Master Year 1 Applied Mathematics and Statistics
Year  Master Year 1 
Program  Applied Mathematics and Statistics 
ECTS Credits  60 
Language  English 
Orientation  Research or Industry 
Location  Palaiseau Campus 
Course duration  12 months, full time 
Course start  September 
Degree awarded  Master’s degree obtained on completion of a second year of Master 
WHY ENROLL IN THIS PROGRAM?
Asset n° 1
Build your own curriculum in mathematical sciences through a wide variety of courses, seminars, projects and internships
Asset n°2
Get ready for a career in research and benefit from close links with the Institut Polytechnique de Paris laboratories
Asset n°3
Prepare for a PhD through PhD tracks in Mathematics for Finance or Data Science and Artificial Intelligence
The mathematical level of this master's program is probably much higher than that of other master's programs offered by other universities in the same "mention" "Applied Mathematics and Statistics". We only recruit students with excellent mathematical skills at BsC level. It is essential that you have taken a course in probability at the level of Ross, "Introduction to Probability Models", and a course in mathematical statistics at the level of Hogg, Mc Kean and Craig 'Introduction to Mathematical Statistics'. Must have taken a solid course in mathematical analysis.
This firstyear Master’s program offers a wide range of basic and more specialized courses in applied mathematics. This allow students to build a personalized curriculum adapted to their academic and professional projects in the following areas:
 Statistics, finance and actuarial science
 Modeling, probability and artificial Intelligence
 Optimization
 Signal, computing and machine learning
 Numerical analysis and EDP
Students can also follow PhD Tracks to guide them towards a doctorate at the end of the Master:
 PhD track Mathematics for Finance
 PhD track Data Science and Artificial Intelligence
Objectives
This program allows students to:
 Acquire a solid foundation in applied mathematics to pursue doctoral studies or directly apply for a mathematicsrelated position in an academy, business or industry
 Delve deeper into applied mathematics by addressing current, open problems
 Develop a strong relationship with research through seminars, mentoring projects and internships in research labs or companies
To complete the Master’s degree, firstyear students can enter the following secondyear programs:
On completing the second year of the Master, graduates can apply for PhD funding in top research labs or a job requiring advanced knowledge of applied mathematics and statistics.
Core courses
4 mandatory courses [14 weeks, 4 hours a week, 7,5 ECTS / Course]
 Elements of functional analysis and measure theory: topology, measure and integration theory – Lebesgue measure, Lebesgue integral abstract measure spaces, classical functional spaces – Banach, Hilbert, bounded and unbounded operators, weak topologies
 Probability theory and stochastic process : Conditional expectations, discretetime stochastic processes [Kolmogorov extension theorem, canonical processes, stopping times], martingales [convergence theorems, main stopping theorems, applications], continuous statespace Markov chains [transition kernels, main convergence theorems for atomic chains]
 Mathematical Statistics : introduction to decision theory, estimators (M and Z estimators, sufficiency, optimality), tests (constructions, Uniformly Most Powerful test for simple and composite alternatives), asymptotic statistics (consistency, asymptotic normality), asymptotic optimality of maximum likelihood estimation, generalized likelihood ratio tests, introduction to linear and nonlinear regression, Bayesian statistics.
 Optimization: Optimization in Rn (general case and convex case), Optimization under equality and inequality constraints, KKT, convex case, Farkas lemma, duality, Dynamic programming techniques: discretetime dynamic programming (finitehorizon problems; infinitehorizon problems with discounted cost), Introduction to optimal control theory (Pontriaguine principle, HamiltonJacobiBellman equation).
