We decided to take a walk in Paris and more specifically around the Latin Quarter, as it is home to many illustrious institutions in France and is rich in history and culture. The walk focuses on those establishments and the mathematicians that attended or were professors at those schools, and the mathematical discoveries that took place in those places.

### Le Collège de France

We started the walk by looking at Le Collège de France. It originated in 1530 when Francis I was advised to start the creation of a school that would teach topics to royal scholars, that were not taught at the Sorbonne, originally greek, hebrew and mathematics, and then later, French law, latin and medicine.

The Collège was composed of multiple colleges in the same location as it is now under Henri II, but it was later decided by Henri IV that the colleges would be reunited into one big college. The constructions were set back when the King was assassinated on the 14th May 1610 and the new King Louis XIII, under Marie de Medici's regency, set down the first stone for the building on the 28th August 1610 with the inscription

This translates to "On the first year of the reign of Louis XIII, King of France and of Navarre, and of the Regency of the Queen Marie de Medici his mother 1610".En l'an premier du Regne de Louis XIII Roy de France et de Navarre, âgé de neuf ans, et de la Regence de la Royne Marie de Médicis sa mère MDCX.

The construction of the building was then continued by architect Chalgrin around the Cour d'Honneur in 1773 and was entirely finished in 1778.

There were since then some modifications during the 19th and 20th Century to allow more space for example for labs.

Nowadays, there are 45 professorships (in French: Chaire), divided into five departments: mathematics and digital sciences, physics and chemistry, sciences of the living, human sciences, history and literature.

In mathematics and digital sciences, the professorships focus on:

Equations with partial derivatives and applications

Algorithms, machines and language

Algebraic geometry

**André-Marie Ampère**(Professor of General and Experimental Physics), who worked on improving mathematics by including them in physics, also discovering the basics of electronics, and creating the vocabulary for electricity.

**Camille Jordan**(Professor of Mathematics), who worked on group theory and analysis based on Weierstrass's work, which lead to the birth of modern analysis in France. The work in analysis focuses on differentials, integrals and differential equation, leading to important results, like Jordan's lemma.

**Szolem Mandelbrot**(Professor of Mathematics), who was one of the founding members of the Bourbaki group. He mainly worked on complex and harmonic analysis.

**Joseph Jérôme Lefrançais de Lalande**(Professor of Mathematics and Astronomy), who mainly worked on the study of astronomy and the sky. He also published a book stating that astronomy was not just for men, but would also interest women. He and his team also nearly discovered Neptune, as it was shown that the planet might have appeared in the observation records.

**Henri Léon Lebesgue**(Professor of Mathematics), who worked mainly on integration, geometry, and trigonometry.

### Lycée Louis-le-Grand

The walk continues with the famous and prestigious high school: Lycée Louis-le-Grand situated Rue St Jacques.

Guillaume du Prat, bishop of Clermont invited Jesuits to create a collège in a Parisian town house, where he would financially support six poor students, with 6000 pounds.

It became the collège de la Compagnie de Jésus.

Being tolerated by the Université de Paris, but without its agreement and authorisation, it received letters from the King, allowing the opening of the establishment in 1563. The success was immediate and unhoped for, as students were coming in large numbers, so much so that the collège needed to expand by buying the adjoining houses.

Because of cases of attempted murder on the King Henri IV by an alumni from the school and the differences with the Université de Paris, the school was forbidden from teaching or closed until 1618, when it finally reopened. After that time, the lycée rose to its apogee, reaching the point where, in 1682, it received the official patronage of King Louis XIV.

In 1762, after the bankruptcy of father Antoine Lavalette, the high school received the immediate order to dismiss its teachers and pupils, also expelling the Jesuits at the same time, allowing the competing high schools (28 in Paris) to take over the building.

Louis XV later became the second royal patron of the school.

During the Revolution, many students left to help in battle and many other schools were closing except Louis-le-Grand, and what was left from those schools was regrouped there. The lycée (official title attributed in June 1848) had changed names many times, due to the different political climates at the times, and officially gained it name Louis-le-Grand in March 1873.

Since then, the school has not changed but has had important construction work at the beginning of the 20th Century.

One very famous student from Louis-le-Grand was **Galois**.

He entered the middle school in the autumn of 1823, at twelve, one year early and quickly won prizes in Latin and greek. But when he was fourteen, in his first year of high school, his lack of interest and tiredness was beginning to show, and it was decided he would repeat a year, in the hope he would gain some maturity. Even with his repeated year, and the fact that he was studying rhetorics at the time, he was able to take some courses in mathematics. Due to his discovery of the subject and interest in it, as well as his ease when learning the notions, he did not work on the subjects that did not interest him and botched then (but still getting good marks). After that year, during which he studied algebra and analysis, he studied very little for his rhetoric course, and was continuing his mathematical journey, studying Gauss's algebra, and the calculations taught from Lagrange and Legendre, even the Abel-Ruffini theorem. At the end of that year, in 1828, he decided to study by himself for the competitive exam to enter Ecole Polytechnique, which he failed. He tried again in 1929, following lessons from Louis Paul Émile Richard, Hermite's professor, in preparatory classes (classes after the baccalaureate to prepare for going into university. Galois didn't get his baccalaureate). He also focused on his own research, publishing in *Annales de mathématiques pures et appliquées* and submitting his work to Cauchy, who was teaching at École Polytechnique at that time. Galois failed the second entrance test there. He later went to École Normale Supérieure.

