The *International Summer School of Mathematics for Young Students* offers 10 days of intense learning and rich interaction with some of the world’s leading mathematicians.

- For talented students in their
**last two years of high school**or in their**first two years at university**. - Open to
**international students**from all countries. - The school will be held in
**English**. - The program features
**plenary talks**and**mini-courses**by leading international mathematicians. - Students and instructors live next to each other on the campus, with ample opportunities for informal interactions.
- Hosted by the Ecole Normale Supérieure of Lyon (ENSL) in Lyon, France, from August 20th to August 30th.

### Summary of Lectures

### “Billiards” by Marie-Claude Arnaud

You play billiard in a smooth and strictly convex table … which kinds of trajectories will you observe?

We will explain:

– Why mathematicians began to look at this problem?

– How to model this? In particular, we will see that the billiard map can be seen as a map of a (bounded) cylinder.

– How to find trajectories? We will see that a lot of orbits are maximizers of a certain function and that this function is the length of the trajectory.

We will prove the existence of periodic orbits and speak of caustics, that are curves that sometimes appear if your ball has an infinite trajectory…

Prerequisites : norm, scalar product, angle, bisector, geometry of the triangle, area (for example area of a parallelogram), derivative, tangent and normal vector to a planar curve, linear momentum and kinetic energy

### “Percolation” by Christophe Garban

The most well-known example of a phase transition is what occurs when water suddenly freezes when the temperature drops below 0 degree Celsius. Such phase transitions are omnipresent both in everyday life and in theoretical physics. A common feature shared by phase transitions is the fact that at equilibrium, when one varies the temperature gently, the structure of the system drastically changes at a precise *critical temperature* (and the two phases only coexist at that precise temperature).

The purpose of this talk will be to introduce a very simple, yet fascinating, mathematical model which will undergo such a phase transition. This model, called **percolation**, can be defined roughly as follows (more details will be given in the talk): consider the square grid in the plane, usually denoted by Z^2 (but you may also think of an infinite chessboard), and keep each edge of this grid independently with probability p (where p is some fixed parameter in the interval [0,1]). If p=1/2, this means that for each edge, one tosses a fair coin to decide whether one keeps that edge or not. If p<1/2, one would have to use a biased coin instead. Just to picture a little bit what this means: the higher p is, the denser the graph is (in fact the proportion of edges that are kept is given by the parameter p, usually called the “intensity”). The striking fact about this simple probabilistic model is that there exists a **critical “temperature”**0<p_c<1 such that when the intensity p is below p_c, then all the connected components of the random graph thus created are finite, while if p>p_c there is a unique infinite connected component.

### “From complex numbers to quaternions and beyond” by Valentin Ovsienko

Algebra? Geometry? Number theory?

Let us call this subject simply : mathematics. The main goal of these lectures is to explain why do mathematicians invent “complicated” algebraic structures. Our main character will be the algebra of quaternions. Invented by Sir William Hamilton in 1843, quaternions extend complex numbers. But, unlike usual numbers, the algebra of quaternions is non-commutative that makes it more complicated.

Non-commutative? Are we sure? We will see that the algebra of quaternions is, in fact, commutative if we understand what the commutativity really means.

### “Polytope algebra” by Gaiane Panina

We will gradually build up a beautiful algebraic object based on purely geometric objects — the polytope (graded) algebra developed by Peter McMulen.

That is, we will introduce addition, multiplication, and even such crazy things as exponent and logarithm for polytopes.

This construction has helped to prove the f-vector problem (I’ll try to hint how) and is directly related to the Chow rings of algebraic toric varieties (this is beyond our course).

Surprisingly, there are no prerequisites. However, it is nice if you know what an abelian (that is, commutative) group, a ring, and the (graded) algebra of polynomials are.

### “Kolmogorov complexity and algorithmic information theory” by Alexander Shen

It is usual to measure the amount of information in a message in bits. However, just the number of bits is not a good measure: message in 8-bit/char encoding and 16-bit/char have essentially the same information content even if the second one has twice more bits. More natural approach is to measure the “compressed size” of a message, but this depends on the compression technique; there are many compressor algorithms and none of them is “the right one”. Andrei Kolmogorov and others (R.Solomonoff, and later G.Chaitin) found that one still have reasonably invariant definition of information content (=algorithmic complexity) and there is a rich theory around this notion. It also allows us to address the philosophical question: what is randomness? why we reject the fair coin hypothesis if we see 0101…0101 (1000 alternating bits) but some other sequence of 1000 bit may look plausible as the outcome of coin tossing? (Note that any two bit strings of length 1000 have the same probability if the coin is symmetric and coin tossings are independent).

### “Circle Packing and Spontaneous Geometry” by Ken Stephenson

Abstract: The topic of “circle packing” is about configurations of circles having specified patterns of tangency. You can specify how many circles you want and which of them are tangent to which others, and then — quite amazingly — find radii and centers for the circles so that they fit together in exactly the desired pattern!

Note that the pattern is combinatorial information, specified via a graph. The circles do an involved dance, adjusting radii until each can fit with its prescribed neighbors, and when they have finished,

their layout gives us geometric information. This is “spontaneous” in that each circle worries only about itself and its immediate neighbors, yet the resulting configuration has rigid global consequences. The connection between combinatorics and packing geometry is what we will study. Fortunately, the software CirclePack does the work of computing and displaying the results, providing an open experimental platform for investigating this fascinating interplay between combinatorics and geometry.

### “Flows and walks on graphs” by Shmuel Weinberger

Suppose that a bacterium splits in two with probability 1/2 or otherwise dies, and we start with a small colony of 1000 bacteria. Can we expect to obtain an immortal colony? If I randomly walk on a line from a position one unit from my home, and I go in each direction with equal probability, how long will it take me to get home? What about in higher dimensional space? Is it possible to rearrange the assets of the points of a graph through trade among neighbors so that all profit?

These are among the problems that I will explain and interrelate in this introduction to the geometric/probabilistic side of graphs.

### “Difference calculus and special numbers” by Tadashi Tokieda

Drop a stone into water. It makes a sound, “glop” for a big stone, “splitch” for a small stone. Can you predict the pitch of the sound from the size of the stone?

The usual teaching of mathematical theories is like a pyramid.

Young people tend to become passive (if passionate) admirers of a structure built by old people, and problems they are taught to solve make them walk straight up to the peak. But what if we want to explore a natural mountain range, whose peaks are invisible among clouds, whose trails among trees are unknown?

The problem of the sound of a stone falling into water is natural, so natural that every child knows the phenomenon and can wonder about it. The mathematics involved is extremely hard, so hard that it is not taught at any mathematics department in the world.

This course tries to teach how to make _some_ progress on _any_ natural problem, when we know _nothing_. We will use no theory more sophisticated than calculus*, which to a passive admirer may seem little, but the _way_ we use it is very robust and powerful and a bit magical, and allows us to solve for example the problem above. In short, we shall learn the first steps in _applied mathematics_.

_______________

*Experience up to having seen simple differential equations

(e.g. y” + ay’ + by = 0) solved, and common sense in mechanics,

are desirable prerequisites.

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