3 mandatory courses :

Markov Processes and applications [7,5 ECTS] : Poisson process, jump process, Markov property, simulations of Markov processes, martingales, Kolmogorov equations, ergodic theorem, long time behavior, Brownian motion
 Introduction to Machine Learning [7,5 ECTS] : Introduction to Supervised Learning, Linear Models, Generalizes Linear Models, Support Vector Machines, Generative Learning (Gaussian discriminant analysis, Mixture models, Naïve Bayes model), Treebased and Ensemble Models (Decision tree, random forests, boosting, bagging), learning theory (Union bound, Hoeffding inequality, Empirical risk, Probably approximately correct framework), Introduction to Unsupervised Learning (Gaussian Mixture Model, ExpectationMaximization, kmeans clustering, Hierarchical clustering), Dimensionality Reduction (Principal component analysis (Eigenvalues/vectors, Spectral theorem, PCA algorithm) ● Independent component analysis (Bell and Sejnowski ICA algorithm, Linear discriminant analysis, Factor analysis), Neural Networks (Basic architecture: multilayer perceptron, Activation functions Sigmoid, tanh, relu, lrelu etc., Losses (crossentropy, l1/l2 loss, binary cross entropy, focal loss, hingeloss etc., Back propagation, learning rate
 Databases [3.5 ECTS] : ● Introduction, Data Models: Relational Model ● Relational Algebra, Relational Calculus ● SQL (Structured Query Language) ● Schema Refinement and Normal Forms ● Storage and Indexes ● Query Processing (Evaluation of Operators)Query Optimization (query plans, algebraic ● equivalence, plan space, cost estimation)
2 elective courses:
The first part of this course is devoted to studying univariate time series: first we present the principal statistical concepts, then estimation methods and tests; we examine the nonstationarity problem by studying the main unit root tests from a practical angle. The course is illustrated with practical examples. The second part of the course is devoted to studying stationary VAR models: we briefly present the general framework of multivariate stationary series before developing the specific case of VAR models. Finally, we take a quick look at the principles of cointegration.
The aim of this course is to present the mathematical concepts used to model and value derivatives in finance. The presentation of the course will be mathematically rigorous, but certain results from stochastic calculus will be admitted as read, to be demonstrated later in the thirdyear stochastic calculus course. After obtaining a mathematical definition of the concept of arbitrage in a financial market, we will study discrete models for each evolutionary tree of assets, giving useful intuitions for the study of continuoustime models. Finally, with the help of the theory of stochastic calculus , we will present asset valuation in the framework of the Black & Scholes model. It is advisable to have taken the course on Markov chains for a better understanding of the concepts from stochastic calculus and to take the simulation course in parallel.
 Numerical Analysis I & 2  TELECOM Paris  5 ECTS
https://synapses.telecomparis.fr/catalogue/20222023/ue/2087/macs205anumericalanalysisparti
Numerical methods come into play in applied mathematics as soon as the numerical value for some quantity of interest is needed, for which no analytic expression is available. This quantity may be e.g. anintegral or the solution of a differential equation.
The scope of application of this course thus embraces such diverse fields as Physics, Biology, Economy or Finance. The role played by numerical
analysis has substantially grown these past few years, together with extensive computing power.
Numerical methods may be divided into two broad classes: deterministic methods on the one hand, based on discrete approximations of integrals or state evolution equations, and stochastic methods on the other hand, which consist in random sampling under an appropriate probability distribution in order to approximate an integral or atrajectory. This may be done by independent sampling (MonteCarlo methods) or by simulating a well chosen Markov Chain (MCMC methods).
The purpose of this course is to introduce the students to essential methods in Numerical Analysis and to give them the key tools for understanding the mathematical principles behind.
Internship
Each M1 student must complete an internship of at least 8 weeks in order to complete their education (credited with 7,5 ECTS).This internship takes place either at a university or industrial research laboratory or in a company. It starts in May. If the internship takes place in a company or abroad and lasts at least 11 weeks, this long internship can be credited with 12,5 ECTS, depending on the decision of the jury, under a contract extended to 35 ECTS with a defense at the end of the summer.
The internship gives students the opportunity to gain professional experience and contacts and helps them define or finalize their choices for M2 and beyond. The internship is evaluated by a report and an oral defense
at the beginning of September. The evaluation will also take into account the assessment of the internship supervisor.
Admission requirements
Academic prerequisites
Completion of a bachelor’s degree in mathematics, mathematical sciences or related field in France or abroad.
The exercises proposed in the following link allow you to evaluate the level expected to enter our master : click here
Language prerequisites
 English
How to apply
Applications can be submitted exclusively online. You will need to provide the following documents:
 Transcript
 Two academic references (added online directly by your referees)
 CV/resume
 Statement of purpose
You will receive an answer in your candidate space within 2 months of the closing date for the application session.
