P**rogramm and abstracts**

Planning of the talks

**Martin Andler (Versailles)**

**Fréchet and the integral with respect to an abstract measure**

In 1915, in the middle of the First World War, Fréchet published a seminal article in the United States in which he defined the integral of a function with respect to a measure on an abstract set. This work of Fréchet, which extends the work of Lebesgue, is in line with his efforts since his thesis to define concepts by reaching "immediately the greatest generality" as Hadamard expressed it in the preface of a later book. The talk will present the content of this 1915 article and its reception.

**Michel Armatte (Paris)**

**Fréchet and mathematical statistics: the campaign on the correlation coefficient.**

Between 1933 and 1937, in a general context of updating the new category of "mathematical statistics", Fréchet led, through a dozen publications, a campaign within the IIS against the misuse of the correlation coefficient. This campaign included a survey of his peers, a debate between several opinions, a motion put to a vote, and finally proposals for new formulas resulting from his own work. This episode reveals Fréchet's double attention to the empirical practices of statisticians of various disciplines and to the mathematical foundations of these practices, in particular his own research on abstract spaces. It also reveals the subtle interplay between truth and opinion that he established within applied mathematics.

**Frédéric Barbaresco (Paris)**

**Maurice Fréchet’s Winter 1939 IHP lecture, Fréchet-Darmois bound, distinguished densities and Information Geometry**

In 1943, two years before C. R. Rao, Maurice Fréchet published a paper making reference to his 1939 winter IHP lecture, in which he introduced the bound which was improperly named Cramer-Rao bound. The paper deals with the monovariate case, but his colleague Georges Darmois published the multivariate version in 1945. It would therefore be more accurate to name this bound the Fréchet-Darmois bound. Beyond the discovery of this bound, Fréchet studied also “distinguished densities” (case where the bound is reached), verifying relations related to Clairaut-Legendre transform. This Clairaut-Legendre transform constitutes a bedrock of the nowadays “Information Geometry” fundamental structure, of which Fréchet half-opened the door, by showing that the inverse of the Fisher matrix is the Hessian of a function. These results were only recovered mathematically during the 1950s by Jean-Louis Koszul, PhD student of Henri Cartan, studying geometry of sharp convex cones associated to homogeneous symmetric bounded domains, developing seminal idea of Elie Cartan.

**Pascal Bertin (Paris) et Frédéric Jaëck (Aix-Marseille) **

** Taming the infinite: Fréchet and compactness.**

From his first works, Fréchet envisages, in a search for generality, the passage from functions defined on the real line to functions defined on abstract spaces. In order to guarantee the generalization of well-known theorems to the real case, Fréchet proposes various (pre-)topological notions and in particular a definition of compactness which is his own. Our philosophical approach is to show how Fréchet's notion of compactness is a particular way of working with and mastering an infinity linked to his notion of abstract space. We will show that this relation to infinity is different from the one we hear nowadays when we talk about compactness in elementary topology.

**Matthias Cléry (Paris)**

**Maurice Fréchet: Professor of Probability at the Faculty of Science in Paris (1928-1949)**

Appointed lecturer in the chair of CPPM in 1928, just after the creation of the Institut Henri Poincaré, and then professor in this chair in 1941, Fréchet was part of a dynamic process initiated by Émile Borel to develop the teaching of probability at the Faculty of Science by linking it to research and to attract young mathematicians to this field. On the basis of Jean-Louis Destouches' lecture notes and the material for the lectures published in the 1940s, we will present Fréchet's teachings at two critical moments for this dynamic: the first years of the IHP and the period of the Occupation.

**Eva Colebunders (Brussels)**

**From metrizable spaces to spaces defined by gauges of (generalized) metrics.**

My talk relies on a joint paper with R.Lowen (Metrically Generated Theories, Proc. Amer. Math. Soc., 133, (5), (2005), 1547-1556). The legacy of Fréchet is impressive and his work had an extremally important impact on the development of general topology. His metric spaces and metrizable topological spaces are the starting point of the emergence of a large array of ”topological categories” studied in topology. We name just a few. The category Top of topological spaces with continuous maps was created as an abstract setting for studying limits or compactness. The category Unif of uniform spaces with uniformly continuous maps became the abstract setting for Cauchyness or completeness. The category Prox of proximity spaces with proximal maps was created to describe nearness between sets and the category App of approach spaces with contractions was constructed to remedy the fact that a product of metrizable spaces is not necessarily metrizable. We show that all these and many other topological categories in a sense are quite close to metric spaces. For each of them a first observation is that for some category of (generalised) metric spaces with non-expansive maps, the functor mapping a (generalised) metric space to the associated metrizable object has the nice properties that the image is initially dense and that the functor preserves initial morphisms. A second observation is that the category can be isomorphically described as a category of sets endowed with a gauge of (generalised) metrics. We show that both observations express essentially the same property which we call metrically generated. The overarching context of gauge spaces enables us to obtain unified theories on subcategories consisting of separated objects, their epimorphisms, existence of initially dense objects, construction of function spaces and completion theory.

