Aleksandr Sergeyevich Merkurjev (russian: Алекса́ндр Сергее́вич Мерку́рьев, born September 25, 1955) is a Russian-American mathematician, who has made major contributions to the field of

algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary a ...

. Currently Merkurjev is a professor at the

University of California, Los Angeles The University of California, Los Angeles (UCLA) is a public land-grant research university in Los Angeles, California. UCLA's academic roots were established in 1881 as a teachers college then known as the southern branch of the California St ...

Work

Merkurjev's work focuses on

algebraic group In mathematics, an algebraic group is an algebraic variety endowed with a group structure which is compatible with its structure as an algebraic variety. Thus the study of algebraic groups belongs both to algebraic geometry and group theory. Man ...

quadratic form In mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example, :4x^2 + 2xy - 3y^2 is a quadratic form in the variables and . The coefficients usually belong to a ...

Galois cohomology In mathematics, Galois cohomology is the study of the group cohomology of Galois modules, that is, the application of homological algebra to modules for Galois groups. A Galois group ''G'' associated to a field extension ''L''/''K'' acts in a nat ...

algebraic K-theory Algebraic ''K''-theory is a subject area in mathematics with connections to geometry, topology, ring theory, and number theory. Geometric, algebraic, and arithmetic objects are assigned objects called ''K''-groups. These are groups in the sense o ...

and

central simple algebra In ring theory and related areas of mathematics a central simple algebra (CSA) over a field ''K'' is a finite-dimensional associative ''K''-algebra ''A'' which is simple, and for which the center is exactly ''K''. (Note that ''not'' every simple a ...

s. In the early 1980s Merkurjev proved a fundamental result about the structure of central simple algebras of

period Period may refer to: Common uses * Era, a length or span of time * Full stop (or period), a punctuation mark Arts, entertainment, and media * Period (music), a concept in musical composition * Periodic sentence (or rhetorical period), a concept ...

dividing 2, which relates the 2-torsion of the

Brauer group Brauer or Bräuer is a surname of German origin, meaning "brewer". Notable people with the name include:- * Alfred Brauer (1894–1985), German-American mathematician, brother of Richard * Andreas Brauer (born 1973), German film producer * Arik ...

with

Milnor K-theory In mathematics, Milnor K-theory is an algebraic invariant (denoted K_*(F) for a field F) defined by as an attempt to study higher algebraic K-theory in the special case of fields. It was hoped this would help illuminate the structure for algebrai ...

. In subsequent work with Suslin this was extended to higher torsion as the Merkurjev–Suslin theorem. The full statement of the

norm residue isomorphism theorem In mathematics, the norm residue isomorphism theorem is a long-sought result relating Milnor ''K''-theory and Galois cohomology. The result has a relatively elementary formulation and at the same time represents the key juncture in the proofs of ...

(also known as the Bloch-Kato conjecture) was proven by Voevodsky. In the late 1990s Merkurjev gave the most general approach to the notion of

essential dimension In mathematics, essential dimension is an invariant (mathematics), invariant defined for certain algebraic structures such as algebraic groups and quadratic forms. It was introduced by Joe P. Buhler, J. Buhler and Zinovy Reichstein, Z. Reichstein an ...

, introduced by Buhler and Reichstein, and made fundamental contributions to that field. In particular Merkurjev determined the essential p-dimension of central simple algebras of degree

p^2

(for a prime p) and, in joint work with Karpenko, the essential dimension of finite ''p''-groups.

Awards

Merkurjev won the Young Mathematician Prize of the

Petersburg Mathematical Society The Saint Petersburg Mathematical Society (russian: Санкт-Петербургское математическое общество) is a mathematical society run by Saint Petersburg mathematicians. Historical notes The St. Petersburg Mathe ...

for his work on algebraic K-theory. In 1986 he was an

invited speaker at the International Congress of Mathematicians This is a list of International Congresses of Mathematicians Plenary and Invited Speakers. Being invited to talk at an International Congress of Mathematicians has been called "the equivalent, in this community, of an induction to a hall of fame." ...

Berkeley, California Berkeley ( ) is a city on the eastern shore of San Francisco Bay in northern Alameda County, California, United States. It is named after the 18th-century Irish bishop and philosopher George Berkeley. It borders the cities of Oakland and Emer ...

, and his talk was entitled "Milnor K-theory and Galois cohomology". In 1995 he won the

Humboldt Prize The Humboldt Prize, the Humboldt-Forschungspreis in German, also known as the Humboldt Research Award, is an award given by the Alexander von Humboldt Foundation of Germany to internationally renowned scientists and scholars who work outside of G ...

, an international prize awarded to renowned scholars. Merkurjev gave a plenary talk at the second

European Congress of Mathematics The European Congress of Mathematics (ECM) is the second largest international conference of the mathematics community, after the International Congresses of Mathematicians (ICM). The ECM are held every four years and are timed precisely betwee ...

Budapest, Hungary Budapest (, ; ) is the capital and most populous city of Hungary. It is the ninth-largest city in the European Union by population within city limits and the second-largest city on the Danube river; the city has an estimated population of ...

in 1996. In 2012 he won the

Cole Prize The Frank Nelson Cole Prize, or Cole Prize for short, is one of twenty-two prizes awarded to mathematicians by the American Mathematical Society, one for an outstanding contribution to algebra, and the other for an outstanding contribution to number ...

in Algebra for his work on the essential dimension of groups. In 2015 a special volume of ''Documenta Mathematica'' was published in honor of Merkurjev's sixtieth birthday.

Bibliography

Books

* Max-Albert Knus, Alexander Merkurjev,

Markus Rost Markus Rost is a German mathematician who works at the intersection of topology and algebra. He was an invited speaker at the International Congress of Mathematicians in 2002 in Beijing, China. He is a professor at the University of Bielefeld. He ...

, Jean-Pierre Tignol: The book of involutions, American Mathematical Society 1998. * Skip Garibaldi,

Jean-Pierre Serre Jean-Pierre Serre (; born 15 September 1926) is a French mathematician who has made contributions to algebraic topology, algebraic geometry, and algebraic number theory. He was awarded the Fields Medal in 1954, the Wolf Prize in 2000 and the ina ...

, Alexander Merkurjev: Cohomological Invariants in Galois Cohomology, American Mathematical Society 2003. * Richard Elman, Nikita Karpenko, Alexander Merkurjev: Algebraic and geometric theory of quadratic forms, American Mathematical Society 2008.

References

External links

Alexander Merkurjev - personal webpage at UCLA
* * {{DEFAULTSORT:Merkurjev, Alexander 1955 births Living people University of California, Los Angeles faculty 20th-century American mathematicians 21st-century American mathematicians Fellows of the American Mathematical Society Algebraists International Mathematical Olympiad participants