90+ years of The Scottish Book
DOI:
https://doi.org/10.4467/2543702XSHS.26.010.22699Keywords:
The Scottish Book, Lwów School of Mathematics, Polish School of Mathematics, open problems in mathematics, mathematics of the first half of the 20th century, mathematical communitiesAbstract
Inspired by the recent 90th anniversary of The Scottish Book we present some reflections about its impact. We first discuss new areas of mathematics it helped launch. Then we argue that it was actively used in stimulating the interests and results of junior mathematicians and students. Also, we summarize the progress during the decade that has passed since the publication of Mauldin (ed.) 2015, which contained a review of solved problems from The Scottish Book. We also provide an overview of collections of open problems related in one way or another to The Scottish Book. All formulations of The Scottish Book problems in English are cited here from Mauldin (ed.) 2015.
References
Alfonseca, M. Ángeles; Saorín Gómez, Eugenia 2023: A counter-example to Ulam’s problem 19 from „the Scottish Book”. La Gaceta de la Real Sociedad Matemática Española 26(3), pp. 457–475.
Anderson, Stuart 2020: Zbigniew Moroń (1904–1971). Wraclow [sic], Poland. URL: http://www.squaring.net/history_theory/z_moron.html (accessed June 15, 2026).
Audin, Michèle 2011: Fatou, Julia, Montel. The Great Prize of mathematical sciences of 1918, and beyond. Transl. from the French. Berlin: Springer. ISBN 978-3-642-17853-5/pbk; 978-3-642-00446-9/ebook (“Lecture Notes in Mathematics 2014. History of Mathematics Subseries”), viii, 332 pp.s
Avila, Artur; Jitomirskaya, Svetlana 2009: The Ten Martini problem. Annals of Mathematics (2) 170(1), pp. 303–342.
Banakh, Iryna; Banakh, Taras; Hryniv, Olena; Prytula, Yaroslav G.; Zarichnyi, Mykhailo 2024: “The Scottish Book”: Continuing traditions. Antiquitates Mathematicae 18, pp. 71–103.
Banakh, Taras 2021: Mini-conference dedicated to the 85th anniversary of the first record in “the Scottish Book”. URL: https://www.youtube.com/watch?v=x51gZonZivw (accessed June 15, 2026).
Banakh, Taras; Cauty, Robert 2011: Topological classification of closed convex sets in Fréchet spaces. Studia Mathematica 205(1), pp. 1–11. DOI: 10.4064/sm205-1-1.
Banakh, Taras.; Plichko, A.M. 1997: On a problem of “Scottish Book” concerning condensation of metric spaces onto compacta. Matematychni Studii 8, pp. 119–122.
Banakh, Taras; Guran, Igor 2013: Perfectly supportable semigroups are σ-discrete in each Hausdorff shift-invariant topology. Topological Algebra and its Applications 1, pp. 1–8.
Barner, Klaus 1997: Paul Wolfskehl and the Wolfskehl Prize. Notices of the American Mathematical Society 44(10), pp. 1294–1303.
Barrow-Green, June 1997: Poincaré and the three body problem. Providence, RI: American Mathematical Society; London: London Mathematical Society. ISBN 0-8218-0367-0 (“History of Mathematics” 11), xv, 272 pp.
Basso, Giuliano 2023: On a problem of Mazur and Sternbach. DOI: 10.48550/arXiv.2308.08400.
Beanland, Kevin; Humke, Paul; Richards, Trevor 2016: On Scottish book problem 157. Real Analysis Exchange 41(2), pp. 331–346.
Biler, Piotr; Krupski, Paweł; Plebanek, Grzegorz; Woyczyński Wojbor, Andrzej 2022: Lwów of the West: a brief history of Wrocław’s New Scottish Book, Part III. [In:] Alexander Blokh, Lex Oversteegen, Vladlen Timorin 2022: Cutpoints of invariant subcontinua of polynomial Julia sets. Arnold Mathematical Journal 8(2), pp. 271–284.
Boroński, Jan Paweł 2017: On a generalization of the Cartwright–Littlewood fixed point theorem for planar homeomorphisms. Ergodic Theory and Dynamical Systems 37(6), pp. 1815–1824. DOI: 10.1017/etds.2015.129.
Boroński, Jan Paweł.; Kupka, Jiri 2020: A Sternbach-type fixed point problem for maps induced on hyperspaces. Topology and its Applications 282, 107304.
Cauty, Robert 1994: Un espace métrique linéaire qui n’est pas un rétracte absolu. Fundamenta Mathematicae 146, pp. 85–99.
