## Resources: summaries & notes

**Summaries, course notes and (forthcoming) scientific productions of students**

**Clément Alleaume**

*Gröbner bases and applications*

This mini-lecture constitutes an introduction to Gröbner bases and their applications. In particular, the algorithmic aspects of Gröbner bases are emphasized. Among the mentioned problems are the division algorithm for polynomials in multiple indeterminates and the resolution of polynomial systems.

**Ivan Bardet**

During this course we discussed how matrix analysis appears in the theory of Markov chains on finite sets, and in particular in the study of their long-time behaviour. We practised on some practical examples, including Google’s PageRank algorithm.

**Oriane Blondel**

*Measure theory*

**Alexandre Bordas, Émilie Devijver and Rémi Molinier**

This lecture is about graph theory. It introduces the main notions in this field, as connectivity, planar graph, Eulerian and Hamiltonian paths.

We also discuss about more recent results, as graph coloring, automata and random walks on graphs.

**Charles-Édouard Bréhier**

*Introduction to numerical methods for Ordinary Differential Equations*

Many numerical schemes (such as the standard Euler methods) can be used to approximate solutions of Ordinary Differential Equations, arising for instance in physics or biology. The aim of these lectures was to discuss the quantitative and the qualitative properties of these schemes, and the principles behind their analysis and their construction.

**Émilie Devijver**

*Asymptotic statistics. An introduction*

This lecture is about asymptotic statistics. We introduce some useful notions in statistics, as a sample, an estimator, and how to evaluate the performance of an estimator. Then, we recall the central limit theorem, and extend it to a well-used family of estimators, called maximum likelihood estimator. This lecture is illustrated on R.

**Matthieu Dussaule**

*An introduction to geometric group theory*

An introduction to geometric group theory. The goal of these lectures is to discuss actions of groups on geometric spaces such as graphs and trees. The main idea is that to understand a group, one should understand how it acts.

**Kevin François**

*Introduction to algebraic topology and covering spaces*

The classical search for complex logarithms gives a starting point to explore covering spaces, to build the first homotopy group of a topological space, and to dive into homotopy theory. A pleasant feature of algebraic topology is that intuition plays an important role, even without expertise in general topology. It is essential to build one’s own representations–that is, pictures–of the situation to follow these notes.

**Idriss Mazari**

An introduction to optimization and to the calculus of variations

In this course, we will try to give a few fundamental ideas underlying the mathematics of optimization. To be more precise, we will focus on heuristics of the Lagrange multipliers and of the Euler-Lagrange equation to try and solve (or at least to say something interesting about) some classical problems, the two-dimensional isoperimetric inequality for instance.

**Petru Mironescu**

*Fine properties of functions. An introduction*

The aim of the lecture is to introduce the main basic tools in the theory of function spaces (maximal function, Calderón-Zygmund decomposition, singular integrals), as well as some applications (Hardy spaces, elliptic regularity).

**Rémi Molinier**

*Introduction to algebraic topology*

This course is an introduction to algebraic topology with a bit of knot theory. This course introduces some classical invariant of homotopy theory (fundamental group) and some classical invariant of knot theory (tricolorability and knot group).

**Pierre Perruchaud**

*Introduction to Ergodic Theory**A first step towards chaos*

This intervention proposes an introduction to dynamical systems, focusing mainly on the measurable approach: what we call ergodic theory. In parallel with Oriane Blondel’s intervention on measure theory, we will introduce some classical ergodic properties and theorems, along with motivating examples. If we are not short on time, we may discuss some topological aspects as well.