### École Polytechnique

The school was created after the Revolution, on the 28th Septembre 1794, by **Jacques- Élie Lamblardie**, **Gaspard Monge** and **Lazare Carnot**, as all the royal engineering schools had previously been closed. The aim of this new establishment was to replace all those schools, but they were later reopened and it was decided that students would have to attend Polytechnique prior to entering those schools. Therefore Polytechnique focused on general knowledge and the theoretical parts, whereas the engineering schools focused on the practical side and specialisation. The school required a high level in sciences and its techniques, and the end rankings became very important, as they determined which corps the graduate went into. The school quickly became very famous for sciences and attracted many scientists from everywhere in Europe.

In 1804, Napoleon gave the school its military status that is still present nowadays, which led the students to help during the many battles from 1814 onwards, to the Worlds Wars. Since then, the reputation of the school has continued to rise and marks the social elevation of its students, and Franc'ois Crouzet, a French historian believes that the students from Polytechnique have had a significant economical effect in France in the 19th Century.

After World War II, the school wanted to renew itself. It therefore created new laboratories and decided to accept more students per year group, which motivated the board to move the location of the school from the Latin Quarter (in the middle of Paris), to Palaiseau, which was inaugurated in 1976.

Some of the very prominent students of Polytechnique were:

**Siméon Denis Poisson**, whose main work was on integrals and Fourier Series. He also worked on statistics, even giving his name to the Poisson distribution. He was also a professor at Polytechnique later on.

**Jacques Philippe Marie Binet**, who worked on Fibonacci numbers, giving a formula for its nth term (Binet's Fibonacci number formula). He also made a lot of progress in number theory and matrices algebra. He replaced Poisson as a professor, teaching mechanics.

**Augustin Louis Cauchy**, who worked mainly on analysis, introducing the criteria for convergence of series and holomorphic functions. His work on permutations was the precursor of group theory. He was friends with Berthollet, Lagrange and Laplace. He firstly became assistant professor and then professor, teaching analysis and mechanics every year until 1830.

**Benoit Mandelbrot**, who contributed greatly to the world of fractals, giving his name to a kind of fractals, the Mandelbrot Set. He was the nephew of Szolem Mandelbrot.

Prominent professors also taught at Polytechnique, such as:

**Joseph-Louis Lagrange**, who was the first analysis professor at Polytechnique. He demonstrated Wilson's theorem on prime numbers, and Bachet's conjecture on whole numbers. He has given his name to a theorem in group theory, worked on continued fractions and Lagrange's differential equation.

**Joseph Fourier**, who is most well known for having calculated the propagation of heat, by decomposing a function into a converging trigonometric series, which is known as a Fourier function, this method being called the Fourier Transform, which is the basis of signaling theory, digital imaging, data compression and systems like 3G and 4G

### Lycée Henri IV

Another famous and prestigious high school is the Lycée Henri IV.

The school is located on the site of the Royal Abbaye of Sainte-Genevieve, created by Clovis, King of Francs in 506. In 1619, the cardinal de La Rochefoucauld created a library which, forty years later, had more than 8000 books. Until 1790, during the Revolution, when the monks were chased away, the location was entirely religious. On 25th October 1795, the abbaye was replaced by a school: école centrale du Panthéon, which changed names many time due again to the political context and obtained its final nowadays name is 1870. During both world wars, the lycée lost many bright students and two plaques are exposed in their memory.

One of the high school's famous pupils was **Cauchy**, whose professor was Jacques Binet.

Another famous mathematician was **Charles Hermite**, who worked on number theory, quadratic forms, orthogonal polynomials, and differential equations. He is also known for being one of the first people to use matrices. In 1873, he showed that the number e is transcendant, which was further extended by Ferdinand von Lindemann, proving the transcendance of π in 1882.

### Institut Henri Poincaré

The next stop is the Institut Poincaré. It was created in 1928 by two mathematicians, a French one, **Émile Borel**, and an American: one, **George Birkhoff**, who both thought that French science needed an international platform for learning and exchanging knowledge in mathematics and physics. They persuaded French and American sponsors (Edmond de Rothschild and the Rockefeller Foundation) to fund the building.

Since then, the institute has become one of the most dynamic structures, modelled after the Mathematical Sciences Research Institute (MSRI) in Berkeley due to the very high level of teaching and its library, and is at the forefront of mathematics and theoretical physics.

