## Master Mathematics and applications

### Mathématiques et Informatique

### Entry requirements

Master 1 in mathematics or in computer science, bachelor level in the other one.

### Benefits of the program

This master is unique in France, with teaching in both mathematics and computer science. The teachers from both fields of research are used to work together in a prestigious long-term project, the Labex Bezout.

### Acquired skills

Master level in areas at the interface of mathematics and computer science: optimization, analysis, geometry, combinatorics and machine learning. Development of research skills: autonomy, bibliography, … Advanced programming skills specialized in application in mathematics and computer science.

### Registration details

Via l'application de candidatures eCandidat : https://candidatures.univ-eiffel.fr

### Course venue

### Schedule of studies

Socle pendant 1 mois, puis 10 semaines de tronc commun, puis 8 semaines de spécialisation et enfin un stage de 3 à 6 mois à partir d’avril.

### Your future career

Continuing in a PhD in mathematics or computer science. Jobs in R&D in areas at the interface of mathematics and computer science, typically optimization and machine learning. Machine learning courses, in particular, are made to be usable directly in the industry.

### Professional integration

Continuing in a PhD in mathematics or computer science. Jobs in R&D in areas between mathematics and computer science, typically optimization and machine learning. Machine learning courses, in particular, are made to be usable directly in the industry.

### Study objectives

At the end of the master, the student should be able to continue with a PhD in mathematics or computer science, with a large knowledge of areas at the interface of the two fields. Some courses, such as optimization and machine learning, are also especially designed to be directly usable in the industry.

### Major thematics of study

Mathematics and computer science: discrete and continuous optimization, algorithms and combinatorics, geometry, data science, large random matrices, random graphs.

### Study organization

Une première période de « socle » de 4 semaines pour faire une remise à niveau en mathématiques et en informatique. Suivi d’un tronc commun de 10 semaines avec 3 UE composée chacune de deux cours. Une seconde période avec deux options de spécialisation à choisir parmi quatre propositions. Un stage de recherche ou en entreprise de 4 mois minimum.

### Modalité d'admission en FC :

Sur avis de la commission pédagogique

### Modalité d'admission en FI :

Sur avis de la commission pédagogique

### Modalité d'admission en Alternance :

Pas d'alternance

### Options

Options may change every year, except « data science ».

Data science: AI, machine learning, applications and use of related softwares

Algebraic combinatorics and computer algebra: operads, Hopf algebra.

Large random matrices and applications: theory and applications to signal processing and statistical testing.

Random graphs and graphons: theory of large dense graphs and applications.

### International

Part of the student benefit from Bezout grants and come from all around the word. English speaking student are welcome.

### Major thematics of Research

This speciality of M2 was created as a natural continuation of the successful Labex Bezout. It therefore benefits from a stimulating research environment, developed by the three excellent laboratories in mathematics and computer science of the « cité Descarte »: CERMICS, LAMA and LIGM.

### Partenariats :

CERMICS

### Co-accréditation :

UPEC

### SEMESTRE 1

Courses | ECTS | CM | TD | TP |
---|---|---|---|---|

Basics in mathematics Basic courses in analysis, algebra, probability and geometry ## Langue de l'enseignementANGLAIS / ENGLISH | 4 | 16h | 16h | |

Basics in computer science Basic courses in complexity, algorithmic, programming and graphs ## Langue de l'enseignementANGLAIS / ENGLISH | 4 | 16h | 16h | |

English English ## Langue de l'enseignementANGLAIS / ENGLISH | 4 | 16h | 16h | |

Discrete and continuous optimization Discrete optimization : Min-max results in combinatorial optimization provide elegant mathematical statements, are often related to the existence of efficient algorithms, and illustrate well the power of duality in optimization. The course will rely on concrete examples taken from industry. Plan of the course: Discrete optimization in bipartite graphs. Chains and antichains in posets. Chordal graphs. Perfect graphs. Lovász theta function. Continuous optimization : The course will cover over theoretic and algorithmic aspects of convex optimization in a finite-dimensional setting. Plan of the course: Linear programming, the simplex algorithm, totally unimodular matrices, convex fuctions, semi-definite programming, convex programming, Karush-Kuhn-Tucker conditions. Weak and strong duality, Farkas Lemma. Sparse solutions via L1 penalization. LASSO method. ## Langue de l'enseignementANGLAIS / ENGLISH | 6 | 20h | 30h | |

