Acta Math. Hungar. DOI: 0 YURI IVANOVICH MANIN ∗ A. NEMETHI´ A. R´enyi Institute of Mathematics, 1053 Budapest, Re´altanoda u. 13–15., Hungary e-mail: [email protected] (Received March 28, 2011; accepted April 11, 2011) In December 2010 the J´anos Bolyai International Mathematical Prize of the Hungarian Academy of Sciences was awarded to Professor Yuri Ivanovich Manin. In this note we wish to present Professor Manin to the readers of Acta Mathematica Hungarica. The first part is a general overview of his sci- entific achievements, in the second part we provide some more details about his broad mathematical work. I. Yuri Ivanovich Manin is widely regarded as one of the outstanding mathe- maticians of the 20th century. His work spans such diverse branches of math- ematics as algebraic geometry, number theory and mathematical physics. But mathematics is not his only interest. He has been interested and pub- lished research or expository papers in literature, linguistics, mythology, semiotics, physics, philosophy of science and history of culture as well. He is one of the few mathematicians who determined essentially the Russian science in the second half of the 20th century. Yuri Manin was born in 1937 in the town of Simferopol, in Crimea. His father died in the war, his mother was a teacher of literature. His mathe- matical abilities became apparent already during his school-years, when he slightly improved Vinogradov’s estimate of the number of lattice points in- side a sphere. Between 1953-58 he studied at the faculty of Mechanics and Mathematics of the Moscow State University, the most prestigious mathe- matical school of in USSR. His class and the following class included sev- eral other talented students: Anosov, Golod, Arnol’d, Kirillov, Novikov and Tyurin. Starting from the second year he became an active member of the seminar led by A. O. Gel’fond and I. R. Shafarevich targeting the work of Hasse and Weil on the ζ-function on algebraic curves over a finite field. ∗ The author is partially supported by OTKA Grant K67928. Key words and phrases: Manin, algebraic geometry, number theory, Mordell conjecture, L¨uroth problem, Brauer–Manin obstruction, modular forms, mathematical physics. 2000 Mathematics Subject Classification: primary 14-00, secondary 4Exx, 11Gxx, 81Txx. 0236-5294/$20.00 c 0 Akad´emiai Kiad´o, Budapest 2 A. NEMETHI´ At this time his first publication appeared containing an elementary proof of Hasse’s theorem. After graduation he continued his postgraduate studies at the Steklov Institute of Mathematics under the supervision of I. R. Shafare- vich. During these years an algebraic geometry seminar led by Shafarevich began to operate, with the active participation of Manin. These two semi- nars determined his major mathematical interests and passions: the interface of algebraic geometry and algebraic number theory. He enriched these fields by numerous fundamental contributions, includ- ing the solutions of major problems and the development of techniques that opened new possibilities for research. The top of the iceberg contains two outstanding results in algebraic geometry. The first one was the proof of the analogue of Mordell’s Conjecture (now Faltings’ Theorem) for algebraic curves over function fields. In Faltings’ classical case the statement is the following: a curve of genus greater than 1 has at most a finite number of ra- tional points. Over a function field, in Manin’s version, the curve depends on parameters. In this proof a new mathematical object played the key role. Later this was named by Grothendieck the Gauss–Manin connection, and it is a basic ingredient of the study of cohomology in families of algebraic va- rieties in modern algebraic geometry. The second outstanding achievement was a joint work with his student V. A. Iskovskih: it provided a negative solution of the L¨uroth problem in dimension 3. They proved the existence of nonrational unirational 3-folds via deep understanding of the geometry of three-dimensional quartics. The early research on the set of rational points of bounded height on cu- bic surfaces was continued and generalized via studying the asymptotics of the distribution of rational points by height on Fano varieties. This gener- ated a sequence of deep conjectures and results, for example the ‘conjecture of linear growth’, developed and finished by Manin in a joint work with some of his students (Batyrev, Franke, Tschinkel). Moreover, motivated by Mordell’s conjecture, Manin and Mumford formulated the so-called Manin– Mumford conjecture which states that any curve, which is different from its Jacobian variety, can only contain a finite number of points that are of fi- nite order in the Jacobian. This problem later solved by M. Raynaud, has developed into the general ‘Manin–Mumford theory’. In number theory, or arithmetic algebraic geometry, Manin developed the so-called Manin–Brauer obstruction to the solvability of Diophantine equa- tions. The Manin obstruction associated with a geometric object measures the failure of the Hasse principle for it; that is, if the value of the obstruc- tion is non-trivial, then the