Alexander Merkurjev
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Alexander Merkurjev
Alexander Merkurjev
BornSeptember 25, 1955 (age 64)
Known forMerkurjev-Suslin theorem, cohomological invariants, canonical dimension, book of involutions, essential dimension
AwardsCole Prize in Algebra (2012)
Petersburg Mathematical Society Prize (1982)
Scientific career
FieldsMathematics
InstitutionsUniversity of California Los Angeles

Aleksandr Sergeyevich Merkurjev (Russian: ? ? ?, born September 25, 1955[1]) is a Russian-American mathematician, who has made major contributions to the field of algebra. Currently Merkurjev is a professor at the University of California, Los Angeles.

## Work

Merkurjev's work focuses on algebraic groups, quadratic forms, Galois cohomology, algebraic K-theory and central simple algebras. In the early 1980s Merkurjev proved a fundamental result about the structure of central simple algebras of period dividing 2, which relates the 2-torsion of the Brauer group with Milnor K-theory.[2] 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 (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, 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 ${\displaystyle p^{2}}$ (for a prime p) and, in joint work with Karpenko, the essential dimension of finite p-groups.[3][4]

## Awards

Merkurjev won the Young Mathematician Prize of the Petersburg Mathematical Society for his work on algebraic K-theory.[5] In 1986 he was an invited speaker at the International Congress of Mathematicians in Berkeley, California, and his talk was entitled "Milnor K-theory and Galois cohomology".[6] In 1995 he won the Humboldt Prize, an international prize awarded to renowned scholars. Merkurjev gave a plenary talk at the 2nd European Congress of Mathematics in Budapest, Hungary in 1996.[7] In 2012 he won the Cole Prize in Algebra for his work on the essential dimension of groups.[8]

In 2015 a special volume of Documenta Mathematica was published in honor of Merkurjev's sixtieth birthday.[9]

## Bibliography

### Books

• Max-Albert Knus, Alexander Merkurjev, Markus Rost, Jean-Pierre Tignol: The book of involutions, American Mathematical Society 1998. ISBN 0-8218-0904-0[10]
• Skip Garibaldi, Jean-Pierre Serre, Alexander Merkurjev: Cohomological Invariants in Galois Cohomology, American Mathematical Society 2003. ISBN 0-8218-3287-5[11]
• Richard Elman, Nikita Karpenko, Alexander Merkurjev: Algebraic and geometric theory of quadratic forms, American Mathematical Society 2008. ISBN 978-0-8218-4329-1[12]

## References

1. ^ Listed in the Library of Congress Online Catalog
2. ^ A. Merkurjev (1981). "On the norm residue symbol of degree 2". Dokl. Akad. Nauk SSSR. 261: 542-547 (English trans. Soviet Math. Dokl. 24 (1982), pp.1546-1551).
3. ^ A. Merkurjev (2010). "Essential p-dimension of PGL(p^2)". JAMS, 23. pp. 693-712.
4. ^ N. Karpenko, A. Merkurjev (2008). "Essential dimension of finite p-groups". Inventiones Mathematicae. 172 (3): 491-508. CiteSeerX 10.1.1.72.8045. doi:10.1007/s00222-007-0106-6.
5. ^
6. ^ "Proceedings of the International Congress of Mathematicians, August 3-11, 1986". International Mathematical Union. Merkurjev's talk: Milnor K-theory and Galois cohomology Archived 2016-03-03 at the Wayback Machine.
7. ^
8. ^ "2012 Cole Prize in Algebra" (PDF).
9. ^ P. Balmer, V. Chernousov, I. Fesenko, E. Friedlander, S. Garibaldi, Z. Reichstein, U. Rehmann (eds.). "Extra volume of Documenta Mathematica (2015) in honour of A. Merkurjev".CS1 maint: multiple names: authors list (link) CS1 maint: extra text: authors list (link)
10. ^ Springer, T. A. (1999). "Review: The book of involutions, by M.-A. Knus, A. Merkurjev, M. Rost, and J.-P. Tignol (with a preface by J. Tits)" (PDF). Bull. Amer. Math. Soc. (N.S.). 36 (3): 383-388. doi:10.1090/S0273-0979-99-00780-6.
11. ^ Swallow, John (2005). "Review: Cohomological invariants in Galois cohomology, by Skip Garibaldi, Alexander Merkurjev, and Jean-Pierre Serre" (PDF). Bull. Amer. Math. Soc. (N.S.). 42 (1): 93-98. doi:10.1090/S0273-0979-04-01033-X.
12. ^ Zaldivar, Felipe (2008). "Review: The Algebraic and Geometric Theory of Quadratic Forms". MAA Reviews.