Victor Ginzburg (born 1957) is a

Russian American Russian Americans ( rus, русские американцы, r=russkiye amerikantsy, p= ˈruskʲɪje ɐmʲɪrʲɪˈkant͡sɨ) are Americans of full or partial Russian ancestry. The term can apply to recent Russian immigrants to the United Stat ...

mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History On ...

who works 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 ...

and in

noncommutative geometry Noncommutative geometry (NCG) is a branch of mathematics concerned with a geometric approach to noncommutative algebras, and with the construction of ''spaces'' that are locally presented by noncommutative algebras of functions (possibly in some ge ...

. He is known for his contributions to geometric representation theory, especially, for his works on representations of

quantum groups In mathematics and theoretical physics, the term quantum group denotes one of a few different kinds of noncommutative algebras with additional structure. These include Drinfeld–Jimbo type quantum groups (which are quasitriangular Hopf algebra ...

and

Hecke algebra In mathematics, the Hecke algebra is the algebra generated by Hecke operators. Properties The algebra is a commutative ring. In the classical elliptic modular form theory, the Hecke operators ''T'n'' with ''n'' coprime to the level acting o ...

s, and on the geometric

Langlands program In representation theory and algebraic number theory, the Langlands program is a web of far-reaching and influential conjectures about connections between number theory and geometry. Proposed by , it seeks to relate Galois groups in algebraic num ...

(Satake

equivalence of categories In category theory, a branch of abstract mathematics, an equivalence of categories is a relation between two categories that establishes that these categories are "essentially the same". There are numerous examples of categorical equivalences fro ...

). He is currently a Professor of Mathematics at the

University of Chicago The University of Chicago (UChicago, Chicago, U of C, or UChi) is a private research university in Chicago, Illinois. Its main campus is located in Chicago's Hyde Park neighborhood. The University of Chicago is consistently ranked among the b ...

Career

Ginzburg received his Ph.D. 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 ...

in 1985, under the direction of

Alexandre Kirillov Alexandre Aleksandrovich Kirillov (russian: Алекса́ндр Алекса́ндрович Кири́ллов, born 1936) is a Soviet and Russian mathematician, known for his works in the fields of representation theory, topological groups a ...

and

Israel Gelfand Israel Moiseevich Gelfand, also written Israïl Moyseyovich Gel'fand, or Izrail M. Gelfand ( yi, ישראל געלפֿאַנד, russian: Изра́иль Моисе́евич Гельфа́нд, uk, Ізраїль Мойсейович Гел� ...

. Ginzburg wrote a textbook ''Representation theory and complex geometry'' with

Neil Chriss Neil A. Chriss is a mathematician, academic, hedge fund manager, philanthropist and a founding board member of the charity organization "Math for America" which seeks to improve math education in the United States. Chriss also serves on the board ...

on geometric representation theory. A paper by

Alexander Beilinson Alexander A. Beilinson (born 1957) 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. In 1 ...

, Ginzburg, and Wolfgang Soergel introduced the concept of

Koszul duality In mathematics, Koszul duality, named after the French mathematician Jean-Louis Koszul, is any of various kinds of dualities found in representation theory of Lie algebras, abstract algebras (semisimple algebra) and topology (e.g., equivariant cohom ...

(cf.

Koszul algebra In abstract algebra, a Koszul algebra R is a graded k-algebra over which the ground field k has a linear minimal graded free resolution, ''i.e.'', there exists an exact sequence: :\cdots \rightarrow R(-i)^ \rightarrow \cdots \rightarrow R(-2)^ ...

) and the technique of "mixed categories" to representation theory. Furthermore, Ginzburg and Mikhail Kapranov developed Koszul duality theory for

operads In mathematics, an operad is a structure that consists of abstract operations, each one having a fixed finite number of inputs (arguments) and one output, as well as a specification of how to compose these operations. Given an operad O, one defin ...

. In

, Ginzburg defined, following earlier ideas of

Maxim Kontsevich Maxim Lvovich Kontsevich (russian: Макси́м Льво́вич Конце́вич, ; born 25 August 1964) is a Russian and French mathematician and mathematical physicist. He is a professor at the Institut des Hautes Études Scientifiques an ...

, the notion of Calabi–Yau algebra. An important role in the theory of motivic Donaldson–Thomas invariants is played by the so-called "Ginzburg dg algebra", a Calabi-Yau (dg)-algebra of dimension 3 associated with any cyclic potential on the

path algebra In graph theory, a quiver is a directed graph where loops and multiple arrows between two vertices are allowed, i.e. a multidigraph. They are commonly used in representation theory: a representation of a quiver assigns a vector space ...

of a

quiver A quiver is a container for holding arrows, bolts, ammo, projectiles, darts, or javelins. It can be carried on an archer's body, the bow, or the ground, depending on the type of shooting and the archer's personal preference. Quivers were trad ...

Selected publications

* * * * * *

References

External links

{{DEFAULTSORT:Ginzburg, Victor 1957 births Living people Russian emigrants to the United States American people of Russian-Jewish descent Soviet Jews Soviet mathematicians 20th-century American mathematicians 21st-century American mathematicians University of Chicago faculty Moscow State University alumni