**Jimmy De Groote (Paris)**

**Logic as a product of experience: the philosophy behind Mathematics and the concrete. **

It is a well-known fact that Fréchet devoted a substantial part of his career to knowing and making known probability. Le calcul des probabilités à la portée de tous (1924), written in four hands with Halbwachs on the basis of courses given together at the Institut commercial supérieur de Strasbourg, illustrates well the pedagogical requirement that accompanied his scientific activity (a requirement that was in particular prompted by the need to compensate for the delays accumulated by France in training in statistical science). The aim was to present the calculation of probabilities by stripping it of its technical trappings in order to encourage its acquisition by "doctors, demographers, actuary, insurance agents, etc.", in short, all persons in a position to use it in their professional or private life. The idea (some would say the bet) underlying this enterprise is that it is possible to develop an intuitive, but mathematically rigorous, understanding of the main results of calculus by applying elementary algebraic notions to concrete cases drawn from experience. This position is based on a strong conviction, which breaks with the thesis that mathematics are purely formal conventions or tautologies: probabilities have an empirical anchor, they refer to something. Moreover, their correspondence with reality is not indifferent to their meaning: it attests to their truth, i.e. their irreducibility to human arbitrariness. Les Mathématiques et le concret, a collection of articles for the general public published in 1955, extends this reflection by embracing a wider perspective: as its title indicates, it is now a question of questioning the link between experience and mathematics in general. It deals with probabilities, of course, but also with analysis, sets, geometry, group theory, etc. These efforts give rise, in particular, in the second article of the first chapter, to what could be called an empirical genealogy of mathematical notions, which ends with the suggestion that "logic itself is a product of our experience". We would like to question the approach that Fréchet pursues in his book in the light of this conclusion : how does he achieve it ? With what arguments? What examples? To what ends? This would be an opportunity for us to discuss the philosophical content of this thesis by comparing it with the positions of Mill and Quine, two other important representatives of mathematical empiricism.

**Georges Fréchet et Marianne Lederer**

**Intimate Maurice Fréchet**

Maurice Fréchet's grandchildren decided to bring to light family and intimate memories of his personality, to describe some aspects of a complex and endearing human being, to put into words how they felt about him and what they learned from him for their own lives. Some of them lived with him for several years, but others who saw him less often, were nonetheless impressed by certain traits of his character. These testimonies will be preceded by a biographical presentation based on unpublished documents, focusing exclusively on non-scientific aspects, in particular his youth, his marriage and his behavior during the two world wars.

**Maria Carla Galavotti (Bologna)**

**Maurice Fréchet and the ‘inductive synthesis’**

Deeply convinced that the notion of probability must be addressed from the standpoint of its applications, Fréchet holds that the mathematics of probability cannot do the whole job and must be backed up by a careful examination of the case under study. To lay a bridge between the abstract, mathematical theory of probability and its applications, a combination of empirical and axiomatic elements is needed. A fundamental role in this regard is played by what Fréchet calls inductive synthesis. The basic idea of this procedure is that the axioms should be preceded by the act of verifying whether in practical situations the axioms apply to the phenomena under study. In other words, the measurement of probability requires a synthesis between the axiomatic theory and a number of considerations pertaining practical situations. Such an inductive synthesis is the cornerstone of Fréchet’s interpretation of probability, which he labels ‘modernised axiomatic theory’. The main features of Fréchet’s conception of probability will be illustrated, together with his criticism of other interpretations, including frequentism and subjectivism.

**Angelo Guerraggio (Varese)**

**Maurice Fréchet and the Italian mathematicians.**

Hadamard's good relations with his Italian colleagues (Volterra in particular) and the 1906 publication of his thesis on the "Rendiconti del Circolo Matematico di Palermo" granted Fréchet privileged access to the most renowned Italian Mathematicians. I analyze here his relationship with Italian analysts (Volterra but also Arzelà, B. Levi, and Tonelli), focusing on the ideas exchange and clarifications made on the first developments of functional analysis, as well as the collaboration developed at an institutional level for the international congresses of Mathematics.

