Victor Gershevich (Grigorievich) Kac (russian: link=no, Виктор Гершевич (Григорьевич) Кац; born 19 December 1943) is a Soviet and American mathematician at

MIT The Massachusetts Institute of Technology (MIT) is a private land-grant research university in Cambridge, Massachusetts. Established in 1861, MIT has played a key role in the development of modern technology and science, and is one of the m ...

, known for his work in

representation theory Representation theory is a branch of mathematics that studies abstract algebraic structures by ''representing'' their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essen ...

. He co-discovered

Kac–Moody algebra In mathematics, a Kac–Moody algebra (named for Victor Kac and Robert Moody, who independently and simultaneously discovered them in 1968) is a Lie algebra, usually infinite-dimensional, that can be defined by generators and relations through a ge ...

s, and used the

Weyl–Kac character formula In mathematics, the Weyl character formula in representation theory describes the characters of irreducible representations of compact Lie groups in terms of their highest weights. It was proved by . There is a closely related formula for the char ...

for them to reprove the

Macdonald identities In mathematics, the Macdonald identities are some infinite product identities associated to affine root systems, introduced by . They include as special cases the Jacobi triple product identity, Watson's quintuple product identity, several identit ...

. He classified the finite-dimensional simple

Lie superalgebra In mathematics, a Lie superalgebra is a generalisation of a Lie algebra to include a Z2 grading. Lie superalgebras are important in theoretical physics where they are used to describe the mathematics of supersymmetry. In most of these theories, the ...

s, and found the

Kac determinant formula In mathematics, the Virasoro algebra (named after the physicist Miguel Ángel Virasoro) is a complex Lie algebra and the unique central extension of the Witt algebra. It is widely used in two-dimensional conformal field theory and in string the ...

for the

Virasoro algebra In mathematics, the Virasoro algebra (named after the physicist Miguel Ángel Virasoro) is a complex Lie algebra and the unique central extension of the Witt algebra. It is widely used in two-dimensional conformal field theory and in string the ...

. He is also known for the Kac–Weisfeiler conjectures with

Boris Weisfeiler Boris Weisfeiler (born 19 April 1941 – disappeared 4-5 January 1985) was a Soviet-born mathematician and professor at Penn State University who lived in the United States before disappearing in Chile in 1985. Declassified US documents sugges ...

Biography

Kac studied mathematics at

Moscow State University M. V. Lomonosov Moscow State University (MSU; russian: Московский государственный университет имени М. В. Ломоносова) is a public research university in Moscow, Russia and the most prestigious ...

, receiving his MS in 1965 and his PhD in 1968. From 1968 to 1976, he held a teaching position at the

Moscow Institute of Electronic Machine Building Moscow Institute of Electronics and Mathematics, MIEM (russian: Московский институт электроники и математики НИУ ВШЭ, МИЭМ; also occasionally referred to as ''Moscow Institute of Electronic Enginee ...

(MIEM). He left the

Soviet Union The Soviet Union,. officially the Union of Soviet Socialist Republics. (USSR),. was a transcontinental country that spanned much of Eurasia from 1922 to 1991. A flagship communist state, it was nominally a federal union of fifteen national ...

in 1977, becoming an associate professor of mathematics at MIT. In 1981, he was promoted to full professor. Kac received a

Sloan Fellowship The Sloan Research Fellowships are awarded annually by the Alfred P. Sloan Foundation since 1955 to "provide support and recognition to early-career scientists and scholars". This program is one of the oldest of its kind in the United States. ...

and the Medal of the

Collège de France The Collège de France (), formerly known as the ''Collège Royal'' or as the ''Collège impérial'' founded in 1530 by François I, is a higher education and research establishment (''grand établissement'') in France. It is located in Paris ne ...

, both in 1981, and a

Guggenheim Fellowship Guggenheim Fellowships are grants that have been awarded annually since by the John Simon Guggenheim Memorial Foundation to those "who have demonstrated exceptional capacity for productive scholarship or exceptional creative ability in the ar ...

in 1986. He received the

Wigner Medal The International Colloquium on Group Theoretical Methods in Physics is an academic conference devoted to applications of group theory to physics. It was founded in 1972 by Henri Bacry and Aloysio Janner. It hosts a colloquium every two years. Th ...

(1996) "in recognition of work on affine

Lie algebras In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identi ...

that has had wide influence in

theoretical physics Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain and predict natural phenomena. This is in contrast to experimental physics, which uses experim ...