Fees and scholarships
Estimated fees for 20222023 are subject to increase
 EU/EEA/Switzerland students: 4243€
 NonEU/EEA/Switzerland students: 6243€
 Engineer students enrolled in one of the five member schools of Institut Polytechnique de Paris (Ecole polytechnique, ENSTA Paris, ENSAE Paris, Télécom Paris and Télécom SudParis): 159€
 Special cases: please refer to the "Cost of studies" of the FAQs
Find out more about scholarships
Applications and admission dates
Coordinators
 Eric Moulines (Ecole polytechnique)
 Olivier Fercoq (Télécom Paris)
Program Office
General enquiries
The mathematical level of this master's program is probably much higher than that of other master's programs offered by other universities in the same "mention" "Applied Mathematics and Statistics". We only recruit students with excellent mathematical skills at BsC level. It is essential that you have taken a course in probability at the level of Ross, "Introduction to Probability Models", and a course in mathematical statistics at the level of Hogg, Mc Kean and Craig 'Introduction to Mathematical Statistics'. Must have taken a solid course in mathematical analysis.
This firstyear Master’s program offers a wide range of basic and more specialized courses in applied mathematics. This allow students to build a personalized curriculum adapted to their academic and professional projects in the following areas:
 Statistics, finance and actuarial science
 Modeling, probability and artificial Intelligence
 Optimization
 Signal, computing and machine learning
 Numerical analysis and EDP
Students can also follow PhD Tracks to guide them towards a doctorate at the end of the Master:
 PhD track Mathematics for Finance
 PhD track Data Science and Artificial Intelligence
Objectives
This program allows students to:
 Acquire a solid foundation in applied mathematics to pursue doctoral studies or directly apply for a mathematicsrelated position in an academy, business or industry
 Delve deeper into applied mathematics by addressing current, open problems
 Develop a strong relationship with research through seminars, mentoring projects and internships in research labs or companies
To complete the Master’s degree, firstyear students can enter the following secondyear programs:
On completing the second year of the Master, graduates can apply for PhD funding in top research labs or a job requiring advanced knowledge of applied mathematics and statistics.
Core courses
4 mandatory courses [14 weeks, 4 hours a week, 7,5 ECTS / Course]
 Elements of functional analysis and measure theory: topology, measure and integration theory – Lebesgue measure, Lebesgue integral abstract measure spaces, classical functional spaces – Banach, Hilbert, bounded and unbounded operators, weak topologies
 Probability theory and stochastic process : Conditional expectations, discretetime stochastic processes [Kolmogorov extension theorem, canonical processes, stopping times], martingales [convergence theorems, main stopping theorems, applications], continuous statespace Markov chains [transition kernels, main convergence theorems for atomic chains]
 Mathematical Statistics : introduction to decision theory, estimators (M and Z estimators, sufficiency, optimality), tests (constructions, Uniformly Most Powerful test for simple and composite alternatives), asymptotic statistics (consistency, asymptotic normality), asymptotic optimality of maximum likelihood estimation, generalized likelihood ratio tests, introduction to linear and nonlinear regression, Bayesian statistics.
 Optimization: Optimization in Rn (general case and convex case), Optimization under equality and inequality constraints, KKT, convex case, Farkas lemma, duality, Dynamic programming techniques: discretetime dynamic programming (finitehorizon problems; infinitehorizon problems with discounted cost), Introduction to optimal control theory (Pontriaguine principle, HamiltonJacobiBellman equation).
3 mandatory courses :

Markov Processes and applications [7,5 ECTS] : Poisson process, jump process, Markov property, simulations of Markov processes, martingales, Kolmogorov equations, ergodic theorem, long time behavior, Brownian motion
 Introduction to Machine Learning [7,5 ECTS] : Introduction to Supervised Learning, Linear Models, Generalizes Linear Models, Support Vector Machines, Generative Learning (Gaussian discriminant analysis, Mixture models, Naïve Bayes model), Treebased and Ensemble Models (Decision tree, random forests, boosting, bagging), learning theory (Union bound, Hoeffding inequality, Empirical risk, Probably approximately correct framework), Introduction to Unsupervised Learning (Gaussian Mixture Model, ExpectationMaximization, kmeans clustering, Hierarchical clustering), Dimensionality Reduction (Principal component analysis (Eigenvalues/vectors, Spectral theorem, PCA algorithm) ● Independent component analysis (Bell and Sejnowski ICA algorithm, Linear discriminant analysis, Factor analysis), Neural Networks (Basic architecture: multilayer perceptron, Activation functions Sigmoid, tanh, relu, lrelu etc., Losses (crossentropy, l1/l2 loss, binary cross entropy, focal loss, hingeloss etc., Back propagation, learning rate
 Databases [3.5 ECTS] : ● Introduction, Data Models: Relational Model ● Relational Algebra, Relational Calculus ● SQL (Structured Query Language) ● Schema Refinement and Normal Forms ● Storage and Indexes ● Query Processing (Evaluation of Operators)Query Optimization (query plans, algebraic ● equivalence, plan space, cost estimation)
2 elective courses:
The first part of this course is devoted to studying univariate time series: first we present the principal statistical concepts, then estimation methods and tests; we examine the nonstationarity problem by studying the main unit root tests from a practical angle. The course is illustrated with practical examples. The second part of the course is devoted to studying stationary VAR models: we briefly present the general framework of multivariate stationary series before developing the specific case of VAR models. Finally, we take a quick look at the principles of cointegration.