**Snezana Lawrence (London)**

**Kurepa’s Networks **

This talk will focus on the work of Đuro Kurepa (1907-1993), one of the two Yugoslavian doctoral students of Maurice Fréchet (second was a statistician Slobodan Zarković). Kurepa was born into a large family; from this family also came another famous Yugoslavian mathematician, Đuro’s nephew, Svetozar (1929–2010). Đuro (George) Kurepa studied mathematics at the University of Zagreb, where he graduated in 1931. Kurepa began his post-graduate education at the Collège de France in 1932 and defended his doctoral thesis Ensembles ordonnés et ramifiés [Ordered and ramified sets] at the Sorbonne in 1935. His main advisor was Maurice Fréchet (1878–1973). Đuro Kurepa’s second advisor was the teacher of Milena Marić-Einstein (1875-1948), and the correspondent of Einstein’s, Vladimir Varićak (1865–1942), professor at the University of Zagreb. Fréchet already had an extensive network of colleagues who valued highly his contributions. In particular, it seems he had made a lasting impact on the colleagues in Eastern Europe, such as Nikolai Luzin (1883–1950), Wacław Sierpiński (1882–1969), and Pavel Aleksandrov (1896–1982). Kurepa continued the cooperation with Sierpiński and went to work at the University of Warsaw with Sierpiński. He returned to Paris in 1937. Whilst in Paris, he was invited to become a professor at the University of Zagreb, where he stayed from 1937 to 1965. After World War II, Kurepa visited the US, and stayed at the five leading US universities—Harvard, Chicago, California at Berkeley, Los Angeles, and Institute for Advanced Study at Princeton. He spent a summer semester of 1959 as a professor at Teachers’ College of the Columbia University, New York, and a spring semester as a visiting professor at Colorado University of Boulder in 1960. Kurepa’s mathematical work is reasonably well known. Perhaps less known, outside of his country (and which eventually disintegrated) however was his work on setting and developing mathematical institutions and his work on values in mathematics education as noted in his writings and addresses in his role as the Vice-president of ICMI. This talk will pay attention to Kurepa’s networks of cooperation, and try to establish an overview of the corresponding points of contact with the networks developed by his PhD supervisor, Fréchet.

**Laurent Mazliak (Paris)**

**Exchanges between Paul Lévy and Maurice Fréchet: at the source of modern probabilities**

After the First World War, Paul Lévy contacted Maurice Fréchet, inaugurating a correspondence which was to last for nearly fifty years. Even if unfortunately we have almost no letters from Fréchet to Lévy, notably because during the Occupation, Lévy's Parisian apartment was looted by the Gestapo, the 120 or so letters from Lévy contained in the Fréchet collection of the Académie des Sciences constitute an invaluable document for following Paul Lévy's trajectory at the most creative moment of his career, when he made an impressive turn from Functional Analysis to probability, of which he was to become one of the world's leaders very quickly. In many cases, the letters to Fréchet gave Lévy a first opportunity to put his ideas in order and to propose sketches of demonstrations of fundamental results. Particularly significant are the first exchanges which show, almost day by day, how Lévy was to begin to think of the concepts of functional analysis in a probabilistic form. Another crucial period was the long months of clandestinity between 1942 and 1944 when Paul Lévy, forbidden to publish because of the racial laws, used this correspondence to take stock of the many results he discovered on Brownian motion, which were to form the framework of his 1948 book. In the presentation, I will give some indications on the genesis of this correspondence, over which the figures of Borel and Hadamard hover, and I will comment on certain important moments.

**Quentin Menet (Mons)**

** Dynamical properties of operators on Fréchet spaces **

The number of mathematical objects with which the name of Fréchet is associated is impressive : Fréchet derivative, Fréchet distribution, Fréchet filter, … As the title suggests, we will be interested during this presentation in one of them : the Fréchet spaces. This family of spaces lies between Baire spaces and Banach spaces and contains important examples in functional analysis : the space of holomorphic functions on the complex plane, the space of infinitely differentiable real functions, the Schwartz space, … Fréchet spaces also form a favorable framework for operator theory by allowing to apply the uniform boundedness principle, the open mapping theorem and the Hahn–Banach continuous extension theorem. During this talk, we will investigate the dynamical properties of operators on Fréchet spaces such as the hypercyclicity (relying on the existence of a dense orbit) or the cyclicity which is linked to the existence of invariant subspaces. We will therefore start from the beginnings of linear dynamics before tackling recent problems related to Fréchet spaces.

**Pierre Mounier-Kuhn (Paris)**

**The IHP, from mechanical calculation to computer science **

The Institut Henri-Poincaré's investment in scientific computing dates back to the 1930s and stems from its vocation at the interface of mathematics and physics. The scientific mobilization of 1939-1940 intensified the need for calculation and modeling for a wide variety of technical applications. It motivated both the development of computing offices and the construction of a large computing machine, which was the subject of two competing projects: that of Maurice Fréchet and that of Louis Couffignal. The military debacle put an end to this and, after the war, it was the CNRS that took over the field of "mechanical calculation". However, the IHP maintained a numerical calculation service that worked at the request of public research establishments and in 1957 it acquired a Bull electronic calculator - one of the very first in French universities. Since 1947, it has also hosted seminars for a numerical calculation group led by Lacroix de Lavalette, an enthusiastic amateur, and Paul Belgodère, the IHP librarian; this group became the learned society of computer scientists. The IHP, in its intellectual and institutional environment, thus appears as one of the cradles of French research in computer science.

