Bernhard Keller

Bernhard Keller
Schmid, Renate · CC BY-SA 2.0 de · source
Name	Bernhard Keller
Birth date	1966
Birth place	Lyon, France
Nationality	French
Fields	Mathematics, Algebra, Representation Theory, Homological Algebra, Category Theory
Workplaces	Université Paris-Sud, Institut des Hautes Études Scientifiques, CNRS, Universität Zürich
Alma mater	École Normale Supérieure de Lyon, Université Paris-Sud
Doctoral advisor	Michel Broué
Known for	Cluster categories, Calabi–Yau categories, derived categories, dg-categories
Awards	Ferran Sunyer i Balaguer Prize, Grand Prix de l'Académie des Sciences

Bernhard Keller is a French mathematician noted for influential work in algebra, representation theory, and homological algebra. His research has shaped modern approaches to derived categorys, dg-categorys, and cluster algebras, with applications across geometry, mathematical physics, and combinatorics. He has held positions at major European institutions and lectured at international conferences and schools.

Early life and education

Keller was born in Lyon and studied at the École Normale Supérieure de Lyon before completing a doctorate at Université Paris-Sud under the supervision of Michel Broué. During his formative years he was influenced by developments in representation theory of finite groups, the work of Pierre Deligne on homological methods, and advances in triangulated category theory originating with Jean-Louis Verdier and Alexander Grothendieck. Early collaborations and visits connected him with researchers at CNRS laboratories and the Institut des Hautes Études Scientifiques.

Academic career

Keller has held research and teaching positions at institutions including Université Paris-Sud, Université de Strasbourg, and Universität Zürich, and he has been affiliated with CNRS and the Institut des Hautes Études Scientifiques. He has served as an invited speaker at the International Congress of Mathematicians, delivered lecture series at the Mathematical Research Institute of Oberwolfach, and organized workshops at the Centre International de Rencontres Mathématiques. His editorial roles include boards of journals that publish in homological algebra and representation theory.

Research contributions and work

Keller's contributions span foundational and structural results in derived categories, dg-categorys, and Calabi–Yau category theory. He introduced and developed the notion of cluster categorys to provide a categorical framework for cluster algebras introduced by Sergey Fomin and Andrei Zelevinsky, linking them to tilting theory and Auslander–Reiten theory. His work on dg-enhancements of triangulated categorys clarified problems about uniqueness of enhancements and led to applications in mirror symmetry and string theory contexts where Calabi–Yau structures are central. Keller established equivalences between derived categories of quiver representations and constructed categorical models for mutation phenomena in cluster algebras, building bridges to Keller–Reiten type results and interactions with Kontsevich's homological mirror symmetry conjecture. He advanced the theory of Hochschild cohomology for dg-algebras and investigated periodicity phenomena related to Calabi–Yau properties and Ginzburg algebra constructions. His survey articles and lecture notes have become standard references for researchers studying perverse sheaf analogues in representation-theoretic settings and for students entering the field through connections to K-theory and algebraic geometry.

Awards and honors

Keller has received recognition including the Ferran Sunyer i Balaguer Prize and the Grand Prix de l'Académie des Sciences. He has been invited to deliver plenary and sectional talks at meetings organized by the European Mathematical Society and has been elected to scientific committees of institutes such as IHÉS and the Max Planck Institute for Mathematics. His work has been cited in award citations for collaborators who received prizes like the Shaw Prize and the Clay Research Award for advances in cluster algebra theory.

Selected publications

- "On differential graded categories" — influential survey and technical development on dg-categorys and enhancements of derived categorys, appearing in lecture notes and proceedings of IAS schools. - "Cluster algebras and derived categories" — papers connecting cluster algebras of Fomin–Zelevinsky with categorical constructions such as cluster categorys and tilting theory. - "Calabi–Yau triangulated categories" — work constructing Calabi–Yau categories from Ginzburg algebras associated to quivers with potential and exploring links to mirror symmetry. - Lecture notes and surveys on Hochschild cohomology, Auslander–Reiten theory, and tilting theory published in proceedings of the European Congress of Mathematics and the ICM satellite events.

Category:French mathematicians Category:Algebraists Category:Representation theorists