Cauty, Robert 2001: Solution du problème de point fixe de Schauder. Fundamenta Mathematicae 170(3), pp. 231–246.
Cauty, Robert 2005: Retractes absolus de voisinage algebriques. Serdica Mathematical Journal 31(4), pp. 309–354.
Chatzakou, Marianna; Sarantopoulos, Yannis 2021: Estimates for polynomial norms on Banach spaces. Dolomites Research Notes on Approximation 14, pp. 40–52.
Ciesielski, Krzysztof; Pogoda, Zdzisław 1988: Conversation with Andrzej Turowicz. The Mathematical Intelligencer 10(4), pp. 13–20.
Dobrowolski, Tadeusz 2003: Revisiting Cauty’s proof of the Schauder conjecture. Abstract and Applied Analysis 7, pp. 407–433.
Dobrowolski, Tadeusz; Toruńczyk, Henryk 1979: On metric linear spaces homeomorphic to l_2 and compact convex sets homeomorphic to Q. Bulletin de l’Académie Polonaise des Sciences. Série des Sciences Mathématiques 27(11–12), pp. 883–887.
Domoradzki, Stanisław; Pawlikowska-Brożek, Zofia 2010: Józef Marcinkiewicz (1910–1940) w świetle dokumentów i wspomnień. W stulecie urodzin. Analecta 19(1–2), pp. 23–62.
Domoradzki, Stanisław; Stawiska, Małgorzata 2015: Distinguished Graduates in Mathematics of Jagiellonian University in the Interwar Period. Part II: 1926–1939. Technical Transactions [Czasopismo Techniczne], Fundamental Sciences [Nauki Podstawowe] 2-NP(20), pp. 117–141.
Domoradzki, Stanisław; Stawiska, Małgorzata; Zarichnyi, Michael 2021: Formation and Activity of the Lwów School of Mathematics. [In:] Martina Bečvářová (ed.) 2021, The Development of Mathematics Between the World Wars. Case Studies, Examples and Analyses, Europe–Singapore: World Scientific Publishnig Ltd. ISBN 978-1-78634-930-9, pp. 227–286.
Duda, Roman 2007/2014: Lwowska szkoła matematyczna. Wrocław: Wydawnictwo Uniwersytetu Wrocławskiego (“Acta Universitatis Wratislaviensis” 2949), 256 pp. / English edition 2014: Pearls from a Lost City: The Lvov School of Mathematics. Translated from the 2007 Polish original by Daniel Davies. Providence, RI: American Mathematical Society. ISBN 978-1-4704-1076-6 (“History of Mathematics” 40), xii + 231 pp.
Duda, Roman 2019: Historia matematyki w Polsce na tle dziejów nauki i kultury. Warszawa: Instytut Historii Nauki PAN and Oficyna Wydawnicza ASPRA-JR.
Duda, Roman (ed.): 2024: Słownik biograficzny matematyków polskich. Kraków: PAU.
Duda, Roman; Terepeta, Małgorzata 2024: Zahorski, Zygmunt Marian. [In:] Roman Duda (ed.), Słownik biograficzny matematyków polskich. Kraków: PAU, pp. 467–468.
Eremenko, Alexandre 2019: A problem of Stanislaw Saks. Matematyka Stosowana 52(2), pp. 173–175.
Feigenbaum, Mitchell 1982: Reflections of the Polish masters: An interview with Stan Ulam and Mark Kac, Los Alamos Science 3(3). ISSN 0273-7116, pp. 54–65.
Fejes Tóth, Gábor 2023: Packing and covering properties of sequences of convex bodies. [In:] László Fejes Tóth, Gábor Fejes Tóth, Włodzimierz Kupenberg (ed.), Lagerungen. Cham: Springer (“Grundlehren der mathematischen Wissenschaften” 360). DOI: 10.1007/978-3-031-21800-2_14.
Fleck, Ludwik 1935: Entstehung und Entwicklung einer wissenschaftlichen Tatsache – Einführung in die Lehre vom Denkstil und Denkkollektiv. Basel: Schwabe und Co., Verlagsbuchhandlung.
Florentin, David I.; Schütt, Carsten; Werner, Elisabeth M.; Zhang, Ning 2022: Convex floating bodies of equilibrium. Proceedings of the American Mathematical Society 150, pp. 3037–3048.
Glicksberg, Irving Leonard 1952: A Further Generalization of the Kakutani Fixed Point Theorem, with Application to Nash Equilibrium Points. Proceedings of the American Mathematical Society 3(1), pp. 170–174.
Guran, Ihor; Prytula Yaroslav 2013: Józef Schreier “On finite base in topological groups”. Matematyka Stosowana 39, pp. 3–9.