**Mathurin Passard (Lyon)**

** Fréchet and the Lyon conference on probability in 1948. Reviving the mathematics of chance in France.**

My talk is devoted to the colloquium "The calculus of probability and its applications" organized in Lyon in 1948 by Maurice Fréchet and Henri Eyrault (director of the Institute of Financial Sciences and Actuarial Science of Lyon). This conference had a unique place in a France that had to rebuild itself after the war while reaffirming its scientific position on the international scene. Fréchet wanted to give French probabilists and statisticians, who were in the majority during the event, an opportunity for international influence and to meet with important foreign actors of recent developments (such as the American mathematician Joseph Doob, who exposed his theory of martingales in Lyon for the first time). The organization of this colloquium was also a pretext for Fréchet to promote the interaction between economics and probability theory by highlighting the ISFA - the host institution of the event - as a research laboratory on these questions.

**Thomas Perfettini (Montpellier)**

**Maurice Fréchet's trip to Eastern Europe in 1935**

In the autumn of 1935, Maurice Fréchet undertook a journey of several weeks in the USSR and in Eastern Europe. From Sofia to Moscow, passing through Lwow or Bucharest, he promoted French mathematics through various conferences and cultivated the links he had developed with the actors of mathematical schools, sometimes still young and in full development. In my talk, I will present the different aspects of the mathematician's journey by specifying the personal and geopolitical conditions in which it occurs. In addition, I will try to give some details on the mathematical themes evoked during the different stages of this journey.

**Julien Pomart (Paris)**

**A presentation of the Maurice Fréchet archives**

What archives has Maurice Fréchet left ? The conference organized on the occasion of the fiftieth anniversary of his death is an opportunity for the Academy of Science Archives to present this collection as well as the processing work recently carried out. Starting from the rapid census of the boxes carried out under the scientist's gaze to arrive at the archival classification and description, we will analyze the general composition and the evolution of the structuring of the collection, as well as the elements particularly likely to attract the attention of the scientific community.

**Gatien Ricotier(Strasbourg)**

**Maurice Fréchet and Bourbaki: cross paths of distant colleagues.**

The geographical, mathematical, and philosophical paths of Maurice Fréchet and Bourbaki's members crossed several times but never really lasted, which cannot be explained only by the gap in the birth years of these mathematicians. This situation reflects the similarities of their careers alongside their opposing points of view. In this talk, I will provide an overview of the interactions between these two generations and explore the origins and consequences of their disagreements.

**Norbert Schappacher (Strasbourg)**

**Maurice Fréchet's years in Strasbourg **

The decade that Maurice Fréchet spent in Strasbourg after the Great War was marked from the outset by his important role in the establishment of the University of Strasbourg within the buildings inherited from the German period. Starting with the way in which Fréchet assumed this role, the presentation will then insist on some of his collegial contacts on the spot, and will end with a look at the orientation of his research during this period.

**Glenn Shafer (Rutgers) **

**Maurice Fréchet named Cournot’s principle. What did he name?**

Jacob Bernoulli inherited from the scholastics the principle that sufficiently high probability provides practical certainty. Antoine Augustin Cournot’s originality was to see the principle as a bridge between mathematics and the objective world. The naming began with Aleksandr Chuprov and culminated with Fréchet. Fréchet advanced two opinions of his own: (1) Cournot’s principle supports Fréchet’s own conception of probability as a physical quantity. (2) The event that is impossible because of its small probability must be specified in advance.

**Reinhard Siegmund-Schultze (Kristiansand) **

**Correspondence between Fréchet and von Mises in the 1930s on the foundations of the theory of probability**

Between 1935 and 1941 Fréchet had a correspondence comprising about 20 letters and postcards in French with the probabilist Richard von Mises (1883-1953). Von Mises was then a refugee in Istanbul and went on to Harvard University in 1939. He was known for his 1919 definition of probability as the limit of the relative frequency of the occurrence of an event which was disputed among probabilists from the beginning and was also one of several points of discussion between the two correspondents. The immediate reason for the correspondence was Fréchet’s two volume book on probability in the Collection Borel, which was published 1937/38. Von Mises read proofs for both volumes, and the discussion beside the definition problem was above all the positions of the two mathematicians vis-à-vis the “method of arbitrary functions”, a main topic of Fréchet’s volume II. Another major point in the correspondence were talks given by von Mises at the Zürich ICM 1932 and by Fréchet at the Oslo ICM 1936. In his sectional talk in Oslo Fréchet was very critical of von Mises presentation in Zürich, not restricting his criticism to the definition of probability. This dispute was continued in the colloquium on probability theory which took place in October 1937 in Geneve under the lead of Fréchet. Von Mises was not attending but submitted two papers which were published in the Proceedings, one of the two being strongly critical of Fréchet’s presentation at the colloquium. The Geneve colloquium was also discussed in the correspondence which had marginal topics such as communication under war conditions and help for Jewish refugees as well.

**Maud Thomas (Paris)**

**Limit distributions of the maximum of a sample**

Fréchet studied the distribution of probability of the maximum of a sample of identically distributed independent variables giving rise to the 1927 article « Sur la loi de probabilité de l’écart maximum". Fréchet's distribution plays an essential role in the extreme value theory. It is one of the three max-stable distributions from which the Fréchet domain is derived, corresponding to the family of heavy-tailed distributions. In this talk, we will study the limit distributions of the maximum of a sequence of random variables and the family of heavy-tailed distributions. We will also see some examples of applications.

