Albert Solomonovich Schwarz ( ; ; born June 24, 1934) is a Soviet and American mathematician and a theoretical physicist educated in the Soviet Union and now a professor at the

University of California, Davis The University of California, Davis (UC Davis, UCD, or Davis) is a Public university, public Land-grant university, land-grant research university in Davis, California, United States. It is the northernmost of the ten campuses of the University ...

Early life and education

Schwarz was born in

Kazan Kazan; , IPA: Help:IPA/Tatar, ɑzanis the largest city and capital city, capital of Tatarstan, Russia. The city lies at the confluence of the Volga and the Kazanka (river), Kazanka Rivers, covering an area of , with a population of over 1. ...

Ashkenazi Jewish Ashkenazi Jews ( ; also known as Ashkenazic Jews or Ashkenazim) form a distinct subgroup of the Jewish diaspora, that Ethnogenesis, emerged in the Holy Roman Empire around the end of the first millennium Common era, CE. They traditionally spe ...

parents,

Soviet Union The Union of Soviet Socialist Republics. (USSR), commonly known as the Soviet Union, was a List of former transcontinental countries#Since 1700, transcontinental country that spanned much of Eurasia from 1922 until Dissolution of the Soviet ...

. His parents were arrested in the

Stalinist purges The Great Purge, or the Great Terror (), also known as the Year of '37 () and the Yezhovshchina ( , ), was a political purge in the Soviet Union that took place from 1936 to 1938. After the assassination of Sergei Kirov by Leonid Nikolae ...

in 1937. Schwarz studied under

Vadim Yefremovich Vadim Arsenyevich Yefremovich (or Efremovich) (; 16 October 1903 – 1 May 1989) was a Soviet mathematician. Yefremovich was a member of the Moscow Topological School and specialized in the geometric aspects of general topology. He introduced the ...

at Ivanovo Pedagogical Institute, having been denied admittance to

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 ...

on the grounds that he was the son of "

enemies of the people The terms enemy of the people and enemy of the nation are designations for the political opponents and the social-class opponents of the power group within a larger social unit, who, thus identified, can be subjected to political repression. ...

Career and later life

After defending his dissertation in 1958, he took a job at Voronezh University. In 1964 he was offered a job at

Moscow Engineering Physics Institute National Research Nuclear University MEPhI (Moscow Engineering Physics Institute) () is a public university, public technical university in Moscow, Russia. It was founded in 1942 as the Moscow Mechanical Institute of Munitions, but was soon renam ...

. He immigrated to the United States in 1989.

Contributions

Schwarz is one of the pioneers of

Morse theory In mathematics, specifically in differential topology, Morse theory enables one to analyze the topology of a manifold by studying differentiable functions on that manifold. According to the basic insights of Marston Morse, a typical differenti ...

and brought up the first example of a

topological quantum field theory In gauge theory and mathematical physics, a topological quantum field theory (or topological field theory or TQFT) is a quantum field theory that computes topological invariants. While TQFTs were invented by physicists, they are also of mathemati ...

. The Schwarz genus, one of the fundamental notions of

topological complexity In mathematics, topological complexity of a topological space ''X'' (also denoted by TC(''X'')) is a topological invariant closely connected to the motion planning problem, introduced by Michael Farber in 2003. Definition Let ''X'' be a topologica ...

, is named after him. Schwarz worked on some examples in

noncommutative geometry Noncommutative geometry (NCG) is a branch of mathematics concerned with a geometric approach to noncommutative algebras, and with the construction of ''spaces'' that are locally presented by noncommutative algebras of functions, possibly in some g ...

. He is the "S" in the AKSZ model (named after Mikhail Alexandrov,

Maxim Kontsevich Maxim Lvovich Kontsevich (, ; born 25 August 1964) is a Russian and French mathematician and mathematical physicist. He is a professor at the Institut des Hautes Études Scientifiques and a distinguished professor at the University of Miami. He ...

, Schwarz, and Oleg Zaboronski).

Recognition

In 1990, Schwarz was an invited speaker of the

International Congress of Mathematicians The International Congress of Mathematicians (ICM) is the largest conference for the topic of mathematics. It meets once every four years, hosted by the International Mathematical Union (IMU). The Fields Medals, the IMU Abacus Medal (known before ...

Kyoto Kyoto ( or ; Japanese language, Japanese: , ''Kyōto'' ), officially , is the capital city of Kyoto Prefecture in the Kansai region of Japan's largest and most populous island of Honshu. , the city had a population of 1.46 million, making it t ...

. He was elected to the 2018 class of

fellow A fellow is a title and form of address for distinguished, learned, or skilled individuals in academia, medicine, research, and industry. The exact meaning of the term differs in each field. In learned society, learned or professional society, p ...

s of the

American Mathematical Society The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, ...

Monographs

* ''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. *''Mathematical Foundations of Quantum Field Theory'', 2020.

Papers (selection)

* A. S. Švarc, A volume invariant of coverings ,

Doklady Akademii Nauk SSSR The ''Proceedings of the USSR Academy of Sciences'' (, ''Doklady Akademii Nauk SSSR'' (''DAN SSSR''), ) was a Soviet journal that was dedicated to publishing original, academic research papers in physics, mathematics, chemistry, geology, and biol ...

, 105 (1955), pp. 32–34. * A. S. Švarc, Род расслоенного пространства, ''Докл''. ''АН СССР'' (The genus of a fiber space (Russian),

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.

Notes

References

*
Biography at ucdavis.edu
* 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. {{DEFAULTSORT:Schwarz, Albert 1934 births 20th-century American mathematicians 21st-century American mathematicians Mathematicians from Kazan 21st-century American physicists University of California, Davis faculty Living people American people of Russian-Jewish descent Fellows of the American Mathematical Society American topologists