Hilbert, David 1902: Mathematical Problems, Bulletin of the American Mathematical Society 8(10), pp. 437–479. DOI: 10.1090/S0002-9904-1902-00923-3.
Huang, Han; Slomka, Boaz A.; Werner, Elisabeth M. 2019: Ulam floating bodies. Journal of the London Mathematical Society 100(2), pp. 425–446.
Jakimowicz, Emilia; Miranowicz, Adam (eds.) 2011: Stefan Banach. Remarkable Life, Brilliant Mathematics. Biographical materials. Third edition. Gdańsk: Gdańsk University Press. ISBN: 978-83-7326-827-2.
Jeśmanowicz, Leon 2011: Wspomnienia. [In:] Janina Marciak-Kozłowska, Emilia Jakimowicz (eds.): Józef Marcinkiewicz. Genialny matematyk – męczennik Katynia. Białystok: Wydawnictwo Buk, pp. 277–280.
Kac, Mark 1981: A Personal History of „the Scottish Book”. [In:] Richard Daniel Mauldin (ed.) 2015, „The Scottish Book”. Mathematics from the Scottish Café. With selected problems from the New Scottish Book. 2nd updated and enlarged edition. Cham: Birkhäuser/Springer. ISBN 978-3-319-22896-9, pp. 13–20.
Kac, Mark 1985: Enigmas of chance. An autobiography. New York: Alfred P. Sloan Foundation; Harper & Row, Publishers. ISBN 0-06-015433-0, xxviii+163 pp.
Kaczmarz, Stefan; Turowicz, Andrzej 1939: Sur l’irrationalité des intégrales indéfinies, Studia Mathematica 8, pp. 129–134.
Kalikinskaya, Ekaterina 2014: Interview with Mark Iosifovich Vishik, Communications on Pure and Applied Analysis 13(5), pp. i–x; DOI: 10.3934/cpaa.2014.13.5i.
Kedlaya, Kiran: The Nonarchimedean Scottish Book. URL: https://scripts.mit.edu/~kedlaya/wiki/index.php?title=The_Nonarchimedean_Scottish_Book (accessed June 15, 2026).
Kordos, Marek 2011: „The Scottish Book”. [In:] Emilia Jakimowicz, Adam Miranowicz (eds.) 2011, Stefan Banach. Remarkable Life, Brilliant Mathematics. Biographical materials. Third edition. Gdańsk: Gdańsk University Press. ISBN: 978-83-7326-827-2, pp. 143–152.
Kos, Anna Maria; Ciesielski, Krzysztof; Ciesielska, Danuta; Dawidowicz, Antoni Leon; Grzesik, Andrzej; Pogoda, Zdzisław; Tutaj, Edward; Valette, Anna, date unknown: Historia Koła. URL: https://kmsuj.matinf.uj.edu.pl/historia_koa/ (accessed June 15, 2026).
Kulpa, Władysław; Szymanski, Andrzej; Turzański, Marian 2018: Fixed point theorems and L^*-operators. Journal of Fixed Point Theory and Applications 20, article number 23. DOI: 10.1007/s11784-018-0494-9.
Kuratowski, Kazimierz 1973/1980: Pół wieku matematyki polskiej 1920–1970. Wspomnienia i refleksje. Warszawa: Wiedza Powszechna. 1973. English translation by Andrzej Kirkor: A half century of Polish mathematics. Remembrances and reflections. With a preface by Ulam Stanisław. Elmsford, N.Y.: Pergamon Press, Inc.; Warsaw: PWN – Polish Scientific Publishers. ISBN 0-08-023046-6 (“International Series in Pure and Applied Mathematics” 108), viii+200 pp.
Lipiński, Jan Stanisław 2000: Zygmunt Zahorski (1914–1988). Wiadomości Matematyczne 36, pp. 73–83.
Liu, Chunyan; Werner, Elisabeth M.; Ye, Deping; Zhang, Ning, 2025: Archimedes’ principle of flotation and floating bodies: construction, extensions and related problems. Acta Mathematica Scientia Ser. B (Engl. ed.) 45(1), pp. 237–256.
Maligranda, Lech 2007: Stefan Kaczmarz (1895–1939). Antiquitates Mathematicae 1, pp. 15–61.
Maligranda, Lech 2011: Józef Marcinkiewicz (1910–1940) – on the centenary of his birth. [In:] Marcinkiewicz Centenary Volume. Warszawa: IM PAN (“Banach Center Publications” 95), pp. 133–234.
Maligranda, Lech 2014: Działalność naukowa Józefa Schreiera. Wiadomości Matematyczne 50(1), pp. 45–68.
