of applications
 Application to the first year of the master

The first year of the Master in Mathematics will supervise 50 students during the 20232024 academic year. The application has to be made via the platform Études en France. The application forms consist of:
 the diploma and mark transcripts from the bachelor's degree (in mathematics or applied mathematics),
 the diplomas and mark transcripts for the last four years,
 a motivation letter, which describes in particular the professional project.
Deadline for the application: April 11^{th}, 2023.  Application to the second year of the master

The second year of the Master in Mathematics will supervise 20 students during the 20232024 academic year. The application has to be made via the platform Études en France. The application forms consist of:
 the diploma and mark transcripts from the first year of the master's degree (in mathematics or applied mathematics),
 the diplomas and mark transcripts for the last four years,
 a motivation letter, which describes in particular the professional project.
Deadline for the application: April 11^{th}, 2023.
Conditions of specific applications
https://cytech.cyu.fr/lecole/institutsciencesettechniques
The completed forms must be submitted to the email addresses
christine.richter@cyu.fr
philippe.gravejat@cyu.fr
Deadline for the application to an excellence scholarship for the second year of the master : April 29^{th}, 2023.
Program
 First year program

 First semester

 Calculus of variations, convex analysis and optimization (6 ECTS)
 Working group 1 (2 ECTS)
 Probability (6 ECTS)
 Programming in Matlab, C, C++ (4 ECTS)
 Dynamical systems (6 ECTS)

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. CauchyLipschitz 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  Topology and functional analysis (6 ECTS)
 1. Open and closed sets
2. Compact and connected sets
3. Fundamental group
4. Banach spaces
5. Duality theorems
6. Hilbert spaces
7. Bounded operators
8. Unitary operators and projections
9. Spectrum of operators
10. Spectral radius
11. Diagonalization
12. Compact operators
 Second semester

 Numerical analysis (5 ECTS)
 Partial differential equations (5 ECTS)
 Working group 2 (2 ECTS)
 Programming in R and Python (2 ECTS)
 Assistance to internship research (1 ECTS)
 Dissertation or internship (5 ECTS)
 Two lectures among the next ones:

 Algebra (5 ECTS)
 Differential geometry (5 ECTS)

I. Advanced differential calculus
1. Basics of differential calculus in R^{n}
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  Continuoustime 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, LehmannScheffé theorem, Fisher information, CramérRao inequality. Exponential families.
5. Confidence regions
6. Statistical tests
 Second year program in pure and applied mathematics

 Refresher lectures

 Algebra and geometry

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  Analysis and probability
 First semester

 Distributions and partial differential equations (8 ECTS)
 Working group 3 (2 ECTS)
 Stochastic processes (8 ECTS)
 Dynamical systems (8 ECTS)
 One lecture among the next ones:

 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. NewtonCotes formulas
b. MonteCarlo method
4. Function approximation
a. Lagrange interpolation
b. Fourier transform
5. Nonlinear systems
a. Iterative methods in dimension one
b. NewtonRaphson 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
 Second semester

 Specialization lecture: Analysis (6 ECTS)

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 selfsimilar 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. Selfsimilar resolution of singularities
4. Heat equation in Lebesgue spaces
a. Semigroup decay estimates
b. Long time behavior
5. Viscous Burgers equation
a. Cauchy problem
b. Convergence towards blowup profiles or solitary waves
6. Weak solutions of the Burgers equation
a. Evanescent viscosity limit
b. Convergence to blowup profile
7. Local Cauchy problem for semilinear parabolic equations
8. Formation of singularities for the KellerSegel system
a. Virial formulas
b. Backward selfsimilar profiles
9. Stability analysis
a. Renormalization and modulation
b. Spectral theory
c. Nonlinear stability  Graduate school lecture : Regularity for partial differential equations (6 ECTS)

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 NavierStokes equations in fluid mechanics, or the finitetime 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 wellknown to the PDE community since (at least) the eighties. However some results (De GiorgiNashMoser theorem, epsilonregularity for the NavierStokes 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 (CalderonZymund 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, Liouvilletype theorems)
5. Epsilonregularity for the NavierStokes 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.
 LemariéRieusset, The NavierStokes Problem in the 21st Century.  Specialization lecture: Geometry and dynamical systems (6 ECTS)
 Dissertation or internship (12 ECTS)
 Second year program in applied mathematics to finance

 Refresher lectures

 Algebra and geometry

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  Analysis and probability
 First semester

 Distributions and partial differential equations (8 ECTS)
 Modelling (8 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. NewtonCotes formulas
b. MonteCarlo method
4. Function approximation
a. Lagrange interpolation
b. Fourier transform
5. Nonlinear systems
a. Iterative methods in dimension one
b. NewtonRaphson 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  Stochastic processes (8 ECTS)
 Three lectures among the next ones:

 Statistical learning (4 ECTS)
 Financial risk management (4 ECTS)
 Time series methods (4 ECTS)
 Numerical methods in finance (4 ECTS)
 Second semester

 Risk measure: Theory and applications (6 ECTS)
 Stochastic modelling (6 ECTS)
 Dissertation or internship (12 ECTS)
Educationnal team
 Master 1 Mathematics

Lectures and tutorials are given by the faculty of the deparment of mathematics:
 Yalcin Aktar
 Smail Alili
 Teodor Banica
 Christian Daveau
 Françoise Demengel
 Philippe Gravejat
 Irina RobertIgnatouk
 Élisabeth Logak
 Marjolaine Puel
 Armen Shirikyan
 Michela Varagnolo
 Michal Wrochna  Master 2 Mathematics

Lectures and tutorials are given by the faculty of the deparment of mathematics:
 Charles Collot
 Christian Daveau
 Françoise Demengel
 Yong Fang
 Aurélien Galateau
 Ruslan Maksimau
 Christophe Prange
 Marjolaine Puel
 Armen Shirikyan
 Michela Varagnolo
 Michal Wrochna
and of the department of economy:
 Tristan Guillaume
 William Kengne
 JeanLuc Prigent
Internship(s)
The research dissertation or the secondyear internship last five to six months. The goal of the dissertation is to prepare for a possible PhD in pure or applied mathematics, the aim of the internship is to prepare to a subsequent professional integration.
Assessment
What's next ?
Further studies
Job opening
Business sector or job
The second year in Applied Mathematics to Finance trains students for jobs in insurance and finance.