Probabilistic algorithms and combinatorics Probabilistic algorithms. This course is about randomized algorithms, which relies on randomness to speed up their running time. The methods: Las Vegas and Monte Carlo algorithmes, complexity classes for randomized algorithms, lower bounds: Yao’s Minimax principle, some probabilistic settings useful in algorithms: coupon collector, birthday problem, …Applications: analysis of Quicksort and Quickselect algorithms, stable marriage problem, probabilistic data structures (hash tables, skip lists, treaps, …), probabilistic counting, graph algorithms. Combinatorics. The lectures on enumerative combinatorics will consist in the study of classical objects: permutations, trees, partitions, parking functions, …- classical sequences: factorial, Catalan, Schroder, …- classical methods: bijections, group actions, induction, generating series.The lectures will be heavily based on the study of various examples, some very easy and others trickier. ## Langue de l'enseignementANGLAIS / ENGLISH | 6 | 20h | 30h | |

Discrete geometry Discrete differential geometry. Extension of curvature to discrete objects such that polyedra or graphs. Geometric flow under constraint preserves a texture mapping along deformation. Programm : Discrete surface theory, topology and Gaussian curvature, Gauss Bonnet theorem. Discrete differential calculus. Discrete mean curvature on triangulated surfaces. Parametrization by line of curvature, Quadrangulation and application to architecture. Intrinsic curvature and application to graph theory. Discrete isoperimetry. We extend the classical isoperimetric problem (which sets have maximal area among sets of given perimeter) to graphs. We present : Brunn-Minkowski and Loomis-Whitney theorems, entropy, Sauer-Shelah, Harper’s theorems, Boolean analysis on the hypercube, Cheeger’s theorem linking the spectral gap to expansion properties of a graph and we give algebraic and probabilistic constructions of families of expander graphs. ## Langue de l'enseignementANGLAIS / ENGLISH | 6 | 20h | 30h |

### SEMESTRE 2

Courses | ECTS | CM | TD | TP |
---|---|---|---|---|

Stage | 18 | |||

Les éléments ci-dessous sont à choix : | ||||

Data Sciences Machine learning. OBJECTIVES: Understanding of the principal Artificial Intelligence algorithms: machine & deep learning. Introduction to the optimization and the stochastic approximation algorithms for learning. Building predictive methods on unstructured datasets such as text data. PROGRAM: Introduction to statistical learning: theoretical and empirical risk, Bias-Variance equilibrium, overfitting; Aggregating methods: random forests; bagging and boosting methods; Kernels methods and Support Vector machine algorithms; Convexification, regularization and penalization technics: Lasso, Ridge, elastic net.. ; Deep learning algorithms: feedforward, convolutional and recurrent networks, dropout regularization; Prediction with unstructured text data: bagofwords, word2vec; Introduction to reinforcement learning. ## Langue de l'enseignementANGLAIS / ENGLISH | 6 | 16h | 16h | |

Random graphs and graphons The objective is to present the theory of large dense graphs, in their analytical, probabilistic and combinatorics aspects. Program : introduction and reminders on finite graphs ; definition of a graphon, graphon as a generator of dense random graphs ; properties of the space of graphons, cut-distance ; convergence of large graphs to a graphon ; sampled graphs; Inequalities of concentration and convergence ; application: classical combinatorial inequalities ; application: epidemic on a graphon, graphs biased by size ; application: degree function, exponential model ## Langue de l'enseignementANGLAIS / ENGLISH | 6 | 16h | 16h | |

Algebraic combinatorics and formal calculus Operads in combinatorics: Informally, an operad is a space of operations having one output and several inputs that can be composed. We present some elementary objects of algebraic combinatorics: combinatorial classes and algebras. We introduce (non-symmetric) operads and study some tools allowing to establish presentations by generators and relations of operads. Koszul duality and generalizations: colored operads, symmetric operads, and pros. We shall also explain how the theory of operads offers a tool to obtain enumerative results. Algebraic combinatorics : study of classical symmetric functions and discussion about representation theory, noncommutative symmetric functions (NCSF), the definition of Hopf algebras, the dual algebra of NCSF, quasi-symmetric functions, the modern generalizations of those algebras and use of all these algebraic properties (transition matrices, expressions in various bases, morphisms of Hopf algebras) to solve (classical) combinatorial questions. ## Langue de l'enseignementANGLAIS / ENGLISH | 6 | 16h | 16h | |

Grandes matrices aléatoires et applications Le but de la th ## Langue de l'enseignementANGLAIS / ENGLISH | 6 | 16h | 16h |

##### Master (en) Mathematics and applications

M2##### Mathématiques et Informatique

### Summary

- Degree
- Master (en)

- Field(s)
- Sciences, technologies, santé

- Thematics of study
- Mathematics and applications

- How to apply
- Initial Education / Continuing Education / Recognition of prior learning

- Course venue

- Departments and Institutes
- UFR Mathématiques

##### Une formation de