nLab Pierre Deligne

Pierre Deligne is a Belgian mathematician who was a student of Alexander Grothendieck, then worked at l’IHÉS and now is emeritus at IAS.

Deligne’s main interests include algebraic geometry especially cohomology of algebraic varieties, Hodge theory, L-functions and automorphic forms, Tannakian theory, representation theory of algebraic groups, and motives, where he extended the conjectural picture from pure to mixed motives (Grothendieck was much earlier aware and pushing toward that extension, though not publishing about it, according to Serre and others).

Deligne has obtained the Fields medal in 1978 for a famous 1973 proof of Weil conjectures.

For the sake of preparatory/foundational steps he wrote a quick amendement for the unfinished volumes of SGA in a form of practical and short (but controversial to Grothendieck) SGA 4124\frac{1}{2}. This work uses a powerful and deep yoga of Hodge filtrations discovered also by Deligne.

Selected writings

Introducing the notion of algebraic stacks and (what came to be called) Deligne-Mumford stacks, and on the example of the moduli space of curves:

On differential equations with regular singular points (and developing local systems, twisted cohomology, twisted de Rham cohomology, Gauss-Manin connections):

Deligne led a seminar on differential systems corresponding to meromorphic connections, whose basic results were explained in a classic in this are:

  • Équations différentielles à points singuliers réguliers, Lect. Notes in Math. 163, Springer-Verlag (1970)

surveyed in:

Introducing Deligne cohomology in complex analytic geometry (by a chain complex of holomorphic differential forms) with applications to Hodge theory and intermediate Jacobians:

  • Pierre Deligne, Théorie de Hodge II , IHES Pub. Math. (1971), no. 40, 5–57 (pdf)

On elliptic curves over general commutative ground rings (in arithmetic geometry):

On the Weil conjectures:

  • La conjecture de Weil : I, Publications Mathématiques de l’IHÉS 43 (1974), p. 273-307 numdam

On rational homotopy theory of Kähler manifolds:

Introducing the notion of perverse sheaves (and of t-structures on triangulated categories):

On Tannakian categories:

On tensor categories and Tannaka duality:

culminating in Deligne's theorem on tensor categories:

  • Pierre Deligne, Catégorie Tensorielle, Moscow Math. Journal 2 (2002) no. 2, 227-248. (pdf)

On mathematical foundations of superalgebra, supergeometry and supersymmetry:

and focus on signs in supergeometry:

category: people

Last revised on March 7, 2024 at 10:54:13. See the history of this page for a list of all contributions to it.