Alexander Yurievich Olshansky (born January 19, 1946 , Saratov ) is a Soviet and Russian mathematician , doctor of physical and mathematical sciences ( 1979 ), laureate of the A. I. Maltsev Prize , professor of mathematics at the University of Vanderbilt (since 1999 ). A specialist in the field of combinatorial and geometric group theory , who also has several papers on Lie and associative algebras.
| Alexander Yurievich Olshansky | |
|---|---|
 | |
| Date of Birth | January 19, 1946 (73 years old) |
| Place of Birth | Saratov |
| A country |  USSR →  Russia |
| Scientific field | group theory |
| Place of work | Vanderbilt University |
| Alma mater | MSU (mehmat) |
| Academic degree | Doctor of Physical and Mathematical Sciences |
| Academic rank | Professor |
| supervisor | Alfred Lvovich Shmelkin |
| Awards and prizes | Prize named after A.I. Maltsev (2000) |
Content
- 1 Biography
- 2 Contribution to science
- 3 Recognition
- 4 References
Biography
Born in the family of a military engineer in the field of aviation weapons, one of three brothers in the family. He graduated from high school in Engels , in 1963 he entered the Faculty of Mechanics and Mathematics of Moscow State University , which he graduated in 1968. There he graduated from graduate school and since 1970 worked as an assistant in the department of higher algebra of Moscow State University, since 1978 - associate professor, since 1985 - professor.
In 1983 - guest speaker at the XIX International Congress of Mathematicians . Since 1999, he has been a professor at the University of Vanderbilt .
He is the author of more than 100 scientific papers, including the monograph “The Geometry of Defining Relations in Groups” (translated into English by Kluwer ). Member of the editorial boards of several mathematical journals. Under his leadership, 22 master's theses were defended at Moscow State University and 6 at Vanderbilt University .
Contribution to Science
In 1969, while still a graduate student, he solved the 1935 Bernard Neumann problem of the existence of an infinite system of group identities that is not equivalent to any finite system. For this achievement, Olshansky received a congratulatory telegram from Neumann, who was then working at Vanderbilt University. Under the influence of his scientific adviser Alfred Lvovich Shmelkin, he was engaged in group varieties during graduate school years, having received a classification of minimal solvable varieties not generated by one finite group, giving a description of varieties where all groups are residually approximate.
In the late 1970s and early 1980s, he adapted van Kampen diagrams , proposed in 1933, but not widely used: he introduced van Kampen graduated diagrams, the use of which allowed him to build the so-called Tarski monsters - infinite groups of a limited period in which all proper subgroups are cyclic. The possibility of constructing such groups was highly doubtful, which explains the statement of the problems by Schmidt (1938), Chernikov (1947), (1956), and all of them were solved by Olshansky, largely changing the idea of infinite groups at that time.
One of the widely known results is counterexamples (1980), which solved the old von Neumann-Day problem: does any non-amenable group contain a non-cyclic free subgroup. Another application of graded diagrams and Olshansky's geometric approach was a new proof of the Novikov – Adyan theorem , which solved the Burnside problem . The original proof required more than three hundred pages, while the Olshansky proof for large odd numbers fit on 32 pages. It is still considered the shortest and is based on clear geometric considerations and global estimates for diagrams.
The groups constructed by Olshansky are the limiting cases of hyperbolic groups , which became in the 1990s under the influence of Gromov the central object in the geometric theory of groups . Olshansky later considered the conditions of small reduction and van Kampen diagrams over hyperbolic groups, expanding his constructions and studying the quotient groups of hyperbolic groups.
As of the 2010s, he is engaged in the asymptotic behavior of groups. He answered a number of questions about the possible behavior of invariants such as Dan functions , distortion, and relative growth of subgroups. Asymptotic invariants are associated with the complexity of algorithmic problems in groups, for example, in a large joint article by Olshansky with Birge, Rips and Sapir, a geometric criterion is obtained when a word problem in a finitely defined group has (non-deterministic) polynomial algorithmic complexity.
Recognition
- Laureate of the Moscow Mathematical Society (1970)
- A. I. Maltsev Prize (2000) - for a series of works on combinatorial and geometric group theory
- Honorary Member of the American Mathematical Society (2015)
Links
- Olshanskiy Alexander on Vanderbilt University website
- Olshansky, Alexander Yuryevich on the official website of the RAS
- Olshansky Alexander Yuryevich (Chronicle of Moscow University) . letopis.msu.ru. Date of treatment March 21, 2016.
- Olshansky Alexander Yuryevich . halgebra.math.msu.su. Date of treatment March 21, 2016.
- Olshansky Alexander Yuryevich - user, employee . istina.msu.ru. Date of treatment March 21, 2016.
- Persons: Olshansky Alexander Yuryevich . mathnet.ru. Date of treatment March 21, 2016.