Michèle Vergne

Michèle Françoise Vergne (born August 29, 1943 in L'Isle-Adam ) is a French mathematician who deals with analysis and representation theory.

Michele Vergne, ICM Madrid 2006

Live and act

Vergne studied from 1962 to 1964 at the École normal supérieure de jeunes filles , which is now part of the ENS. She wrote her doctoral thesis in 1966 with Claude Chevalley , she was entitled "Variété des algèbres de Lie nilpotentes" and her habilitation in 1971 with Jacques Dixmier ("Recherches sur les groupes et les algèbres de Lie") at the University of Paris. From 1967 she was Attachée de Recherches at the CNRS . In 1971/2 she was a lecturer at the University of California, Berkeley . From 1972 she was Chargée de Recherches of the CNRS, from 1976 Maitre de Recherches of the CNRS and from 1981 Directeur de Recherche. From 1975/6 and 1977 to 1979 she was visiting professor at the Massachusetts Institute of Technology (MIT), from 1979 associate professor and 1981 to 1986 professor at MIT. From 1986 to 2008 she was the CNRS Director of Research.

Vergne was concerned with the construction of unitary representations of Lie groups using coadjoint orbits of Lie algebra . She proved a generalized Poisson sum formula (called the Poisson-Plancherel formula), which relates the integrals of a function on adjoint orbits with integrals of its Fourier transforms on coadjoint "quantized" orbits. In the 1970s and 1980s she organized a series of conferences at Luminy (University of Marseille) on harmonic analysis on Lie groups with Jacques Carmona .

Furthermore, she dealt with the index theory of elliptic differential operators and generalizations of them and with equivariant cohomology, in which the cohomological aspects of a Lie group operating on a manifold are related to the cohomology of the manifold. In 1985 she and Nicole Berline connected Atiyah-Bott fixed point formulas with the character formula of the Lie group by Kirillov . The theory also has applications in physics (e.g. Edward Witten ).

In addition, she also dealt with the geometry of numbers, subject was the number of integer points in convex polyhedra.

With Masaki Kashiwara , she formulated a conjecture about the combinatorial structure of the envelope algebras of Lie algebras (fully proven in 2006 by Anton Alekseev and Eckhard Meinrenken ).

Since 1997 she has been a member of the Académie des Sciences , whose Prix Bordin she received in 1980 and whose Prix ​​Ampère she received in 1997. She is a member of the American Academy of Arts and Sciences . In 1992 he gave a plenary lecture at the first European Congress of Mathematicians in Paris (Cohomologie equivariante et formules de caracteres) . In 2006 she gave a plenary lecture at the International Congress of Mathematicians in Madrid (Applications of Equivariant Cohomology) and in 1983 she was invited speaker at the ICM in Warsaw (Formule de Kirilov et indice de l'opérateur de Dirac) . In 2008 she was an Emmy Noether visiting professor at the University of Göttingen. She is a Fellow of the American Mathematical Society .

Fonts

• with G. Lion: The Weil representation, Maslov Index and Theta Series , Birkhäuser 1980
• with Nicole Berline , Ezra Getzler : Heat kernels and Dirac operators , Springer, Grundlehren der Mathematischen Wissenschaften, 1992, 2004
• Quantification geometrique et reduction symplectique , Bourbaki seminar 2000/1
• Représentations unitaires des groupes de Lie résolubles. , Bourbaki Seminar, 1973/4
• with Michel Duflo , Jacques Dixmier : Sur la représentation coadjointe d'une algèbre de Lie , Compositio Mathematica 1974
• Applications of Equivariant Cohomology , ICM 2006

Web links

• Forgive me on the pages of the French Academy
• Homepage with CV

Individual evidence

1. ↑ today, the Val-d'Oise department
2. ^ Nicole Berline, Michele Vergne: The Equivariant Index and Kirillov's Character Formula. In: American Journal of Mathematics . Vol. 107, No. 5, 1985, pp. 1159-1190, JSTOR 2374350 .