Institut Polytechnique de Paris
Ecole Polytechnique ENSTA ENSAE Télécom Paris Télécom SudParis

Master Year 1 Applied Mathematics and Statistics

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

This first-year 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 mathematics-related 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, first-year students can enter the following second-year 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, discrete-time stochastic processes [Kolmogorov extension theorem, canonical processes, stopping times], martingales [convergence theorems, main stopping theorems, applications], continuous state-space 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 non-linear regression, Bayesian statistics.
  • Optimisation: Optimization in Rn (general case and convex case), Optimization under equality and inequality constraints, KKT, convex case, Farkas lemma, duality, Dynamic programming techniques: discrete-time dynamic programming (finite-horizon problems; infinite-horizon problems with discounted cost), Introduction to optimal control theory (Pontriaguine principle, Hamilton-Jacobi-Bellman equation).

 

 

2 mandatory courses :

  • Processus de Markov 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), Tree-based 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, Expectation-Maximization, k-means 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: multi-layer perceptron, Activation functions -Sigmoid, tanh, relu, lrelu etc.-, Losses (cross-entropy, l1/l2 loss, binary cross entropy, focal loss, hinge-loss etc.-, Back propagation, learning rate

2 elective courses: 

  • Introduction to Mathematical Finance (ENSAE) - 3 ECTS
  • Liner Time Series Models (ENSAE) - 3 ECTS
  • Numerical Analysis (TP) - 5 ECTS
  • Advanced Statistics (TP) - 2.5 ECTS

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.

Language prerequisites

  • French
  • English (for some courses)

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 2022-2023

  • EU/EEA/Switzerland students: 4243€
  • Non-EU/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

Program Office

Stéphanie Clevenot

General enquiries

master-admission@ip-paris.fr

Description

This first-year 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 mathematics-related 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, first-year students can enter the following second-year 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, discrete-time stochastic processes [Kolmogorov extension theorem, canonical processes, stopping times], martingales [convergence theorems, main stopping theorems, applications], continuous state-space 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 non-linear regression, Bayesian statistics.
  • Optimisation: Optimization in Rn (general case and convex case), Optimization under equality and inequality constraints, KKT, convex case, Farkas lemma, duality, Dynamic programming techniques: discrete-time dynamic programming (finite-horizon problems; infinite-horizon problems with discounted cost), Introduction to optimal control theory (Pontriaguine principle, Hamilton-Jacobi-Bellman equation).

 

 

2 mandatory courses :

  • Processus de Markov 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), Tree-based 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, Expectation-Maximization, k-means 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: multi-layer perceptron, Activation functions -Sigmoid, tanh, relu, lrelu etc.-, Losses (cross-entropy, l1/l2 loss, binary cross entropy, focal loss, hinge-loss etc.-, Back propagation, learning rate

2 elective courses: 

  • Introduction to Mathematical Finance (ENSAE) - 3 ECTS
  • Liner Time Series Models (ENSAE) - 3 ECTS
  • Numerical Analysis (TP) - 5 ECTS
  • Advanced Statistics (TP) - 2.5 ECTS

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.

Language prerequisites

  • French
  • English (for some courses)

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 2022-2023

  • EU/EEA/Switzerland students: 4243€
  • Non-EU/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

Program Office

Stéphanie Clevenot

General enquiries

master-admission@ip-paris.fr