Manin gained a doctorate in 1960 at the Steklov Mathematics Institute as a student of Igor Shafarevich. He is now a Professor at the Max-Planck-Institut für Mathematik in Bonn, and a professor emeritus at Northwestern University.^[2][3]

Manin's early work included papers on the arithmetic and formal groups of abelian varieties, the Mordell conjecture in the function field case, and algebraic differential equations. The Gauss–Manin connection is a basic ingredient of the study of cohomology in families of algebraic varieties. He wrote a book on cubic surfaces and cubic forms, showing how to apply both classical and contemporary methods of algebraic geometry, as well as nonassociative algebra. He also indicated the role of the Brauer group, via Grothendieck's theory of global Azumaya algebras, in accounting for obstructions to the Hasse principle, setting off a generation of further work. He pioneered the field of arithmetic topology (along with John Tate, David Mumford, Michael Artin and Barry Mazur). He also formulated the Manin conjecture, which predicts the asymptotic behaviour of the number of rational points of bounded height on algebraic varieties. He has further written on Yang–Mills theory, quantum information, and mirror symmetry.

Manin had over 40 doctoral students, including Vladimir Berkovich, Mariusz Wodzicki, Alexander Beilinson, Ivan Cherednik, Alexei Skorobogatov, Vladimir Drinfeld, Mikhail Kapranov, Vyacheslav Shokurov, Arend Bayer and Victor Kolyvagin, as well as foreign students including Hà Huy Khoái.

He was awarded the Brouwer Medal in 1987, the first Nemmers Prize in Mathematics in 1994, the Schock Prize of the Royal Swedish Academy of Sciences in 1999, the Cantor Medal of the German Mathematical Society in 2002, the King Faisal International Prize in 2002 and the Bolyai Prize of the Hungarian Academy of Sciences in 2010.

In 1990 he became a foreign member of the Royal Netherlands Academy of Arts and Sciences.^[4]

• Manin: Selected works with commentary, World Scientific 1996
• Manin: Mathematics as metaphor - selected essays, American Mathematical Society 2009
• Manin: Rational points of algebraic curves over function fields. AMS translations 1966 (Mordell conjecture for function fields)
• Manin: Algebraic topology of algebraic varieties. Russian Mathematical Surveys 1965
• Manin: Modular forms and Number Theory. International Congress of Mathematicians, Helsinki 1978
• Manin: Frobenius manifolds, quantum cohomology, and moduli spaces, American Mathematical Society 1999^[5]
• Manin: Quantum groups and non commutative geometry, Montreal, Centre de Recherches Mathématiques, 1988
• Manin: Topics in non-commutative geometry, Princeton University Press 1991^[6]
• Manin: Gauge field theory and complex geometry. Springer 1988 (Grundlehren der mathematischen Wissenschaften)^[7]
• Manin: Cubic forms - algebra, geometry, arithmetics, North Holland 1986
• Manin: A course in mathematical logic, Springer 1977,^[8] second expanded edition with new chapters by the author and Boris Zilber, Springer 2010.
• Manin: The provable and the unprovable (Russ.), Moscow 1979
• Manin: Computable and Uncomputable (Russ.), Moscow 1980 arXiv:quant-ph/0005003
• Manin: Mathematics and physics, Birkhäuser 1981
• Manin: New dimensions in geometry. in Arbeitstagung Bonn 1984, Lectures Notes in Mathematics Vol. 1111, Springer Verlag
• Manin, Alexei Ivanovich Kostrikin: Linear algebra and geometry, Gordon and Breach 1989
• Manin, Sergei Gelfand: Homological algebra, Springer 1994 (Encyclopedia of mathematical sciences).
• Manin, Sergei Gelfand: Methods of Homological algebra, Springer 1996
• Manin, Igor Kobzarev: Elementary Particles: mathematics, physics and philosophy, Dordrecht, Kluwer, 1989 (This book is introductory.)
• Manin, Alexei A. Panchishkin: Introduction to Number theory, Springer Verlag 1995, 2nd edn. 2005
• Manin Moduli, Motives, Mirrors, 3. European Congress Math. Barcelona 2000, Plenary talk
• Manin Classical computing, quantum computing and Shor´s factoring algorithm, Bourbaki Seminar 1999
• Manin Von Zahlen und Figuren 2002
• Manin, Matilde Marcolli Holography principle and arithmetic of algebraic curves, 2002
• Manin 3-dimensional hyperbolic geometry as infinite-adic Arakelov geometry, Inventiones Mathematicae 1991^{[permanent dead link]}
• Manin: Mathematik, Kunst und Zivilisation, e-enterprise, 2014

• ADHM construction
• Arithmetic topology
• Cartier-Manin operator
• CH-quasigroup
• Dieudonné–Manin classification theorem
• Gauss–Manin connection
• Modular symbol
• Manin–Drinfeld theorem
• Manin matrices
• Manin–Mumford conjecture
• Manin obstruction
• Manin triple
• Manin conjecture

• Némethi, A. (April 2011). "Yuri Ivanovich Manin" (PDF). Acta Mathematica Hungarica. 133 (1–2): 1–13. doi:10.1007/s10474-011-0151-x.
• Jean-Paul Pier (1 January 2000). Development of Mathematics 1950–2000. Springer Science & Business Media. p. 1116. ISBN 978-3-7643-6280-5.

• Yuri Manin at the Mathematics Genealogy Project
• Manin's page at Max-Planck-Institut für Mathematik website
• Good Proofs are Proofs that Make us Wiser, interview by Martin Aigner and Vasco A. Schmidt
• Biography