". In 1978 he was an Invited Speaker (''Highest weight representations of infinite dimensional Lie algebras'') at the

International Congress of Mathematicians The International Congress of Mathematicians (ICM) is the largest conference for the topic of mathematics. It meets once every four years, hosted by the International Mathematical Union (IMU). The Fields Medals, the Nevanlinna Prize (to be rename ...

(ICM) in

Helsinki Helsinki ( or ; ; sv, Helsingfors, ) is the Capital city, capital, primate city, primate, and List of cities and towns in Finland, most populous city of Finland. Located on the shore of the Gulf of Finland, it is the seat of the region of U ...

. Kac was a plenary speaker at the 1988

American Mathematical Society The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, ...

centennial conference. In 2002 he gave a plenary lecture, ''Classification of Supersymmetries'', at the ICM in Beijing. Kac is a Fellow of the

, an Honorary member of the

Moscow Mathematical Society The Moscow Mathematical Society (MMS) is a society of Moscow mathematicians aimed at the development of mathematics in Russia. It was created in 1864, and Victor Vassiliev is the current president. History The first meeting of the society wa ...

, Fellow of the

American Academy of Arts and Sciences The American Academy of Arts and Sciences (abbreviation: AAA&S) is one of the oldest learned societies in the United States. It was founded in 1780 during the American Revolution by John Adams, John Hancock, James Bowdoin, Andrew Oliver, and ...

and a Member of the

National Academy of Sciences The National Academy of Sciences (NAS) is a United States nonprofit, non-governmental organization. NAS is part of the National Academies of Sciences, Engineering, and Medicine, along with the National Academy of Engineering (NAE) and the Nati ...

. The research of Victor Kac primarily concerns

and

mathematical physics Mathematical physics refers to the development of mathematics, mathematical methods for application to problems in physics. The ''Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and t ...

. His work appears in mathematics and physics and in the development of

quantum field theory In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...

string theory In physics, string theory is a theoretical framework in which the point-like particles of particle physics are replaced by one-dimensional objects called strings. String theory describes how these strings propagate through space and interac ...

and the theory of

integrable systems In mathematics, integrability is a property of certain dynamical systems. While there are several distinct formal definitions, informally speaking, an integrable system is a dynamical system with sufficiently many conserved quantities, or first i ...

. Kac has published 13 books and over 200 articles in mathematics and physics journals and is listed as an

ISI highly cited researcher The Institute for Scientific Information (ISI) was an academic publishing service, founded by Eugene Garfield in Philadelphia in 1956. ISI offered scientometric and bibliographic database services. Its specialty was citation indexing and analysis, ...

. Victor Kac was awarded the 2015 AMS

Leroy P. Steele Prize The Leroy P. Steele Prizes are awarded every year by the American Mathematical Society, for distinguished research work and writing in the field of mathematics. Since 1993, there has been a formal division into three categories. The prizes have b ...

for Lifetime Achievement. He was married with Michèle Vergne and they have a daughter, Marianne Kac-Vergne, who is a professor of American civilization at the university of Picardie. His brother Boris Katz is a principal research scientist at MIT.

Kac–Moody algebra

"Almost simultaneously in 1967, Victor Kac in the USSR and

Robert Moody Robert Vaughan Moody, (; born November 28, 1941) is a Canadian mathematician. He is the co-discover of Kac–Moody algebra, a Lie algebra, usually infinite-dimensional, that can be defined through a generalized root system. "Almost simultaneou ...

in Canada developed what was to become

. Kac and Moody noticed that if

Wilhelm Killing Wilhelm Karl Joseph Killing (10 May 1847 – 11 February 1923) was a German mathematician who made important contributions to the theories of Lie algebras, Lie groups, and non-Euclidean geometry. Life Killing studied at the University of Mü ...

's conditions were relaxed, it was still possible to associate to the

Cartan matrix In mathematics, the term Cartan matrix has three meanings. All of these are named after the French mathematician Élie Cartan. Amusingly, the Cartan matrices in the context of Lie algebras were first investigated by Wilhelm Killing, whereas the Ki ...

a Lie algebra which, necessarily, would be infinite dimensional." – A.J. ColemanColeman, A. John, "The Greatest Mathematical Paper of All Time",
The Mathematical Intelligencer
'' vol. 11, no. 3, pp. 29–38.

Bibliography

* * * * * * *

References

External links

* *Victor Kac, A Biographical Interview

{{DEFAULTSORT:Kac, Victor 1943 births Living people People from Buguruslan 20th-century American mathematicians Russian mathematicians Soviet mathematicians Fellows of the American Academy of Arts and Sciences American people of Russian-Jewish descent Fellows of the American Mathematical Society Moscow State University alumni 21st-century American mathematicians