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Albert Schwarz
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<h2 id="early-life-and-education">Early life and education</h2> <p>Schwarz was born in <a href="/facts/Kazan/dpeeCpGH">Kazan</a> to <a href="/facts/Ashkenazi_Jewish/B9PIYy71">Ashkenazi Jewish</a> parents, <a href="/facts/Soviet_Union/m7wWsPto">Soviet Union</a>. His parents were arrested in the <a href="/facts/Stalinist_purges/KSu23l0v">Stalinist purges</a> in 1937.<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> </p><p>Schwarz studied under <a href="/facts/Vadim_Yefremovich/Wu6TvLoX">Vadim Yefremovich</a> at Ivanovo Pedagogical Institute, having been denied admittance to <a href="/facts/Moscow_State_University/xHPG4oau">Moscow State University</a> on the grounds that he was the son of "<a href="/facts/Enemies_of_the_people/LdBl7ySQ">enemies of the people</a>."<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> </p> <h2 id="career-and-later-life">Career and later life</h2> <p>After defending his dissertation in 1958, he took a job at Voronezh University. In 1964 he was offered a job at <a href="/facts/Moscow_Engineering_Physics_Institute/94h5plBI">Moscow Engineering Physics Institute</a>.<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> He immigrated to the United States in 1989.<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a> </p> <h2 id="contributions">Contributions</h2> <p>Schwarz is one of the pioneers of <a href="/facts/Morse_theory/bJRqH8BQ">Morse theory</a> and brought up the first example of a <a href="/facts/Topological_quantum_field_theory/zHjrp3b8">topological quantum field theory</a>.<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a> The Schwarz genus, one of the fundamental notions of <a href="/facts/Topological_complexity/u23T6ug7">topological complexity</a>, is named after him.<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a> Schwarz worked on some examples in <a href="/facts/Noncommutative_geometry/WSwggL1p">noncommutative geometry</a>. He is the "S" in the AKSZ model (named after Mikhail Alexandrov, <a href="/facts/Maxim_Kontsevich/7No1RaTQ">Maxim Kontsevich</a>, Schwarz, and Oleg Zaboronski).<a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a> </p> <h2 id="recognition">Recognition</h2> <p>In 1990, Schwarz was an invited speaker of the <a href="/facts/International_Congress_of_Mathematicians/qQX4kVpe">International Congress of Mathematicians</a> in <a href="/facts/Kyoto/gUWtvjfd">Kyoto</a>. He was elected to the 2018 class of <a href="/facts/Fellow/DLThQbkq">fellows</a> of the <a href="/facts/American_Mathematical_Society/fr5gotCt">American Mathematical Society</a>.<a class="footnote-ref" id="fnref:9" href="#fn:9"><sup>9</sup></a> </p> <h2 id="monographs">Monographs</h2> <ul><li><i>Topology for physicists</i>, Springer, 1996.</li> <li><i>Quantum field theory and topology</i>, Grundlehren der Math. Wissen. 307, Springer 1993 (translated from Russian original <i>Kvantovaja teorija polja i topologija, Nauka</i>, Moscow, 1989).</li> <li>A. S. Švarc, <i>Математические основы квантовой теории поля (Mathematical aspects of quantum field theory)</i>, Atomizdat, Moscow, 1975.</li> <li><i>Mathematical Foundations of Quantum Field Theory</i>, 2020.</li></ul> <h2 id="papers-selection">Papers (selection)</h2> <ul><li>A. S. Švarc, A volume invariant of coverings (in Russian), <a href="/facts/Doklady_Akademii_Nauk_SSSR/dqes3C9S">Doklady Akademii Nauk SSSR</a>, 105 (1955), pp. 32–34.</li> <li>A. S. Švarc, Род расслоенного пространства, <i>Докл</i>. <i>АН СССР</i> (The genus of a fiber space (Russian), <a href="/facts/Doklady_Akademii_Nauk_SSSR/dqes3C9S">Doklady Akademii Nauk SSSR</a> 119 (1958), no. 2, 219–222.</li> <li>A. Schwarz, O. Zaboronsky, Supersymmetry and localization, <i>Comm. Math. Phys.</i> 183(2) (1997), 463–476.