The institute is now run by Cedric Villani, Fields Medal 2010, and specializes in bringing mathematicians from all around the world by organizing seminars, conferences and meetings. Another of the institute's aim is to popularize science (mathematics, physics, biology and computer science) to the general public using fun and interactive regular events: high level doctoral lectures, short terms collaborations etc. ... About 11,000 mathematicians go the IHP every year.

### École Normale Supérieure

The Ecole Normale supérieure was created in 1794 and the first inaugural lecture was given on the 20th January 1795 . The aim was to provide teachers for schools, which meant that the level of pupils in those schools across France would be homogeneous in sciences and humanity subjects. The male students at the Ecole had a very dense syllabus to go through, so scientists like Monge, Vandermonde, Daubenton and Berthollet and writers and philosophers like Bernardin de Saint-Pierre and Volney were brought there to teach them.

The school was then reformed in 1808.

The new format of the school opened in 1810, created by Napoleon I, to "train in the art of teaching sciences and literature". Until 1818, students were chosen based on their grades during high school.

On the 4th November 1847, the school changed locations, to Rue d'Ulm, where is it now, and was expanded in 1937 to allow for the study of experimental sciences.

In 1903, the École Normale Supérieure was united with the Université de Paris, until in 1953 it gained financial autonomy and personality. The school nowadays results from the combination of École Normale Supérieure and l'École Normale Supérieure for young women.

Ecole Normale Supérieure has ten Fields Medallists, which makes it the second most prolific institution after Princeton:

**Laurent Schwartz**, Fields Medal 1950

Year Group 1943:

**René Thom**, Fields Medal 1958

Year Group 1945:

**Jean-Pierre Serre**, Fields Medal 1954

Year Group 1966:

**Alain Connes**, Fields Medal 1982

Year Group 1975:

**Pierre-Louis Lions**, Fields Medal 1994

Year Group 1975:

**Jean-Christophe Yoccoz**, Fields Medal 1994

Year Group 1986:

**Laurent Lafforgue**, Fields Medal 2002

Year Group 1987:

**Wendelin Werner**, Fields Medal 2006

Year Group 1992:

**Ngô Bả Châu**, Fields Medal 2010

Year Group 1992:

**Cédric Villani**, Fields Medal 2010

**Nicolas Bourbaki**, an imaginary mathematician, composed of André Weil, Jean Delsarte, Henri Cartan, Jean Coulomb, René de Possel, Jean Dieudonné, Charles Ehresmann, Claude Chevalley and Szolem Mandelbrojt, all alumni from the school, except Mandelbrojt.

The group worked on:

*Set theory*, making Zorn's lemma more popular

*Algebra*, especially algebraic structures, linear algebra and multilinear

algebra

*Topology*, in particular topological structures and uniform structures, topological groups with natural numbers, and functional space

*Topological Vector Spaces*

*Integration*

*Commutative Algebra*(they wanted to explore the basics of the new algebraic geometry made by Grothendieck)

*Notation in mathematics*, such as the use of ∀ and ∃ in France and ⇒ , ⇐ , ⇔ in logic, as well as the popularisation of Ø (for the empty set) and ⊗ (for the tensor product).

### Port-Royal

The last stop on the walk is the Abbey of Port-Royal.

The Abbaye was a high place for Jansenism, a form of catholicism, and was *built to relieve pressure of numbers on the mother house at Port-Royal-des-Champs*. It was regularly used as a place for debate and meetings for the writers and philosophers of the time, where they could work and discuss on grammar, theology, and translations from Hebrew and Greek.

In 1662, Pierre Nicole, a Janseist theologian and **Antoine Arnauld**, members of the Jansenism movement, wrote *La Logique ou l'art de penser*, also called *La logique de Port-Royal*, Pascal probably contributing a lot to the project. It focuses on rules of grammar and logic, which was, until mid 19th Century, a reference in terms of philosophy and logic; logic, language, epistemology and theology being intertwined and very closely correlated notions in this book.

Antoine Arnault was a priest, theologian, philosopher and mathematician, and was called the Euclid of the 17th Century by critics, due to his mathematical work.

**Blaise Pascal** was a mathematician, physicist, philosopher and theologian of the 17th Century. His mathematical study includes projective geometry, calculus, and series of whole numbers, as well as working on Pascal's triangle, for binomial coefficients, in his book *Traité du triangle arithmétique*. The contents of the book were an important training for Leibniz's later work on calculus, and, in there, Pascal used for the first time the principle of mathematical induction. When he was 19, he created the first calculating machine, the *Pascaline*, after three years of research and development and more than 50 prototypes.

This concludes our walk in the streets of the Quartier Latin, having looked at many historical places, important in the study of mathematics and the discovery of notions. Many more mathematical places can be discovered in other areas of Paris, on another walk, on a sunny summer day ...

**Article by: **Diane Paya-Monet University of St Andrews