object might have points over all local fields but not over the a global field. For torsors of abelian varieties the Manin ob- struction characterizes completely the failure of the local-to-global principle (provided that the Tate–Shafarevich group is finite). He also obtained fundamental results in the theory of modular forms and modular symbols, and the theory of p-adic L-functions, classification of Acta Mathematica Hungarica YURI IVANOVICH MANIN 3 isogeny classes of formal p-divisible groups; he proved the Weil conjecture for unirational projective 3-folds. Manin obtained a series of outstanding results in mathematical physics as well, including Yang–Mills theory, string theory, quantum groups, quan- tum information theory, and mirror symmetry. These papers show the strong symbiosis of mathematics and physics and how they strongly influence each other. For example, an article of Atiyah, Drinfel’d, Hitchin and Manin pro- vided a complete description of instantons by algebro-geometrical methods emphasizing the potential power of algebro-geometrical tools in theoretical physics. Symmetrically, ideas from physics solved crucial open problems in algebraic geometry, see for example the papers of Kontsevich and Manin about quantum cohomology of algebraic varieties. His work with Kontse- vich on Gromov–Witten invariants and work on Frobenius manifolds (later on ‘F-manifolds’ developed with C. Hertling) created new areas of mathe- matics with strong mathematical machinery and several applications. He has also written famous papers on formal groups, noncommutative algebraic geometry and mathematical logic. Professor Manin is the author and coauthor of 11 monographs and about 235 articles in algebraic geometry, number theory, mathematical physics, history of culture and psycholinguistics. In some of his books he created and developed new theories (like in the first one in the next list). Some titles of his books and monographs emphasizing the diversity of subjects: Cubic forms: algebra, geometry, arithmetic published in 1972, A course in mathematical logic (1977), Computable and noncomputable (1980), Linear algebra and geometry with A.I. Kostrikin published in 1980, Gauge fields and complex geometry (1984), Methods in homological algebra and Homological Algebra with Sergei Gelfand (1988-89), Quantum groups and noncommutative geometry (1988), Elementary particles with I. Yu. Kobzarev (1989), Introduction in Number Theory with A.A. Panchishkin (1990), Topics in noncommutative geometry (1991), Frobenius manifolds, quantum cohomology and moduli spaces (1999). The famous book Mathematics and physics or the selected essays Math- ematics as Metaphor provide a deep insight in his philosophy of science. Manin’s pedagogical activity started in 1957 at Moscow State University, and he remained there until the early 90’s. At parallel, he was principal researcher at Steklov Mathematical Institute. In the period 1968–86 he was not allowed to travel abroad, but starting from 88 he was visiting professor at several Universities including Berkeley, Harvard, Columbia, MIT, IHES. He accepted a professorship at Northwestern University in the United States, Acta Mathematica Hungarica 4 A. NEMETHI´ and the position in the Board of Directors of the Max Planck Institute in Bonn in 1993. He became Professor emeritus at the Max Planck Institute in 2005. During these years he was continuously surrounded by a large number of students, the most talented ones wished to be guided under his supervision. He was advisor of 49 students, some of them became celebrated mathemati- cians. Just a few of them: Kapranov, Beilinson, Zarhin, Danilov, Iskovskih, Shokurov, Drinfeld, Wodzicki, Tsygan, Tschinkel. Manin was a very popular professor with a lot of energy with vivid presentations and real pedagogical vein. Professor Manin was six times invited speaker at international con- gresses, he was invited plenary speaker at European Congress of Mathe- matics. He received several international honors. He was awarded several prices: Highest USSR National Prize (the so-called Lenin Prize) in 1967 for work in algebraic geometry, Brouwer Gold Medal in Number Theory from the Dutch Royal Society and Mathematical Society in 1987, Frederic Esser Nemmers Prize in Mathematics from Northwestern Uni- versity in 1994, Rolf Schock Prize in Mathematics of the Swedish Royal Academy of Sci- ences in 1999, King Faisal International Prize for Mathematics from Saudi Arabia in 2002, Georg Cantor Medal of the German Mathematical Society in 2002, Order Pour le M´erite and Great Cross of Merit with Star from Germany in 2007 and 2008; and the J´anos Bolyai International Mathematical Prize of the Hungarian Academy of Sciences in 2010. He is elected member of several scientific academies: Academy of Sci- ences, Russia; Academy of Natural Sciences, Russia; Royal Academy of Sciences, the Netherlands; Academia Europaea, Max-Planck-Society for Scientific Research, Germany; G¨ottingen Academy of Sciences; Pontifical Academy of Sciences, Vatican; German Academy of Sciences; American Academy of Arts and Sciences; Acad´emia des Sciences de l’Institut de France.