**Laura E. Turner (Monmouth) **

**Fréchet in America: Transmitting Modern Ideas to Students**

One facet of the transfer of Fréchet’s work and ideas to the United States involves American students. Sometimes this transfer was indirect, through a book, a class taught by an American professor, or a report on Fréchet’s work by a fellow student at a mathematics club meeting. Other times, however, students came into contact with Fréchet, himself. Fréchet’s interactions with American students spanned several decades — and hence several stages of his career — beginning in the 1920s, with a planned visit in 1914 thwarted by the First World War. What is more, his visits to the United States brought him to various institutions across the country, including Columbia University, the Illinois Institute of Technology, and Oregon State University. Notably, during his 1924 visit to the University of Chicago he taught two advanced courses, one each on the theory of abstract sets and the theory of probability, in the summer quarter. Beyond this, the transfer of his ideas, whether directly or indirectly, took place within different contexts and at different levels, ranging from expository mathematical articles and descriptions of the French academic system to discussions of advanced work. In this talk, we will touch upon a number of Fréchet’s interactions, both direct and indirect, with American students, in an attempt to gain a more complete understanding of his mathematical activities and their many contexts.

In 1915, in the middle of the First World War, Fréchet published a seminal article in the United States in which he defined the integral of a function with respect to a measure on an abstract set. This work of Fréchet, which extends the work of Lebesgue, is in line with his efforts since his thesis to define concepts by reaching "immediately the greatest generality" as Hadamard expressed it in the preface of a later book. The talk will present the content of this 1915 article and its reception.

Between 1933 and 1937, in a general context of updating the new category of "mathematical statistics", Fréchet led, through a dozen publications, a campaign within the IIS against the misuse of the correlation coefficient. This campaign included a survey of his peers, a debate between several opinions, a motion put to a vote, and finally proposals for new formulas resulting from his own work. This episode reveals Fréchet's double attention to the empirical practices of statisticians of various disciplines and to the mathematical foundations of these practices, in particular his own research on abstract spaces. It also reveals the subtle interplay between truth and opinion that he established within applied mathematics.

In 1943, two years before C. R. Rao, Maurice Fréchet published a paper making reference to his 1939 winter IHP lecture, in which he introduced the bound which was improperly named Cramer-Rao bound. The paper deals with the monovariate case, but his colleague Georges Darmois published the multivariate version in 1945. It would therefore be more accurate to name this bound the Fréchet-Darmois bound. Beyond the discovery of this bound, Fréchet studied also “distinguished densities” (case where the bound is reached), verifying relations related to Clairaut-Legendre transform. This Clairaut-Legendre transform constitutes a bedrock of the nowadays “Information Geometry” fundamental structure, of which Fréchet half-opened the door, by showing that the inverse of the Fisher matrix is the Hessian of a function. These results were only recovered mathematically during the 1950s by Jean-Louis Koszul, PhD student of Henri Cartan, studying geometry of sharp convex cones associated to homogeneous symmetric bounded domains, developing seminal idea of Elie Cartan.

From his first works, Fréchet envisages, in a search for generality, the passage from functions defined on the real line to functions defined on abstract spaces. In order to guarantee the generalization of well-known theorems to the real case, Fréchet proposes various (pre-)topological notions and in particular a definition of compactness which is his own. Our philosophical approach is to show how Fréchet's notion of compactness is a particular way of working with and mastering an infinity linked to his notion of abstract space. We will show that this relation to infinity is different from the one we hear nowadays when we talk about compactness in elementary topology.

Appointed lecturer in the chair of CPPM in 1928, just after the creation of the Institut Henri Poincaré, and then professor in this chair in 1941, Fréchet was part of a dynamic process initiated by Émile Borel to develop the teaching of probability at the Faculty of Science by linking it to research and to attract young mathematicians to this field. On the basis of Jean-Louis Destouches' lecture notes and the material for the lectures published in the 1940s, we will present Fréchet's teachings at two critical moments for this dynamic: the first years of the IHP and the period of the Occupation.

My talk relies on a joint paper with R.Lowen (Metrically Generated Theories, Proc. Amer. Math. Soc., 133, (5), (2005), 1547-1556). The legacy of Fréchet is impressive and his work had an extremally important impact on the development of general topology. His metric spaces and metrizable topological spaces are the starting point of the emergence of a large array of ”topological categories” studied in topology. We name just a few. The category Top of topological spaces with continuous maps was created as an abstract setting for studying limits or compactness. The category Unif of uniform spaces with uniformly continuous maps became the abstract setting for Cauchyness or completeness. The category Prox of proximity spaces with proximal maps was created to describe nearness between sets and the category App of approach spaces with contractions was constructed to remedy the fact that a product of metrizable spaces is not necessarily metrizable. We show that all these and many other topological categories in a sense are quite close to metric spaces. For each of them a first observation is that for some category of (generalised) metric spaces with non-expansive maps, the functor mapping a (generalised) metric space to the associated metrizable object has the nice properties that the image is initially dense and that the functor preserves initial morphisms. A second observation is that the category can be isomorphically described as a category of sets endowed with a gauge of (generalised) metrics. We show that both observations express essentially the same property which we call metrically generated. The overarching context of gauge spaces enables us to obtain unified theories on subcategories consisting of separated objects, their epimorphisms, existence of initially dense objects, construction of function spaces and completion theory.

