Fedor Bogomolov
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Fedor Alekseyevich Bogomolov (born 26 September 1946) (Фёдор Алексеевич Богомолов) is a Russian and American
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, mathematical structure, structure, space, Mathematica ...
, known for his research in
algebraic geometry Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ...
and
number theory Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...
. Bogomolov worked at the
Steklov Institute Steklov Institute of Mathematics or Steklov Mathematical Institute () is a premier research institute based in Moscow, specialized in mathematics, and a part of the Russian Academy of Sciences. The institute is named after Vladimir Andreevich Ste ...
in
Moscow Moscow is the Capital city, capital and List of cities and towns in Russia by population, largest city of Russia, standing on the Moskva (river), Moskva River in Central Russia. It has a population estimated at over 13 million residents with ...
before he became a professor at the
Courant Institute The Courant Institute of Mathematical Sciences (commonly known as Courant or CIMS) is the mathematics research school of New York University (NYU). Founded in 1935, it is named after Richard Courant, one of the founders of the Courant Institute ...
in
New York New York most commonly refers to: * New York (state), a state in the northeastern United States * New York City, the most populous city in the United States, located in the state of New York New York may also refer to: Places United Kingdom * ...
. He is most famous for his pioneering work on
hyperkähler manifold In differential geometry, a hyperkähler manifold is a Riemannian manifold (M, g) endowed with three integrable almost complex structures I, J, K that are Kähler with respect to the Riemannian metric g and satisfy the quaternionic relations I^2= ...
s. Born in Moscow, Bogomolov graduated from
Moscow State University Moscow State University (MSU), officially M. V. Lomonosov Moscow State University,. is a public university, public research university in Moscow, Russia. The university includes 15 research institutes, 43 faculties, more than 300 departments, a ...
, Faculty of Mechanics and Mathematics, and earned his doctorate (''"candidate degree"'') in 1973, at the Steklov Institute. His
doctoral advisor A doctoral advisor (also dissertation director, dissertation advisor; or doctoral supervisor) is a member of a university faculty whose role is to guide graduate students who are candidates for a doctorate, helping them select coursework, as well ...
was Sergei Novikov.


Geometry of Kähler manifolds

Bogomolov's Ph.D. thesis was entitled ''Compact Kähler varieties''. In his early papers Bogomolov studied the manifolds which were later called Calabi–Yau and hyperkähler. He proved a
decomposition theorem In mathematics, especially algebraic geometry, the decomposition theorem of Beilinson, Bernstein and Deligne or BBD decomposition theorem is a set of results concerning the cohomology of algebraic varieties. It was originally conjectured by Gelfand ...
, used for the
classification of manifolds In mathematics, specifically geometry and topology, the classification of manifolds is a basic question, about which much is known, and many open questions remain. Main themes Overview * Low-dimensional manifolds are classified by geometric struct ...
with trivial
canonical class In mathematics, the canonical bundle of a non-singular variety, non-singular algebraic variety V of dimension n over a field is the line bundle \,\!\Omega^n = \omega, which is the nth exterior power of the cotangent bundle \Omega on V. Over the c ...
. It has been re-proven using the
Calabi–Yau theorem In the mathematical field of differential geometry, the Calabi conjecture was a conjecture about the existence of certain kinds of Riemannian metrics on certain complex manifolds, made by . It was proved by , who received the Fields Medal and Oswal ...
and Berger's classification of Riemannian holonomies, and is foundational for modern
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 intera ...
. In the late 1970s and early 1980s Bogomolov studied the
deformation theory In mathematics, deformation theory is the study of infinitesimal conditions associated with varying a solution ''P'' of a problem to slightly different solutions ''P''ε, where ε is a small number, or a vector of small quantities. The infinitesima ...
for manifolds with trivial canonical class. He discovered what is now known as Bogomolov–Tian–Todorov theorem, proving the smoothness and un-obstructedness of the deformation space for hyperkaehler manifolds (in 1978 paper) and then extended this to all Calabi–Yau manifolds in the 1981 IHES preprint. Some years later, this theorem became the mathematical foundation for
Mirror Symmetry In mathematics, reflection symmetry, line symmetry, mirror symmetry, or mirror-image symmetry is symmetry with respect to a Reflection (mathematics), reflection. That is, a figure which does not change upon undergoing a reflection has reflecti ...
. While studying the deformation theory of hyperkähler manifolds, Bogomolov discovered what is now known as the Bogomolov–Beauville–Fujiki form on H^2(M). Studying properties of this form, Bogomolov erroneously concluded that compact hyperkaehler manifolds do not exist, with the exception of
K3 surface In mathematics, a complex analytic K3 surface is a compact connected complex manifold of dimension 2 with а trivial canonical bundle and irregularity of a surface, irregularity zero. An (algebraic) K3 surface over any field (mathematics), field ...
s, tori, and their products. Almost four years passed since this publication before Akira Fujiki found a counterexample.


