What puzzles you most about France?

Why has France historically produced so many renowned mathematicians, such as: Descartes, Fermat, Pascal, Rolle, L’Hôpital, de Moivre, d’Alembert, Lagrange, Monge, Laplace, Legendre, Fourier, Poisson, Cauchy, Liouville, Galois, Hermite, Jordan, Darboux, Poincaré, Picard, Hadamard, Élie Cartan, Borel, Lebesgue, Henri Cartan, André Weil, Jean Leray, Laurent-Moïse Schwartz, Jean-Pierre Serre, and Pierre-Louis Lions…

Ding Jiu: Hadamard’s Mathematics, Seminars, and Affection for China

The same year brought Hadamard personal joy: he married a Jewish woman. They had three sons and two daughters; one daughter, Jacqueline Hadamard (1902–1988), would decades later play an active role in strengthening his ties with the People’s Republic of China. Tragically, however, all three of Hadamard’s sons perished in German artillery fire during the two world wars—an occurrence perhaps unparalleled among internationally renowned mathematicians’ families. Two years after his marriage, Hadamard moved to the ancient University of Bordeaux to take up a teaching position. During his first year there, he proved the famous determinant inequality now known as “Hadamard’s inequality,” publishing it that same year. This mathematical formula—applicable to complex matrices—has a geometric interpretation for real 3×3 matrices: among all parallelepipeds defined by three adjacent edge vectors, the volume is maximized when those three vectors are pairwise orthogonal. In investigating when equality holds, Hadamard was inspired to consider a special class of matrices—square matrices whose entries are either 1 or −1 and whose rows are pairwise orthogonal—a type now named “Hadamard matrices” in his honor. 1896 proved to be Hadamard’s annus mirabilis. At just thirty years old, he proved the Prime Number Theorem, using tools from analytic function theory—the field in which he excelled. This celebrated theorem in number theory originated as a conjecture formulated at the end of the eighteenth century by the French mathematician Adrien-Marie Legendre (1752–1833); it addresses the asymptotic distribution of prime numbers among the natural numbers. A prime number is a natural number greater than 1 that has no positive integer divisors other than 1 and itself (e.g., 5 is prime, whereas 12 is not). The Prime Number Theorem states that the number of primes less than or equal to a real number x ≥ 1 is asymptotically equal to x divided by its natural logarithm, as x tends to infinity. Major figures in number theory—including the German mathematicians Carl Friedrich Gauss (1777–1855) and Peter Gustav Lejeune Dirichlet (1805–1859), and the Russian mathematician Pafnuty Lvovich Chebyshev (1812–1894)—all made significant contributions toward establishing this theorem. Beyond the Prime Number Theorem, Hadamard’s second major mathematical achievement that year concerned his research on differential geometry of surfaces and geodesics in dynamical systems. For this work, he received the Bordin Prize of the French Academy of Sciences. That same year, he was appointed Professor of Astronomy and Rational Mechanics at the University of Bordeaux. Two years later, owing to his continued research in geometry and symbolic dynamics—especially concerning geodesics on negatively curved surfaces—he was awarded another prestigious honor by the French Academy of Sciences: the Poncelet Prize. 02 A Life of Honors Having achieved fame and distinction, Hadamard returned to Paris in 1897, taking positions successively at the University of Paris (Sorbonne) and the Collège de France. In 1909, he was appointed Professor of Mechanics at the latter institution.

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