It is a well-known fact that Fréchet devoted a substantial part of his career to knowing and making known probability. Le calcul des probabilités à la portée de tous (1924), written in four hands with Halbwachs on the basis of courses given together at the Institut commercial supérieur de Strasbourg, illustrates well the pedagogical requirement that accompanied his scientific activity (a requirement that was in particular prompted by the need to compensate for the delays accumulated by France in training in statistical science). The aim was to present the calculation of probabilities by stripping it of its technical trappings in order to encourage its acquisition by "doctors, demographers, actuary, insurance agents, etc.", in short, all persons in a position to use it in their professional or private life. The idea (some would say the bet) underlying this enterprise is that it is possible to develop an intuitive, but mathematically rigorous, understanding of the main results of calculus by applying elementary algebraic notions to concrete cases drawn from experience. This position is based on a strong conviction, which breaks with the thesis that mathematics are purely formal conventions or tautologies: probabilities have an empirical anchor, they refer to something. Moreover, their correspondence with reality is not indifferent to their meaning: it attests to their truth, i.e. their irreducibility to human arbitrariness. Les Mathématiques et le concret, a collection of articles for the general public published in 1955, extends this reflection by embracing a wider perspective: as its title indicates, it is now a question of questioning the link between experience and mathematics in general. It deals with probabilities, of course, but also with analysis, sets, geometry, group theory, etc. These efforts give rise, in particular, in the second article of the first chapter, to what could be called an empirical genealogy of mathematical notions, which ends with the suggestion that "logic itself is a product of our experience". We would like to question the approach that Fréchet pursues in his book in the light of this conclusion : how does he achieve it ? With what arguments? What examples? To what ends? This would be an opportunity for us to discuss the philosophical content of this thesis by comparing it with the positions of Mill and Quine, two other important representatives of mathematical empiricism.

Maurice Fréchet's grandchildren decided to bring to light family and intimate memories of his personality, to describe some aspects of a complex and endearing human being, to put into words how they felt about him and what they learned from him for their own lives. Some of them lived with him for several years, but others who saw him less often, were nonetheless impressed by certain traits of his character. These testimonies will be preceded by a biographical presentation based on unpublished documents, focusing exclusively on non-scientific aspects, in particular his youth, his marriage and his behavior during the two world wars.

Deeply convinced that the notion of probability must be addressed from the standpoint of its applications, Fréchet holds that the mathematics of probability cannot do the whole job and must be backed up by a careful examination of the case under study. To lay a bridge between the abstract, mathematical theory of probability and its applications, a combination of empirical and axiomatic elements is needed. A fundamental role in this regard is played by what Fréchet calls inductive synthesis. The basic idea of this procedure is that the axioms should be preceded by the act of verifying whether in practical situations the axioms apply to the phenomena under study. In other words, the measurement of probability requires a synthesis between the axiomatic theory and a number of considerations pertaining practical situations. Such an inductive synthesis is the cornerstone of Fréchet’s interpretation of probability, which he labels ‘modernised axiomatic theory’. The main features of Fréchet’s conception of probability will be illustrated, together with his criticism of other interpretations, including frequentism and subjectivism.

Hadamard's good relations with his Italian colleagues (Volterra in particular) and the 1906 publication of his thesis on the "Rendiconti del Circolo Matematico di Palermo" granted Fréchet privileged access to the most renowned Italian Mathematicians. I analyze here his relationship with Italian analysts (Volterra but also Arzelà, B. Levi, and Tonelli), focusing on the ideas exchange and clarifications made on the first developments of functional analysis, as well as the collaboration developed at an institutional level for the international congresses of Mathematics.