The aim of this course is to present the mathematical concepts used to model and value derivatives in finance. The presentation of the course will be mathematically rigorous, but certain results from stochastic calculus will be admitted as read, to be demonstrated later in the thirdyear stochastic calculus course. After obtaining a mathematical definition of the concept of arbitrage in a financial market, we will study discrete models for each evolutionary tree of assets, giving useful intuitions for the study of continuoustime models. Finally, with the help of the theory of stochastic calculus , we will present asset valuation in the framework of the Black & Scholes model. It is advisable to have taken the course on Markov chains for a better understanding of the concepts from stochastic calculus and to take the simulation course in parallel.
 Numerical Analysis I & 2  TELECOM Paris  5 ECTS
https://synapses.telecomparis.fr/catalogue/20222023/ue/2087/macs205anumericalanalysisparti
Numerical methods come into play in applied mathematics as soon as the numerical value for some quantity of interest is needed, for which no analytic expression is available. This quantity may be e.g. anintegral or the solution of a differential equation.
The scope of application of this course thus embraces such diverse fields as Physics, Biology, Economy or Finance. The role played by numerical
analysis has substantially grown these past few years, together with extensive computing power.
Numerical methods may be divided into two broad classes: deterministic methods on the one hand, based on discrete approximations of integrals or state evolution equations, and stochastic methods on the other hand, which consist in random sampling under an appropriate probability distribution in order to approximate an integral or atrajectory. This may be done by independent sampling (MonteCarlo methods) or by simulating a well chosen Markov Chain (MCMC methods).
The purpose of this course is to introduce the students to essential methods in Numerical Analysis and to give them the key tools for understanding the mathematical principles behind.
Internship
Each M1 student must complete an internship of at least 8 weeks in order to complete their education (credited with 7,5 ECTS).This internship takes place either at a university or industrial research laboratory or in a company. It starts in May. If the internship takes place in a company or abroad and lasts at least 11 weeks, this long internship can be credited with 12,5 ECTS, depending on the decision of the jury, under a contract extended to 35 ECTS with a defense at the end of the summer.
The internship gives students the opportunity to gain professional experience and contacts and helps them define or finalize their choices for M2 and beyond. The internship is evaluated by a report and an oral defense
at the beginning of September. The evaluation will also take into account the assessment of the internship supervisor.
Admission requirements
Academic prerequisites
Completion of a bachelor’s degree in mathematics, mathematical sciences or related field in France or abroad.
The exercises proposed in the following link allow you to evaluate the level expected to enter our master : click here
Language prerequisites
 English
How to apply
Applications can be submitted exclusively online. You will need to provide the following documents:
 Transcript
 Two academic references (added online directly by your referees)
 CV/resume
 Statement of purpose
You will receive an answer in your candidate space within 2 months of the closing date for the application session.
Fees and scholarships
Estimated fees for 20222023 are subject to increase
 EU/EEA/Switzerland students: 4243€
 NonEU/EEA/Switzerland students: 6243€
 Engineer students enrolled in one of the five member schools of Institut Polytechnique de Paris (Ecole polytechnique, ENSTA Paris, ENSAE Paris, Télécom Paris and Télécom SudParis): 159€
 Special cases: please refer to the "Cost of studies" of the FAQs
Find out more about scholarships
Applications and admission dates
Coordinators
 Eric Moulines (Ecole polytechnique)
 Olivier Fercoq (Télécom Paris)