Charles Hermite

Charles Hermite was a French mathematician renowned for deep contributions to algebra, number theory, and analysis during the 19th century. He established foundational results on algebraic forms, elliptic functions, and the theory of transcendental numbers, influencing contemporaries and successors such as Karl Weierstrass, Henri Poincaré, and Leopold Kronecker. Hermite's work on the transcendence of the constant e marked a milestone linking Joseph Liouville's earlier constructions to modern transcendence theory, and his name endures in many objects across mathematics, including Hermite polynomials and Hermitian matrices.

Biography

Born in Dieuze in 1822, Hermite was orphaned young and raised by his mother before moving to Paris for education at the Lycée Louis-le-Grand and the École Polytechnique. He studied under influential figures such as Augustin-Louis Cauchy and encountered the functional analysis trends shaped by Niels Henrik Abel and Carl Gustav Jacob Jacobi. After early struggles in finding academic positions, Hermite held teaching posts at the École Polytechnique and later the Collège de France, where he succeeded Joseph Liouville as a leading voice in the Académie des Sciences. His mentorship of students included Henri Poincaré, Jules Tannery, and Charles Émile Picard, and he maintained correspondence with European mathematicians like Bernhard Riemann, Sophus Lie, and Camille Jordan. Hermite received national honors such as the Legion of Honour and remained active in mathematical life until his death in Paris in 1901.

Mathematical Work

Hermite's research spanned elliptic functions, theta functions, algebraic equations, and the early development of linear algebra via forms that later were framed as Hermitian matrix theory. He produced influential analyses of binary forms and reduction theory related to the work of Carl Friedrich Gauss and Arthur Cayley, and he advanced methods in interpolation—now called Hermite interpolation—paralleling techniques by Isaac Newton and Joseph-Louis Lagrange. In complex analysis he engaged with the theories of Karl Weierstrass and Bernhard Riemann, clarifying convergence and representation of entire functions. His studies on quadratic and cubic forms connected to the arithmetic of Pafnuty Chebyshev's predecessors and anticipated later advances by David Hilbert and Emil Artin.

Major Theorems and Concepts

Hermite proved the transcendence of the number e in 1873, building on ideas from Joseph Liouville and influencing later proofs such as those by Felix Lindemann (who proved π transcendental). He introduced Hermite polynomials, which play a central role in solutions to the Hermite differential equation and in quantum mechanics via connections to the harmonic oscillator studied by Erwin Schrödinger. Hermite forms and reduction theory led to classification results that informed William Rowan Hamilton's and Arthur Cayley's matrix formulations, later formalized in linear algebra by Camille Jordan and Issai Schur. The Hermite–Hadamard inequality and concepts bearing his name—Hermite interpolation, Hermite normal form, and Hermite's identity—appear across approximation theory, algebraic number theory, and computational algebra. His techniques for expressing roots of algebraic equations using elliptic and modular functions influenced the work of Niels Henrik Abel, Carl Gustav Jacobi, and later Felix Klein on the icosahedron and modular equations.

Publications and Lectures

Hermite published numerous papers in the Journal de Mathématiques Pures et Appliquées and communicated extensively through the Comptes rendus de l'Académie des Sciences. His lecture courses at the Collège de France and addresses to the Société Mathématique de France disseminated ideas on functions, forms, and number theory that shaped curricula adopted by figures like Émile Picard and Henri Poincaré. Though he wrote relatively few long treatises, his collected works and correspondence were influential; he often reviewed and promoted work by Karl Weierstrass, Leopold Kronecker, and Georg Cantor in public forums, thereby affecting the reception of emerging theories such as set theory and complex function theory. Posthumous compilations gathered his research papers, lectures, and letters, preserving exchanges with contemporaries including George Gabriel Stokes and Augustin-Jean Fresnel.

Honors and Influence

Hermite was elected to the Académie des Sciences and received national recognition like the Legion of Honour. His role as examiner and mentor shaped the careers of leading mathematicians including Henri Poincaré, Paul Émile Appell, and Jules Tannery, and his methods influenced later formalizations by David Hilbert, Emil Artin, and Élie Cartan. Mathematical objects bearing his name—Hermite polynomials, Hermite interpolation, Hermite normal form, Hermitian matrices—remain central in modern studies across quantum mechanics, signal processing, cryptography, and computational number theory. Internationally, his correspondence linked the French school with mathematicians across Germany, England, Italy, and Russia, fostering development that led toward 20th-century advances by André Weil, Hermann Minkowski, and Emmy Noether.

Category:19th-century mathematicians Category:French mathematicians