This talk will focus on the work of Đuro Kurepa (1907-1993), one of the two Yugoslavian doctoral students of Maurice Fréchet (second was a statistician Slobodan Zarković). Kurepa was born into a large family; from this family also came another famous Yugoslavian mathematician, Đuro’s nephew, Svetozar (1929–2010). Đuro (George) Kurepa studied mathematics at the University of Zagreb, where he graduated in 1931. Kurepa began his post-graduate education at the Collège de France in 1932 and defended his doctoral thesis Ensembles ordonnés et ramifiés [Ordered and ramified sets] at the Sorbonne in 1935. His main advisor was Maurice Fréchet (1878–1973). Đuro Kurepa’s second advisor was the teacher of Milena Marić-Einstein (1875-1948), and the correspondent of Einstein’s, Vladimir Varićak (1865–1942), professor at the University of Zagreb. Fréchet already had an extensive network of colleagues who valued highly his contributions. In particular, it seems he had made a lasting impact on the colleagues in Eastern Europe, such as Nikolai Luzin (1883–1950), Wacław Sierpiński (1882–1969), and Pavel Aleksandrov (1896–1982). Kurepa continued the cooperation with Sierpiński and went to work at the University of Warsaw with Sierpiński. He returned to Paris in 1937. Whilst in Paris, he was invited to become a professor at the University of Zagreb, where he stayed from 1937 to 1965. After World War II, Kurepa visited the US, and stayed at the five leading US universities—Harvard, Chicago, California at Berkeley, Los Angeles, and Institute for Advanced Study at Princeton. He spent a summer semester of 1959 as a professor at Teachers’ College of the Columbia University, New York, and a spring semester as a visiting professor at Colorado University of Boulder in 1960. Kurepa’s mathematical work is reasonably well known. Perhaps less known, outside of his country (and which eventually disintegrated) however was his work on setting and developing mathematical institutions and his work on values in mathematics education as noted in his writings and addresses in his role as the Vice-president of ICMI. This talk will pay attention to Kurepa’s networks of cooperation, and try to establish an overview of the corresponding points of contact with the networks developed by his PhD supervisor, Fréchet.

After the First World War, Paul Lévy contacted Maurice Fréchet, inaugurating a correspondence which was to last for nearly fifty years. Even if unfortunately we have almost no letters from Fréchet to Lévy, notably because during the Occupation, Lévy's Parisian apartment was looted by the Gestapo, the 120 or so letters from Lévy contained in the Fréchet collection of the Académie des Sciences constitute an invaluable document for following Paul Lévy's trajectory at the most creative moment of his career, when he made an impressive turn from Functional Analysis to probability, of which he was to become one of the world's leaders very quickly. In many cases, the letters to Fréchet gave Lévy a first opportunity to put his ideas in order and to propose sketches of demonstrations of fundamental results. Particularly significant are the first exchanges which show, almost day by day, how Lévy was to begin to think of the concepts of functional analysis in a probabilistic form. Another crucial period was the long months of clandestinity between 1942 and 1944 when Paul Lévy, forbidden to publish because of the racial laws, used this correspondence to take stock of the many results he discovered on Brownian motion, which were to form the framework of his 1948 book. In the presentation, I will give some indications on the genesis of this correspondence, over which the figures of Borel and Hadamard hover, and I will comment on certain important moments.

The number of mathematical objects with which the name of Fréchet is associated is impressive : Fréchet derivative, Fréchet distribution, Fréchet filter, … As the title suggests, we will be interested during this presentation in one of them : the Fréchet spaces. This family of spaces lies between Baire spaces and Banach spaces and contains important examples in functional analysis : the space of holomorphic functions on the complex plane, the space of infinitely differentiable real functions, the Schwartz space, … Fréchet spaces also form a favorable framework for operator theory by allowing to apply the uniform boundedness principle, the open mapping theorem and the Hahn–Banach continuous extension theorem. During this talk, we will investigate the dynamical properties of operators on Fréchet spaces such as the hypercyclicity (relying on the existence of a dense orbit) or the cyclicity which is linked to the existence of invariant subspaces. We will therefore start from the beginnings of linear dynamics before tackling recent problems related to Fréchet spaces.

The Institut Henri-Poincaré's investment in scientific computing dates back to the 1930s and stems from its vocation at the interface of mathematics and physics. The scientific mobilization of 1939-1940 intensified the need for calculation and modeling for a wide variety of technical applications. It motivated both the development of computing offices and the construction of a large computing machine, which was the subject of two competing projects: that of Maurice Fréchet and that of Louis Couffignal. The military debacle put an end to this and, after the war, it was the CNRS that took over the field of "mechanical calculation". However, the IHP maintained a numerical calculation service that worked at the request of public research establishments and in 1957 it acquired a Bull electronic calculator - one of the very first in French universities. Since 1947, it has also hosted seminars for a numerical calculation group led by Lacroix de Lavalette, an enthusiastic amateur, and Paul Belgodère, the IHP librarian; this group became the learned society of computer scientists. The IHP, in its intellectual and institutional environment, thus appears as one of the cradles of French research in computer science.

My talk is devoted to the colloquium "The calculus of probability and its applications" organized in Lyon in 1948 by Maurice Fréchet and Henri Eyrault (director of the Institute of Financial Sciences and Actuarial Science of Lyon). This conference had a unique place in a France that had to rebuild itself after the war while reaffirming its scientific position on the international scene. Fréchet wanted to give French probabilists and statisticians, who were in the majority during the event, an opportunity for international influence and to meet with important foreign actors of recent developments (such as the American mathematician Joseph Doob, who exposed his theory of martingales in Lyon for the first time). The organization of this colloquium was also a pretext for Fréchet to promote the interaction between economics and probability theory by highlighting the ISFA - the host institution of the event - as a research laboratory on these questions.

