S2 Elective | Master of Applied Mathematics - Grenoble

Graduate School project

Mon, 01 Jan 0001 00:00:00 +0000

Credits

6 ECTS

Person in charge

Faouzi Triki (Faouzi.Triki@univ-grenoble-alpes.fr)

Description

A student selected in the Graduate School track will have to work with a researcher from the Grenoble environment on a research project.

Please contact Faouzi Triki if you have any question.

Introduction to cryptology

Mon, 01 Jan 0001 00:00:00 +0000

Credits

6 ECTS, CTD 36h, TP 18h

Instructor

Bruno Grenet and Clément Pernet

Description

The goal of this course is to acquire the main theoretical and practical notions of modern cryptography: from notions in algorithmic complexity and information theory, to a general overview on the main algorithms and protocols in symmetric and asymmetric cryptography.

Content

Part 1. Cryptology

Symmetric cryptography: encryption (AES, block ciphers), Message Authentication Codes (HMAC), hash functions (SHA-2, SHA-3)
Public-Key cryptography: key exchange (Diffie-Hellman), encryption (ElGamal, RSA), signatures (Schnorr, RSA)
TLS protocol

Part 2. Algebraic Algorithms for cryptology

Integer and polynomial arithmetic (multiplication, GCD, exponentiation, …)
Finite groups, rings and fields (mathematical context algorithms and implementation aspects)
Applications: error correcting codes, asymmetric ciphers

Operations Research

Mon, 01 Jan 0001 00:00:00 +0000

Credits

6 ECTS, CTD 36h, TP 18h

Instructor

Nadia Brauner, Jérôme Malick

Description

Operations Research offers scientific methods for better decisions. The idea is to develop and use mathematics and informatics tools to solve complex organization problems. Historical applications are in the management of large systems of humans, machines, materials in industry, service, humanitarian aid, environment…

At the end of this course, students should be able to propose a modelization and implement practical solutions (dedicated or industrial tools) to solve a decision or optimization problem. Interested students can continue in master 2 Operations Research, Combinatorics and Optimization (ORCO).

This course is divided into two parts : an introduction to operations Research modelling and solving methods (common with M1 Mosig) and a complementary part for M1 Math (AM and MG) students only.

Part 1: introduction to OR (common with M1 Mosig)

Skills

Recognize a situation where Operations Research is relevant.
Know the main tools of Operations Research.
Have the methodological elements to choose the solution methods and the tools the most adapted for a given practical problem.
Know how to manipulate the software tools to solve a discrete optimization problem.

The course covers various topics:

Linear Programming (modelling, solving, duality)
Mixed Integer Linear Programming (modelling techniques, solving with Branch and Bound)
Dynamic Programming
Bonus (riddles, elsewhere on the web, OR News)

Prerequisites

Classical algorithms (sort, divide and conquer)
Algorithms complexity calculation
Programming: basic notions (variables, fonctions, if, for, while, tables)
Language Python or Java
Basic notions on graphs (basic definitions, graph search, trees, shortest paths)
Basic notions on linear algebra and matrix analysis (matrix multiplication, invertible matrix definition)
Basics of statistics and probability

More details for the first part : https://moodle.caseine.org/course/view.php?id=42

Part 2: OR Complementary for M1 AM students

In this part, we will investigate in more details some mathematical notions related to operations research. We will focus on three aspects: Spectral graph theory, Game theory, and Numerical Optimal transport.

Statistical learning and applications

Mon, 01 Jan 0001 00:00:00 +0000

Credits

6 ECTS, CTD 36h, TP 18h

Instructor

Pedro Rodrigues (Lectures) + Razan Mhanna (Labs)

Description

The aim of this course is to present the statistical approaches for analysing multivariate data. The information age has resulted in masses of multivariate data in many different field: finance, marketing, economy, biology, environmental sciences. The theoretical and practical aspects of multivariate data analysis are given equal importance. This balance is achieved through practicals involving actual data analysis using Python.

Content

Multiple linear regression. Least squares, Gaussian linear model, test of linear hypotheses, one-way analysis of variance.
Principal Components Analysis (PCA).
Classification, linear discriminant analysis, perceptron, Naive Bayes
Text mining, numeric representation of texts, connexion with graph clustering.

Prerequisites

Elementary notions in probability theory (probability distribution, joint probability density function for random vectors, conditional distribution, expectation, variance, covariance, Gaussian distribution)

Elementary notions in mathematical statistics (estimator, confidence interval, statistical tests).

As a bonus: simple linear regression, linear algebra (matrix reductions), elementary notions in Rstudio and the R software.

Variational methods applied to modelling

Mon, 01 Jan 0001 00:00:00 +0000

Credits

6 ECTS, CTD 36h, TP 18h

Instructor

Clément Jourdana and Frédérique Charles

Description

The aim of this course is to get deep knowledge of PDE modelling and their numerical resolution, in particular using variational methods such as the Finite Elements method.

Content

Introduction to modelling with examples.
Boundary problem in 1D, variational formulation, Sobolev spaces.
Stationary problem, elliptic equations.
Finite element method: algorithm, errors…
Evolution models, parabolic equations, splitting methods
Extensions and applications, FreeFEM++

This course include practical sessions.

Prerequisites

notions of distribution theory, linear algebra, integral calculus, some notions of programming in some high level language, basic numerical analysis, as numerical integration of differential equations, basic notions on Hilbert spaces, usual partial differential operators (gradient, divergence, laplacian…)