</li> <li>M. Alexandrov, M. Kontsevich, A. Schwarz, O. Zaboronsky, The geometry of the master equation and topological quantum field theory, <i>Int. J. Modern Phys</i>. A12(7):1405–1429, 1997.</li> <li>V. Kac, A. Schwarz, Geometric interpretation of the partition function of 2D gravity, <i>Phys. Lett.</i> B257 (1991), nos. 3–4, 329–334.</li> <li>A. A. Belavin, A. M. Polyakov, A. S. Schwartz, Yu. S. Tyupkin, Pseudoparticle solutions of the Yang-Mills equations, <i>Phys. Lett.</i> B59 (1975), no. 1, 85–87.</li> <li>V. N. Romanov, A. S. Švarc, Anomalies and elliptic operators (Russian), <i>Teoret. Mat. Fiz.</i> 41 (1979), no. 2, 190–204.</li> <li>S. N. Dolgikh, A. A. Rosly, A. S. Schwarz, Supermoduli spaces, <i>Comm. Math. Phys.</i> 135 (1990), no. 1, 91–100.</li></ul> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/ADHM_construction/k7zqDPWL">ADHM construction</a></li> <li><a href="/facts/BPST_instanton/0Iq9qjLl">BPST instanton</a></li> <li><a href="/facts/Instanton/IuXhVcRF">Instanton</a></li> <li><a href="/facts/Chern%25E2%2580%2593Simons_theory/AQ3Kayn3">Chern–Simons theory</a></li> <li><a href="/facts/Schwarz-type_TQFTs/zHjrp3b8">Schwarz-type TQFTs</a></li> <li><a href="/facts/%25C5%25A0varc%25E2%2580%2593Milnor_lemma/0LqSHReN">Švarc–Milnor lemma</a></li> <li><a href="/facts/Supermanifold/mapsm1Qb">Supermanifold</a></li></ul> <h2 id="notes">Notes</h2> <ul><li><a href="https://mathgenealogy.org/id.php?id=12690">Albert Schwarz</a> at the <a href="/facts/Mathematics_Genealogy_Project/J7AFo1Do">Mathematics Genealogy Project</a></li> <li><a href="https://web.archive.org/web/20101125124406/http://www.math.ucdavis.edu/~schwarz/">Biography at ucdavis.edu</a></li> <li>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.</li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Credited as Schwartz in A. A. Belavin et al (1975). <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>"My life in science" (PDF). ucdavis.edu. Retrieved 18 May 2023. <a href="https://www.math.ucdavis.edu/~schwarz/bion.pdf" target="_blank">https://www.math.ucdavis.edu/~schwarz/bion.pdf</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>"Albert Schwarz". The Mathematics Genealogy Project. Retrieved 18 May 2023. <a href="https://www.genealogy.math.ndsu.nodak.edu/id.php?id=12690" target="_blank">https://www.genealogy.math.ndsu.nodak.edu/id.php?id=12690</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>"Knots and Quantum Theory - Ideas | Institute for Advanced Study". 5 August 2011. <a href="https://www.ias.edu/ideas/2011/witten-knots-quantum-theory" target="_blank">https://www.ias.edu/ideas/2011/witten-knots-quantum-theory</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>Schwarz, Albert (2020). Mathematical Foundations of Quantum Field Theory. ISBN 978-9813278639. <a href="978-9813278639" target="_blank">978-9813278639</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>"Albert Schwarz in nLab". <a href="https://ncatlab.org/nlab/show/Albert+Schwarz" target="_blank">https://ncatlab.org/nlab/show/Albert+Schwarz</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> <li id="fn:7"><p>Vassiliev, V. A. (2011). "Topological complexity and Schwarz genus of general real polynomial equation". Moscow Mathematical Journal. 11 (3): 617–625, 632. MR 2894434. <a href="/wiki/MR_(identifier)" target="_blank">/wiki/MR_(identifier)</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li> <li id="fn:8"><p>Cattaneo, Alberto S.; Felder, Giovanni (2001). "On the AKSZ formulation of the Poisson sigma model". Letters in Mathematical Physics. 56 (2): 163–179. doi:10.1023/A:1010963926853. MR 1854134. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li> <li id="fn:9"><p>2018 Class of the Fellows of the AMS, American Mathematical Society, retrieved 2017-11-03 <a href="http://ams.org/profession/ams-fellows/new-fellows" target="_blank">http://ams.org/profession/ams-fellows/new-fellows</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li> </ol>