Schedule and courses

All times are Paris times. Cells with a blue background denote activities reserved for participants of the full school.

	9:00	11:00	13:00	15:00	17:00	19:00	21:00
Monday 26th		(12:40) Opening	Beukers		van Hoeij
Tuesday 27th		Moussafir	Lima	Chambert-Loir	van Hoeij (Problem session)	Melczer
Wednesday 28th	Labbé	Labbé		SOLEIL virtual tour
Thursday 29th			Bouttier	Cohen	Beukers (problem session)		Barnard
Friday 30th	Li (10:00!!)	Moussafir (problem session, 12:00!!)		Chambert-Loir (Problem session)		Melczer (problem session)

Titles and abstracts

Emily Barnard

Frits Beukers, Sums of squares and cubes, a history in modern number theory

One of the surprising results in 17th century number theory was the statement that any prime number of the form 4k+1 can be written as the sum of two squares. It was discovered through the study of so-called Pythagorean triangles, which are right-angled triangles with integer sides, like 3^2+ 4^2 = 5^2. From this result followed the interest in the question: which integers can be written as the sum of two squares or two cubes of rational numbers. This question is still not quite answered, despite the many efforts and the many beautiful results that have been discovered. In this presentation we give a glimpse of these developments, which go right up to 2023. 

Jérémie Bouttier, On the enumeration of domino and lozenge tilings

A classical question in enumerative combinatorics is to determine the number of ways to tile a rectangle of size m times n (with m,n two integers) by “dominos”, i.e. rectangles of size 2 times 1 (arranged in two possible orientations, namely “horizontal” or “vertical”).

A general formula was given in 1961 by Kasteleyn and, independently, by Temperley and Fisher. Interestingly, one of the key motivations for these people actually came from physics, and more precisely from statistical mechanics. Indeed, domino tilings are in bijection with so-called dimer configurations, which are used to model the adsorption of diatomic molecules on the surface of a crystal.

I will present a number of variations on this problem and, if time allows, discuss the fascinating phenomenon of “arctic curve”.

Antoine Chambert-Loir, Electrostatics and number theory

I will consider the following question that combines analysis and number theory: Given a domain D of the complex plane containing the origin, what can one say about the power series with integral coefficients which define a function on D (possibly with poles) ? The first result in this direction was due to the mathematician Émile Borel in a 1893 paper with the mysterious title “Sur un théorème de M. Hadamard” (On a theorem of Mr Hadamard”): there the domain D was simply a disk centered at 0 with radius strictly bigger than 1, and the answer is that then the power series is a rational function.

We will see how its proof combines classical 19th century linear algebra, via simple estimates of determinants provided by that theorem of Hadamard. For generalizations of that result to more general domains D, it appears fruitful to imagine that the complement of D is made of conducting material, and to consider the electric field generated by a single charge at the origin.

If time permits, I would like to explain how more complicated elaborations of this results reappear in some proofs of the transcendence of π.

Serge Cohen

Brenna Conin, Virtual tour of synchrotron SOLEIL

Sébastien Labbé, Introduction to Sage

Cheuk Ting Li, The Mathematical Aspects of Data Compression

Yuri LIma, Billiards and symbolic dynamics

One of the classical examples in dynamical systems are billiards, which were introduced more than a century ago as a simplification of Boltzmann’s model in statistical mechanics. Billiards can represent various facets in dynamics, going from the simplest ones (totally integrable) to the most complicated ones (chaotic). In this talk, we will introduce these billiards, and discuss a technique, called symbolic dynamics, that allows us to understand chaotic billiards.

Stephen Melczer, An Invitation to Enumerative Combinatorics

From DNA sequences, to computer science algorithms, to the network of websites making up the internet, much of the modern world can be modelled by discrete structures (large structures comprised of distinct parts). The field of enumerative combinatorics adapts techniques from pure mathematics to the study of discrete structures, making it perfectly situated both to see the amazing depth and breadth of math, and to illustrate the wide variety of interesting and impactful applications it can address. This talk gives an introduction to the area of enumerative combinatorics, and surveys some of its basic methods. Applications mentioned include topics in theoretical computer science, queuing theory, the complexity of biological networks, non-standard game dice, the shape of biomembranes, lace art, and algebraic statistics.

Jacques-Olivier Moussafir, Some ideas and problems from mathematical finance

First part of the presentation will be about historical background, key steps in the setting of current mathematical finance and some basic facts from probability, in particular the construction of Brownian motion.

Second part will be about the most important themes, in particular Pricing and Arbitrage, Factor modelling and Hidden Factor Models and finally Limit Order Book modelling. On theme will be about the numerous connections with Physics, in particular Heat equation, Fokker-Planck equation etc.

The largest part of Mathematical Finance makes a large use of relatively technical items, like Lebesgue integral, Probability theory, Continuous Time Random Processes, Martingales, etc. I’ll try to be very light on these technicalities, and present ideas as simply as I possibly can.

Mark van Hoeij, An example of group theory in mathematics: How symmetries in the zometool construction toy determine which pieces can fit together in a loop.

Slides from Mark van Hoeij’s lecture

Problems for Mark van Hoeij’s lecture