- Groups : group actions, orbit-stabilizer theorem, $p$-group, 3 Sylow theorems, permutable subgroups, direct product, structure of finite abelian groups. - Rings : Ideal, maximal, prime, quotient, integral domain, euclidian ring, principal ideal domain, unique factorization domain, polynomial ring. - Fields : subfield, caracteristic, division ring, non commutative field, finite field filed, field extension, field of fractions. - Analysis on groups : Fourier transform on a finite group, characters, linear representation of a finite group.

- Revisions of topology, continuous linear maps and Hilbert spaces. - Space of continuous fonctions on a compact set, Ascoli theorem. - Fondamental theorems : Baire, Closed graph, open mapping, Banach-Steinhaus. - Hahn-Banach (geometric and analytic forms).- Theory of compact operators, spectrum, Fredholm's theory. - Holomorphic functional calculus and applications.

Introduction to discrete time stochastic processes. Dynamical modelisation of random behaviour via Markov chains and Poisson processes: Markov chains : transition functions, Markov graph, stopping time and return time, recurrence, transience, irreductibility, decomposition of the state space, stationary law, asymptotic behaviour, aperiodicity, convergence to the stationary measure. Poisson process : properties of the counting function, consequences for simulating and estimating.

Introduction to parametric statistics and the utilisation of the R software. Usual methods in parametric estimates. Bayesian methods. Hypothesis testing.

Ordinary differential equations: Cauchy’s problem, Cauchy-Lipschitz’ theorem, analytic resolution of linear ODE. Examples of modelisation using ODE (mechanics, chemistry, population’s dynamics). Explicit methods (Euler’s method, one step methods of Heun, of the middle point), methods of high order (Runge-Kutta, multiple steps), convergence (consistancy, stability, order). Implicit methods (Euler, Crank-Nicolson and middle point’s methods). Stiff problems and absolute stability, convergence. Partial differential equations in 1D : Poisson’s equation, heat equation, wave equation, advection equation. Physical interpretation. Examples of analytic resolution of PDE. Approximation by finite differences of Poisson’s equation (3 points’ scheme and convergence) and the heat equation (explicit and implicit Euler’s scheme, Crank-Nicolson. Convergence). The course includes programming sessions and a project in Python.

Infinitely derivable functions, convolution. Distribution: derivative, product by a function, support, convolution, examples. Operations on distributions. Infinitely derivable functions which are rapidly decreasing. Tempered distributions. Fourier transform. Paley-Wiener theorems. Linear PDE. Malgrange-Ehrenpreis theorem.

The TER is an initiation to the research in mathematics done alone or in a group of two students under the guidance of a member of the maths departments, from January to May. The student has to give a written report of approximatively 20 pages and there is an oral examination of 45 minutes with questions, in front of a jury constituted of members of the maths departments.

- Curves : in the plane and the space, length, parametrization by length, curvature, torsion. - Surfaces : parametrized, regular, impicitely defined. - line integral and surfaces integral, length, area, volume. Tangent space, vector fields, differential forms, exact, closed, integration, Stokes' theorem.

Financial models in discrete time: arbitrage, self-financing strategies. Conditional expectation. Martingales and arbitrage: notion of martingale, stopping theorem. Complete market. Binomial model. Brownian motion. Stochastic integral, Itô formula, Black-Scholes model.

Links between optimization and statistic, maximum likelihood, gradient descent, conjugate gradient, constrained optimization, penalization, Lagrange’s multipliers. Loss function, arbitrage, bias, variance, overlearning. Decision tree, Support-vector machine, random forest, boosting. Crossed validation. Arbitrage between interpretability and predictive power. Introduction to neural network. Implementation in python and participation to a Hackathon.