Yuri I. Manin
Encyclopedia
Yuri Ivanovitch Manin is a Soviet/Russia
n/German
mathematician, known for work in algebraic geometry
and diophantine geometry
, and many expository works ranging from mathematical logic
to theoretical physics
.
. He is now a Professor at the Max-Planck-Institut für Mathematik
in Bonn
, and a professor at Northwestern University
.
Manin's early work included papers on the arithmetic and formal group
s of abelian varieties, the Mordell conjecture in the function field
case, and algebraic differential equation
s. The Gauss–Manin connection is a basic ingredient of the study of cohomology
in families of algebraic varieties. He wrote an influential book on cubic surface
s and cubic form
s, 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 algebra
s, in accounting for obstructions to the Hasse principle
, setting off a generation of further work.
He has also written on Yang-Mills theory, quantum information
, and mirror symmetry
.
Manin had over 40 doctoral students, including Mariusz Wodzicki, Alexander Beilinson
, Ivan Cherednik
, Alexei Skorobogatov
, Vladimir Drinfeld
, Vyacheslav Shokurov
, Arend Bayer and Victor Kolyvagin
. He
was awarded the Schock Prize
in 1999 and the Cantor Medal
in 2002. In 1994, he was awarded the Nemmers Prize in Mathematics
. In 2010, he received the Bolyai Prize
of the Hungarian Academy of Sciences
.
Russia
Russia or , officially known as both Russia and the Russian Federation , is a country in northern Eurasia. It is a federal semi-presidential republic, comprising 83 federal subjects...
n/German
Germany
Germany , officially the Federal Republic of Germany , is a federal parliamentary republic in Europe. The country consists of 16 states while the capital and largest city is Berlin. Germany covers an area of 357,021 km2 and has a largely temperate seasonal climate...
mathematician, known for work in algebraic geometry
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
and diophantine geometry
Diophantine geometry
In mathematics, diophantine geometry is one approach to the theory of Diophantine equations, formulating questions about such equations in terms of algebraic geometry over a ground field K that is not algebraically closed, such as the field of rational numbers or a finite field, or more general...
, and many expository works ranging from mathematical logic
Mathematical logic
Mathematical logic is a subfield of mathematics with close connections to foundations of mathematics, theoretical computer science and philosophical logic. The field includes both the mathematical study of logic and the applications of formal logic to other areas of mathematics...
to theoretical physics
Theoretical physics
Theoretical physics is a branch of physics which employs mathematical models and abstractions of physics to rationalize, explain and predict natural phenomena...
.
Biography
Manin gained a doctorate in 1960 at the Steklov Mathematics Institute as a student of Igor ShafarevichIgor Shafarevich
Igor Rostislavovich Shafarevich is a Soviet and Russian mathematician, founder of a school of algebraic number theory and algebraic geometry in the USSR, and a political writer. He was also an important dissident figure under the Soviet regime, a public supporter of Andrei Sakharov's Human Rights...
. He is now a Professor at the Max-Planck-Institut für Mathematik
Max-Planck-Institut für Mathematik
The Max-Planck-Institut für Mathematik is a mathematical research institute located in Bonn, Germany. It is named in honour of the German physicist Max Planck...
in Bonn
Bonn
Bonn is the 19th largest city in Germany. Located in the Cologne/Bonn Region, about 25 kilometres south of Cologne on the river Rhine in the State of North Rhine-Westphalia, it was the capital of West Germany from 1949 to 1990 and the official seat of government of united Germany from 1990 to 1999....
, and a professor at Northwestern University
Northwestern University
Northwestern University is a private research university in Evanston and Chicago, Illinois, USA. Northwestern has eleven undergraduate, graduate, and professional schools offering 124 undergraduate degrees and 145 graduate and professional degrees....
.
Manin's early work included papers on the arithmetic and formal group
Formal group
In mathematics, a formal group law is a formal power series behaving as if it were the product of a Lie group. They were introduced by . The term formal group sometimes means the same as formal group law, and sometimes means one of several generalizations. Formal groups are intermediate between...
s of abelian varieties, the Mordell conjecture in the function field
Function field
Function field may refer to:*Function field of an algebraic variety*Function field...
case, and algebraic differential equation
Algebraic differential equation
In mathematics, an algebraic differential equation is a differential equation that can be expressed by means of differential algebra. There are several such notions, according to the concept of differential algebra used....
s. The Gauss–Manin connection is a basic ingredient of the study of cohomology
Cohomology
In mathematics, specifically in algebraic topology, cohomology is a general term for a sequence of abelian groups defined from a co-chain complex. That is, cohomology is defined as the abstract study of cochains, cocycles, and coboundaries...
in families of algebraic varieties. He wrote an influential book on cubic surface
Cubic surface
A cubic surface is a projective variety studied in algebraic geometry. It is an algebraic surface in three-dimensional projective space defined by a single polynomial which is homogeneous of degree 3...
s and cubic form
Cubic form
In mathematics, a cubic form is a homogeneous polynomial of degree 3, and a cubic hypersurface is the zero set of a cubic form.In , Boris Delone and Dmitriĭ Faddeev showed that binary cubic forms with integer coefficients can be used to parametrize orders in cubic fields...
s, 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
Brauer group
In mathematics, the Brauer group of a field K is an abelian group whose elements are Morita equivalence classes of central simple algebras of finite rank over K and addition is induced by the tensor product of algebras. It arose out of attempts to classify division algebras over a field and is...
