Dmitry B. Fuchs
AMS Math Reviews
Professor Dmitry Borisovich Fuchs's research ranges from topology and the theory of foliations to homological algebra and representation theory. His most important work is on the representations and cohomology of infinite-dimensional Lie algebras. This work has consequences in string theory and conformal quantum field theory as codified in the mathematical theory of vertex operator algebras. However, Professor Fuchs' overall perspective in his work is that of a pure mathematician and not a mathematical physicist.
A good example of an infinite-dimensional Lie algebra is the Virasoro algebra. In , B.L. Feigin and D.B. Fuchs described the structure of Verma modules and modules of semi-infinite forms over the Virasoro algebra. In particular they showed that all submodules of a Verma module are generated by singular vectors and they described the relation between different submodules. The role of the Virasoro algebra in mathematical physics is that it describes the infinitesimal symmetries of a closed circle, in particular a closed string in string theory.
In another series of papers, Professor Fuchs calculated the Gelfand-Fuchs cohomology of Lie algebras and Lie superalgebras. Gelfand-Fuchs cohomology, also called continuous cohomology, is an interesting relative of the ordinary cohomology of a Lie algebra or Lie group devised by Gelfand and Fuchs  for tractable computations. He has also authored a book on the cohomology of infinite dimensional Lie algebras  where one can find most of the known results and methods of calculating Lie algebra cohomology. These calculations and treatises have applications to other areas of mathematics. For example, they imply some of the MacDonald identities, notorious coincidences from enumerative combinatorics which count certain kinds of lattice paths. They also give rise to characteristic classes of foliations, which are tools in classification problems for foliations.
Professor Fuchs is the author of several mathematics textbooks. A Course in Homotopic Topology, by Fuchs and Fomenko, deserves special mention. In this graduate topology book, Fuchs' conceptual approach is complemented by Fomenko's modern artistic renditions. The book not only cover the main concepts of algebraic topology, but also mentions applications to other areas of mathematics and to physics. It was reviewed positively in Math Reviews.
Professor Fuchs has advised several graduate students who are now noted mathematicians, including Boris Feigin, Fedor Malikov, and Vladimir Rokhlin.
- Cohomologies of the Lie algebra of tangential vector fields of a smooth
manifold, I (with Israel M. Gelfand), Funct. Anal. Appl. 3 (1969), 194-210.
- Cohomologies of the Lie algebra of formal vector fields (with Israel M.
Gelfand), Math. USSR Izv. 4 (1970), 327-342.
- Singular vectors of Verma modules over Kac-Moody algebras (with Boris L.
Feigin and Fedor G. Malikov), Funct. Anal. Appl. 20 (1986), 25-37.
- Cohomologies of infinite dimensional Lie algebras, Plenum Publ., New York,
- Representations of the Virasoro algebra (with Boris L. Feigin), in
Representations of Lie Groups and Related Topics, 465--554,
Adv. Stud. Contemp. Math., 7, Gordon and Breach, New York, 1990.
Last updated: 0000-00-00