Master Year 1 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.