Albert Schwarz

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Albert Solomonovich Schwarz[1] (/ʃwɔːrts/; Russian: А. С. Шварц; born June 24, 1934) is a mathematician and a theoretical physicist educated in the Soviet Union and now a professor at the University of California, Davis.

Schwarz was born in Kazan, Soviet Union.

Schwarz is one of the pioneers of Morse theory and brought up the first example of a topological quantum field theory. The Schwarz genus, one of the fundamental notions of topological complexity, is named after him. Schwarz worked on some examples in noncommutative geometry. He is the "S" in the famous AKSZ model (named after Mikhail Alexandrov, Maxim Kontsevich, Schwarz, and Oleg Zaboronsky).

In 1990, Schwarz was an invited speaker of the International Congress of Mathematicians in Kyoto. He was elected to the 2018 class of fellows of the American Mathematical Society.[2]


  • Topology for physicists, Springer, 1996.
  • Quantum field theory and topology, Grundlehren der Math. Wissen. 307, Springer 1993 (translated from Russian original Kvantovaja teorija polja i topologija, Nauka, Moscow, 1989).
  • A. S. Švarc, Математические основы квантовой теории поля (Mathematical aspects of quantum field theory), Atomizdat, Moscow, 1975.

Papers (selection)[edit]

  • A. S. Švarc, Род расслоенного пространства, Докл. АН СССР (The genus of a fiber space (Russian), Dokl. Akad. Nauk SSSR (N.S.)) 119 (1958), no. 2, 219–222.
  • A. Schwarz, O. Zaboronsky, Supersymmetry and localization, Comm. Math. Phys. 183(2) (1997), 463–476.
  • M. Alexandrov, M. Kontsevich, A. Schwarz, O. Zaboronsky, The geometry of the master equation and topological quantum field theory, Int. J. Modern Phys. A12(7):1405–1429, 1997.
  • V. Kac, A. Schwarz, Geometric interpretation of the partition function of 2D gravity, Phys. Lett. B257 (1991), nos. 3–4, 329–334.
  • A. A. Belavin, A. M. Polyakov, A. S. Schwartz, Yu. S. Tyupkin, Pseudoparticle solutions of the Yang-Mills equations, Phys. Lett. B59 (1975), no. 1, 85–87.
  • V. N. Romanov, A. S. Švarc, Anomalies and elliptic operators (Russian), Teoret. Mat. Fiz. 41 (1979), no. 2, 190–204.
  • S. N. Dolgikh, A. A. Rosly, A. S. Schwarz, Supermoduli spaces, Comm. Math. Phys. 135 (1990), no. 1, 91–100.

See also[edit]


  1. ^ Credited as Schwartz in A. A. Belavin et al (1975).
  2. ^ 2018 Class of the Fellows of the AMS, American Mathematical Society, retrieved 2017-11-03


  • Albert Schwarz at the Mathematics Genealogy Project
  • Biography at
  • Шварц Альберт Соломонович // Воронежская энциклопедия : в 2 т. / Гл. ред. М. Д. Карпачёв. — Воронеж : Центр духовного возрождения Чернозёмного края, 2008. — Т. 2 : Н — Я. — 524 с. : ил., карты.