Luc Illusie

Luc Illusie
Name	Luc Illusie
Birth date	1940
Birth place	Paris, France
Nationality	French
Fields	Algebraic geometry, Number theory, Homological algebra
Workplaces	Université Paris-Sud, École Normale Supérieure (Paris), Institut des Hautes Études Scientifiques
Alma mater	École Normale Supérieure (Paris), Université Paris-Sud
Doctoral advisor	Jean-Pierre Serre
Known for	Deformation theory, Cotangent complex, Crystalline cohomology, Logarithmic geometry

Luc Illusie is a French mathematician renowned for foundational work in algebraic geometry, cohomology theory, and number theory. His contributions include development and applications of the cotangent complex, advances in crystalline cohomology, and influential expositions on logarithmic geometry and deformations. He has been associated with leading French institutions and international research bodies and has collaborated with prominent mathematicians across Europe and North America.

Early life and education

Illusie was born in Paris and educated at the École Normale Supérieure (Paris), where he was immersed in a milieu connected to figures such as Jean-Pierre Serre, Alexander Grothendieck, Jean-Louis Verdier, and Henri Cartan. He completed doctoral studies at Université Paris-Sud under the supervision of Jean-Pierre Serre, situating him amid developments following the Séminaire de Géométrie Algébrique du Bois Marie and the work of Grothendieck on schemes and cohomological methods. During his formative years he interacted with researchers from institutions including the Collège de France, Institut des Hautes Études Scientifiques, and international centers such as Harvard University, Massachusetts Institute of Technology, and the University of Cambridge.

Mathematical career and positions

Illusie held research and teaching posts at Université Paris-Sud and was affiliated with the Institut des Hautes Études Scientifiques and the Centre National de la Recherche Scientifique. He participated in and organized seminars linked to the Séminaire de Géométrie Algébrique, the Seminar on the Moduli of Curves, and conferences associated with the International Congress of Mathematicians. His professional network includes collaborations and exchanges with mathematicians from Princeton University, University of Chicago, Université Grenoble Alpes, École Polytechnique, and research groups at École Normale Supérieure (Lyon). Illusie also served on editorial boards for journals connected to Elsevier, Springer, and societies such as the Société Mathématique de France and contributed to proceedings of gatherings at Mathematical Sciences Research Institute and the Institut Mittag-Leffler.

Research contributions and major results

Illusie's research developed central tools in modern algebraic geometry and arithmetic geometry. He formulated and applied the cotangent complex in the context of deformation theory building on ideas of André Weil, Alexander Grothendieck, and Jean-Pierre Serre. His work on the cotangent complex systematized obstructions to deformations and linked with derived categories and techniques employed by Pierre Deligne and Jean-Louis Verdier. Illusie made fundamental contributions to crystalline cohomology, extending methods initiated by Berthelot and connected with Monsky–Washnitzer cohomology and the de Rham–Witt complex. He clarified the relationship between crystalline cohomology and de Rham cohomology, influencing subsequent developments by Kazuya Kato and Christopher D. Hacon.

A major theme in his work is logarithmic geometry: Illusie advanced the formalism of log structures that interacts with the work of Kazuya Kato and Fumiharu Kato, enabling refined study of degenerations, compactifications, and monodromy phenomena tied to Hodge theory and p-adic Hodge theory. He also investigated the cotangent complex in the logarithmic setting and cohomological operations, influencing research by Lucien Szpiro, Gerd Faltings, and Shigefumi Mori. His expository and research monographs synthesized results related to the Hodge–de Rham spectral sequence, comparison theorems in étale cohomology, and the structure of flat cohomology.

Illusie's papers introduced techniques now standard in the study of moduli problems, stacks, and deformation of morphisms, interfacing with the work of Michael Artin, Deligne–Mumford, Pierre Deligne, and Gérard Laumon. He also contributed to the formalism of spectral sequences and exact triangles in derived categories, tools employed by scholars such as Amnon Neeman and Joseph Lipman.

Awards and honors

Illusie received national and international recognition including fellowships and invitations to speak at venues such as the International Congress of Mathematicians and institutes like the Mathematical Sciences Research Institute. He was awarded honors from French institutions including appointments within the Centre National de la Recherche Scientifique and distinctions associated with the Société Mathématique de France and the Académie des Sciences. His selection for dedicated conferences and festschrifts placed him among contemporaries such as Jean-Pierre Serre, Alexander Grothendieck, Pierre Deligne, and Jean-Louis Verdier.

Selected publications

- Illusie, L., "Complexe cotangent et déformations", Lecture Notes in Mathematics, Springer — foundational work connecting cotangent complexes with deformation theory and derived functors; dialogues with the works of Jean-Pierre Serre and Alexander Grothendieck. - Illusie, L., "Grothendieck's existence theorem in formal geometry" — studies linked to Grothendieck and applications in moduli theory; used in research by Michael Artin and Pierre Deligne. - Illusie, L., expository articles on crystalline cohomology and the de Rham–Witt complex, engaging with results of P. Berthelot and Jean-Michel Fontaine. - Illusie, L., papers on logarithmic geometry and the cotangent complex in the log setting, interacting with the research of Kazuya Kato and Fumiharu Kato. - Illusie, L., survey lectures collected in volumes associated with seminars at Institut des Hautes Études Scientifiques and proceedings of the International Congress of Mathematicians.

Category:French mathematicians Category:Algebraic geometers Category:1930s births