Other works in algebraic geometry

Bogomolov's paper on "Holomorphic tensors and vector bundles on projective manifolds" proves what is now known as the Bogomolov–Miyaoka–Yau inequality, and also proves that a stable bundle on a surface, restricted to a curve of sufficiently big degree, remains stable. In "Families of curves on a surface of general type", Bogomolov laid the foundations to the now popular approach to the theory of
diophantine equations ''Diophantine'' means pertaining to the ancient Greek mathematician Diophantus. A number of concepts bear this name: *Diophantine approximation In number theory, the study of Diophantine approximation deals with the approximation of real n ...
through geometry of hyperbolic manifolds and
dynamical systems In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space, such as in a parametric curve. Examples include the mathematical models ...
. In this paper Bogomolov proved that on any
surface of general type In algebraic geometry, a surface of general type is an algebraic surface with Kodaira dimension 2. Because of Algebraic geometry and analytic geometry#Chow.27s theorem, Chow's theorem any compact complex manifold of dimension 2 and with Kodaira ...
with c_1^2>c_2, there is only a finite number of curves of bounded genus. Some 25 years later, Michael McQuillan extended this argument to prove the famous Green–Griffiths conjecture for such surfaces. In "Classification of surfaces of class VII_0 with b_=0", Bogomolov made the first step in a famously difficult (and still unresolved) problem of classification of surfaces of Kodaira class VII. These are compact complex surfaces with b_2=1. If they are in addition minimal, they are called ''class VII_0''.
Kunihiko Kodaira was a Japanese mathematician known for distinguished work in algebraic geometry and the theory of complex manifolds, and as the founder of the Japanese school of algebraic geometers. He was awarded a Fields Medal in 1954, being the first Japanese ...
classified all compact complex surfaces except class VII, which are still not understood, except the case b_=0 (Bogomolov) and b_=1 (Andrei Teleman, 2005).


Later career

Bogomolov obtained his
Habilitation Habilitation is the highest university degree, or the procedure by which it is achieved, in Germany, France, Italy, Poland and some other European and non-English-speaking countries. The candidate fulfills a university's set criteria of excelle ...
(Russian ''"Dr. of Sciences"'') in 1983. In 1994, he emigrated to the United States and became a full professor at the Courant Institute. He is very active in algebraic geometry and number theory. From 2009 till March 2014 he served as the Editor-in-Chief of the Central European Journal of Mathematics. Since 2014 he serves as the Editor-in-Chief of the European Journal of Mathematics. Since 2010 he is the academic supervisor of the HSE Laboratory of algebraic geometry and its applications. Bogomolov has extensively contributed to the revival of Russian mathematics. Three major international conferences commemorating his 70th birthday were held in 2016: at the
Courant Institute The Courant Institute of Mathematical Sciences (commonly known as Courant or CIMS) is the mathematics research school of New York University (NYU). Founded in 1935, it is named after Richard Courant, one of the founders of the Courant Institute ...
, the
University of Nottingham The University of Nottingham is a public research university in Nottingham, England. It was founded as University College Nottingham in 1881, and was granted a royal charter in 1948. Nottingham's main campus (University Park Campus, Nottingh ...
, and the
Higher School of Economics HSE University (), officially the National Research University Higher School of Economics () is a public research university founded in 1992 and headquartered in Moscow, Russia. Along with its main campus located in the capital, the university ...
in Moscow.


See also

* Bogomolov–Sommese vanishing theorem


References


External links


Official NYU home page
* {{DEFAULTSORT:Bogomolov, Fedor 1946 births 20th-century Russian mathematicians 21st-century Russian mathematicians Living people Soviet mathematicians Academic staff of the Higher School of Economics Courant Institute of Mathematical Sciences faculty Algebraic geometers Moscow State University alumni Silver professors