, via Grothendieck's theory of global Azumaya algebra
Azumaya algebra
In mathematics, an Azumaya algebra is a generalization of central simple algebras to R-algebras where R need not be a field. Such a notion was introduced in a 1951 paper of Goro Azumaya, for the case where R is a commutative local ring...
s, in accounting for obstructions to the Hasse principle
Hasse principle
In mathematics, Helmut Hasse's local-global principle, also known as the Hasse principle, is the idea that one can find an integer solution to an equation by using the Chinese remainder theorem to piece together solutions modulo powers of each different prime number...
, setting off a generation of further work.
He has also written on Yang-Mills theory, quantum information
Quantum information
In quantum mechanics, quantum information is physical information that is held in the "state" of a quantum system. The most popular unit of quantum information is the qubit, a two-level quantum system...
, and mirror symmetry
Mirror symmetry
In physics and mathematics, mirror symmetry is a relation that can exist between two Calabi-Yau manifolds. It happens, usually for two such six-dimensional manifolds, that the shapes may look very different geometrically, but nevertheless they are equivalent if they are employed as hidden...
.
Manin had over 40 doctoral students, including Mariusz Wodzicki, Alexander Beilinson
Alexander Beilinson
Alexander A. Beilinson is the David and Mary Winton Green University Professor at the University of Chicago and works on mathematics. His research has spanned representation theory, algebraic geometry and mathematical physics...
, Ivan Cherednik
Ivan Cherednik
Ivan Cherednik is a Russian mathematician. He introduced double affine Hecke algebras, and used them to prove Macdonald's constant term conjecture in . He has also dealt with algebraic geometry, number theory and Soliton equations. His research interests include representation theory,...
, Alexei Skorobogatov
Alexei Skorobogatov
Alexei Nikolaievich Skorobogatov is a mathematician and currently a Professor in Pure Mathematics at Imperial College London. He specialises in algebraic geometry....
, Vladimir Drinfeld
Vladimir Drinfel'd
Vladimir Gershonovich Drinfel'd is a Ukrainian and Soviet mathematician at the University of Chicago.The work of Drinfeld related algebraic geometry over finite fields with number theory, especially the theory of automorphic forms, through the notions of elliptic module and the theory of the...
, Vyacheslav Shokurov
Vyacheslav Shokurov
Vyacheslav Vladimirovich Shokurov, born on 18 May 1950, is aRussian mathematician best known for his research inalgebraic geometry. The proof of the Noether–Enriques–Petri theorem,the Cone theorem, the existence of a line on smooth Fano varieties...
, Arend Bayer and Victor Kolyvagin
Victor Kolyvagin
Victor Alexandrovich Kolyvagin is a Russian mathematician who wrote a series of papers on Euler systems, leading to breakthroughs on the Birch and Swinnerton-Dyer conjecture, and Iwasawa's conjecture for cyclotomic fields. His work also influenced Andrew Wiles's work on Fermat's Theorem.Kolyvagin...
. He
was awarded the Schock Prize
Schock prize
The Rolf Schock Prizes were established and endowed by bequest of philosopher and artist Rolf Schock . The prizes were first awarded in Stockholm, Sweden, in 1993 and have been awarded every two years since...
in 1999 and the Cantor Medal
Cantor medal
The Cantor medal of the Deutsche Mathematiker-Vereinigung is named in honor of Georg Cantor. It is awarded at most every second year during the yearly meetings of the society...
in 2002. In 1994, he was awarded the Nemmers Prize in Mathematics
Nemmers Prize in Mathematics
The Frederic Esser Nemmers Prize in Mathematics is awarded biennially from Northwestern University. It was initially endowed along with a companion prize, the Erwin Plein Nemmers Prize in Economics, as part of a $14 million donation from the Nemmers brothers. They envisioned creating an award that...
. In 2010, he received the Bolyai Prize
Bolyai Prize
The International Bolyai János Prize of Mathematics is an international prize for mathematicians founded by the Hungarian Academy of Sciences. The prize is awarded in every five years to mathematicians having published their monograph describing their own highly important new results in the past 10...
of the Hungarian Academy of Sciences
Hungarian Academy of Sciences
The Hungarian Academy of Sciences is the most important and prestigious learned society of Hungary. Its seat is at the bank of the Danube in Budapest.-History:...
.
External links
- Manin's page at Max-Planck-Institut für Mathematik website
- Good Proofs are Proofs that Make us Wiser, interview by Martin AignerMartin AignerMartin Aigner is an Austrian mathematician, professor at Freie Universität Berlin since 1974, with interests in combinatorial mathematics and graph theory....
and Vasco A. Schmidt - Biography