In the autumn of 1935, Maurice Fréchet undertook a journey of several weeks in the USSR and in Eastern Europe. From Sofia to Moscow, passing through Lwow or Bucharest, he promoted French mathematics through various conferences and cultivated the links he had developed with the actors of mathematical schools, sometimes still young and in full development. In my talk, I will present the different aspects of the mathematician's journey by specifying the personal and geopolitical conditions in which it occurs. In addition, I will try to give some details on the mathematical themes evoked during the different stages of this journey.

What archives has Maurice Fréchet left ? The conference organized on the occasion of the fiftieth anniversary of his death is an opportunity for the Academy of Science Archives to present this collection as well as the processing work recently carried out. Starting from the rapid census of the boxes carried out under the scientist's gaze to arrive at the archival classification and description, we will analyze the general composition and the evolution of the structuring of the collection, as well as the elements particularly likely to attract the attention of the scientific community.

The geographical, mathematical, and philosophical paths of Maurice Fréchet and Bourbaki's members crossed several times but never really lasted, which cannot be explained only by the gap in the birth years of these mathematicians. This situation reflects the similarities of their careers alongside their opposing points of view. In this talk, I will provide an overview of the interactions between these two generations and explore the origins and consequences of their disagreements.

The decade that Maurice Fréchet spent in Strasbourg after the Great War was marked from the outset by his important role in the establishment of the University of Strasbourg within the buildings inherited from the German period. Starting with the way in which Fréchet assumed this role, the presentation will then insist on some of his collegial contacts on the spot, and will end with a look at the orientation of his research during this period.

Jacob Bernoulli inherited from the scholastics the principle that sufficiently high probability provides practical certainty. Antoine Augustin Cournot’s originality was to see the principle as a bridge between mathematics and the objective world. The naming began with Aleksandr Chuprov and culminated with Fréchet. Fréchet advanced two opinions of his own: (1) Cournot’s principle supports Fréchet’s own conception of probability as a physical quantity. (2) The event that is impossible because of its small probability must be specified in advance.

Between 1935 and 1941 Fréchet had a correspondence comprising about 20 letters and postcards in French with the probabilist Richard von Mises (1883-1953). Von Mises was then a refugee in Istanbul and went on to Harvard University in 1939. He was known for his 1919 definition of probability as the limit of the relative frequency of the occurrence of an event which was disputed among probabilists from the beginning and was also one of several points of discussion between the two correspondents. The immediate reason for the correspondence was Fréchet’s two volume book on probability in the Collection Borel, which was published 1937/38. Von Mises read proofs for both volumes, and the discussion beside the definition problem was above all the positions of the two mathematicians vis-à-vis the “method of arbitrary functions”, a main topic of Fréchet’s volume II. Another major point in the correspondence were talks given by von Mises at the Zürich ICM 1932 and by Fréchet at the Oslo ICM 1936. In his sectional talk in Oslo Fréchet was very critical of von Mises presentation in Zürich, not restricting his criticism to the definition of probability. This dispute was continued in the colloquium on probability theory which took place in October 1937 in Geneve under the lead of Fréchet. Von Mises was not attending but submitted two papers which were published in the Proceedings, one of the two being strongly critical of Fréchet’s presentation at the colloquium. The Geneve colloquium was also discussed in the correspondence which had marginal topics such as communication under war conditions and help for Jewish refugees as well.

Fréchet studied the distribution of probability of the maximum of a sample of identically distributed independent variables giving rise to the 1927 article « Sur la loi de probabilité de l’écart maximum". Fréchet's distribution plays an essential role in the extreme value theory. It is one of the three max-stable distributions from which the Fréchet domain is derived, corresponding to the family of heavy-tailed distributions. In this talk, we will study the limit distributions of the maximum of a sequence of random variables and the family of heavy-tailed distributions. We will also see some examples of applications.

One facet of the transfer of Fréchet’s work and ideas to the United States involves American students. Sometimes this transfer was indirect, through a book, a class taught by an American professor, or a report on Fréchet’s work by a fellow student at a mathematics club meeting. Other times, however, students came into contact with Fréchet, himself. Fréchet’s interactions with American students spanned several decades — and hence several stages of his career — beginning in the 1920s, with a planned visit in 1914 thwarted by the First World War. What is more, his visits to the United States brought him to various institutions across the country, including Columbia University, the Illinois Institute of Technology, and Oregon State University. Notably, during his 1924 visit to the University of Chicago he taught two advanced courses, one each on the theory of abstract sets and the theory of probability, in the summer quarter. Beyond this, the transfer of his ideas, whether directly or indirectly, took place within different contexts and at different levels, ranging from expository mathematical articles and descriptions of the French academic system to discussions of advanced work. In this talk, we will touch upon a number of Fréchet’s interactions, both direct and indirect, with American students, in an attempt to gain a more complete understanding of his mathematical activities and their many contexts.