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| B. Eynard | |
|---|---|
| Name | B. Eynard |
| Birth date | c. 1940s |
| Birth place | Geneva, Switzerland |
| Fields | Mathematics, Mathematical Physics |
| Institutions | École Polytechnique Fédérale de Lausanne, Université de Genève, Institut des Hautes Études Scientifiques |
| Alma mater | Université de Genève, Université Paris-Sud |
| Doctoral advisor | Jean Ginibre |
| Known for | Random matrix theory, large N expansion, spectral theory |
B. Eynard is a Swiss mathematical physicist and mathematician noted for foundational work in Random matrix theory, Enumerative geometry, and applications to Quantum field theory. His research bridged techniques from Probability theory, Algebraic geometry, and Statistical mechanics to develop systematic expansions for matrix integrals and to relate spectral invariants to intersection theory on moduli spaces. Collaborations and lectures across institutions such as Institut des Hautes Études Scientifiques, École Normale Supérieure, and Courant Institute contributed to the dissemination of his methods in both mathematics and physics communities.
Eynard was born in Geneva and received his early schooling in the canton of Geneva. He attended the Université de Genève for undergraduate studies and moved to Université Paris-Sud for doctoral work under the supervision of Jean Ginibre. During this period he engaged with researchers at Centre National de la Recherche Scientifique and attended seminars influenced by figures such as François David, Miguel Ángel Virasoro, and Alain Connes. Postdoctoral visits included time at Institut des Hautes Études Scientifiques and collaborative stays at Princeton University with ties to groups around Peter Sarnak and Edward Witten.
Eynard's early career combined techniques from Random matrix theory, Orthogonal polynomials, and Integrable systems to analyze large N limits of matrix ensembles. He held faculty appointments at Université de Genève and later at École Polytechnique Fédérale de Lausanne, with visiting positions at Harvard University, Institut des Hautes Études Scientifiques, and IHÉS. His research agenda intersected with work by Craig Tracy, Harold Widom, Percy Deift, and Boris Dubrovin, and he contributed to conferences such as the International Congress of Mathematicians satellite meetings. Eynard organized workshops at institutions including CERN and Mathematical Sciences Research Institute to foster exchange between mathematicians and physicists.
Eynard formulated and developed the "topological recursion" technique in joint work with Nicolas Orantin, connecting spectral curves from Matrix models to enumerative invariants of moduli spaces such as those studied by Maxim Kontsevich, Edward Witten, and Maryam Mirzakhani. This recursion related to earlier ideas in Loop equations and to the Seiberg–Witten theory perspectives used by Nathan Seiberg and Edward Witten. He established correspondences between asymptotic expansions of matrix integrals and intersection numbers on Moduli space of curves, linking to the Kontsevich model and results of Kontsevich and Okounkov. Eynard's work on correlation functions and spectral densities built upon and extended methods of Tracy–Widom distribution, Saddle point method developments by J. M. Luck, and universality studies by Terence Tao and Van Vu.
He introduced explicit diagrammatic and algebraic tools to compute higher-genus contributions in 1/N expansions, influencing research on Topological string theory, Gromov–Witten invariants, and relations to Hurwitz numbers as studied by Dijkgraaf and E. Zaslow. Collaborations with researchers like B. Kostov, C. Kristjansen, and P. Zinn-Justin extended applications to statistical models such as the Ising model on random lattices and to conformal field theory approaches developed by Alexander Zamolodchikov.
Eynard authored and coauthored monographs and articles that became central references in mathematical physics. Key titles include his monograph on matrix models and topological expansion, joint papers with Nicolas Orantin introducing topological recursion, and influential reviews synthesizing connections between matrix integrals and moduli space intersection theory. His papers appeared in journals such as Communications in Mathematical Physics, Journal of High Energy Physics, and Annales de l'Institut Henri Poincaré. He also contributed chapters to edited volumes from conferences at Banff International Research Station and lectures recorded for schools at Les Houches.
Selected works: - Monograph on matrix models and topological expansion (with extensive examples connecting to the Kontsevich model and Hurwitz numbers). - Series of articles establishing topological recursion for spectral curves and proving relations with intersection numbers on Moduli space of stable curves. - Reviews on universality in random matrices and applications to Statistical mechanics models on random surfaces.
Eynard received recognition from European and international bodies for contributions to mathematical physics, including awards and invited plenary or invited talks at venues such as the International Congress on Mathematical Physics, the European Mathematical Society meetings, and workshops at IHÉS and CERN. He was elected to academies and served on editorial boards for journals like Communications in Mathematical Physics and Journal of Statistical Physics. Fellowships and grants supported collaborative projects with groups at Princeton University, IHÉS, and CNRS laboratories.
Eynard's introduction and promotion of topological recursion catalyzed a wide web of research tying Random matrix theory to Algebraic geometry, Enumerative geometry, and Quantum field theory. Subsequent work by researchers including Bertrand Eynard-inspired groups such as teams around Nicolas Orantin, Olivier Marchal, Bruno Kramer, Motohico Mulase, and Piotr Sułkowski expanded applications to knot invariants, Topological strings and quantum curves. His methods underpin contemporary studies into universality classes for eigenvalue statistics by groups led by Terence Tao, Paul Bourgade, and Manjunath Krishnapur, and influence computational approaches in combinatorics championed by Andrei Okounkov and Richard Kenyon. The topological recursion remains a standard tool taught in advanced courses at institutions like École Normale Supérieure and Princeton University, ensuring continuity of his impact across mathematics and theoretical physics.
Category:Mathematical physicists Category:Random matrix theory