# Dmitry B. Fuchs

** Professor Emeritus ****Topology and algebra**

C.Sc., 1964, Moscow State University; D.Sc., 1987, Tbilisi State University**Refereed publications:** Via Math Reviews

Email: fuchs@math.ucdavis.edu

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 **[5]**, 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 **[1]** for tractable computations. He has also authored a
book on the cohomology of infinite dimensional Lie algebras **[4]** 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.

### Selected publications

**[1]**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.

**[2]** Cohomologies of the Lie algebra of formal vector fields (with Israel M.
Gelfand), Math. USSR Izv. 4 (1970), 327-342.

**[3]** Singular vectors of Verma modules over Kac-Moody algebras (with Boris L.
Feigin and Fedor G. Malikov), Funct. Anal. Appl. 20 (1986), 25-37.

**[4]** Cohomologies of infinite dimensional Lie algebras, Plenum Publ., New York,
1986.

**[5]** 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.

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