Maligranda, Lech 2015: Meier (Maks) Eidelheit (1910–1943). Wiadomości Matematyczne 51(1), pp. 31–59.
Maligranda, Lech 2020: Doktorat i doktoranci Stefana Banacha. Antiquitates Mathematicae 14, pp. 89–149.
Maligranda, Lech 2022: Juliusz Schauder (1899–1943). Antiquitates Mathematicae 16, pp. 27–149.
Maligranda, Lech; Mykhaylyuk, Volodymyr; Plichko, Anatolij 2011: On a problem of Eidelheit from “the Scottish Book” concerning absolutely continuous functions. Journal of Mathematical Analysis and Applications 375(2), pp. 401–411.
Maligranda, Lech; Prytuła, Jarosław G. 2017: Uniwersytet we Lwowie w latach 1939–1941. Matematyka, fizyka i astronomia. Wiadomości Matematyczne 53(2), pp. 303–329.
Maligranda, Lech; Wnuk, Witold 2000: Władysław Orlicz (1903–1990). Wiadomości Matematyczne 36, pp. 85–147.
Mauldin, Richard Daniel (ed.) 2015: The Scottish Book. Mathematics from the Scottish Café. With selected problems from the New Scottish Book. 2nd updated and enlarged edition. Cham: Birkhäuser/Springer, ISBN 978-3-319-22896-9.
Marciak-Kozłowska, Janina; Jakimowicz, Emilia (eds.) 2011: Józef Marcinkiewicz. Genialny matematyk – mȩczennik Katynia. Białystok: Wydawnictwo Buk.
Mayen, Józef 1934: Gawȩdy o lwowskich kawiarniach. Chwila 18 czerwca – 9 lipca. URL: http://www.lwow.com.pl/rocznik/kawiarnie.html (accessed June 15, 2026).
van Mill, Jan; Reed, George M. (eds.) 1990: Open Problems in Topology. Amsterdam: North-Holland.
Montejano, Luis 2024: Three old problems from the Polish school of mathematics. Boletín de la Sociedad Matemática Mexicana 30, article number 68, 9 pp. DOI: 10.1007/s40590-024-00637-0.
Mori, Michiya 2024: On the Scottish Book Problem 155 by Mazur and Sternbach. Comptes Rendus. Mathématique 362, pp. 813–816. DOI: 10.5802/crmath.572.
Moroń, Zdzisław 1925: O rozkładach prostoka̧tów na kwadraty. Przegląd Matematyczno-Fizyczny 3, pp. 152–153.
Moroń, Zdzisław 1956: O prawie-doskonałych [sic] rozkładach prostokątów. Wiadomości Matematyczne 1(2), pp. 175–179.
Mycielski, Jan; Steinhaus, Hugo 1962: A mathematical axiom contradicting the axiom of choice. Bulletin de l’Académie Polonaise des Sciences. Série des Sciences Mathématiques, Astronomiques et Physiques 10, pp. 1–3.
Nadler Sam B., Jr., Quinn J., Stavrokas Nick M. 1979: Hyperspace of compact convex sets. Pacific Journal of Mathematics 83(2), pp. 441–462.
von Neumann, Johan 1928: Zur Theorie der Gesellschaftsspiele. Mathematische Annalen 100, pp. 295–320.
Ornstein, Daniel 1971: A characterization of monotone functions. Illinois Journal of Mathematics 15, pp. 73–76.
Osipov, Alexander V. 2023: Note on the Banach Problem 1 of condensations of Banach spaces onto compacta. Filomat 37(7), pp. 2183–2186. DOI: https://doi.org/10.2298/FIL2307183O.
Osipov, Alexander V. 2024: On the first Banach problem concerning condensations of absolute κ-Borel sets onto compacta. Acta Mathematica Hungarica 173, pp. 169–175. DOI: https://doi.org/10.1007/s10474-024-01428-9.
Osipov, Aleksander V.; Pytkeev, Evgenii G. 2023: Every metric space of weight λ=λ^(ℵ_0 ) admits a condensation onto a Banach space. Topology and its Applications 330, 108486.
Pearl, Elliott M. (ed.) 2007: Open Problems in Topology II. Amsterdam: Elsevier.
Pytkeev, Evgenii 1976: Upper bounds of topologies. Mathematical Notes 20, pp. 831–837.
Robatian, Damoon 2016: The fixed-point property under induced interval maps of continua. Journal of Mathematical Sciences 215, pp. 376–386.
Rosa, Alessandro 2026: The origins of holomorphic dynamics. A historical study of the topic selected for the Grand Prix des Mathématiques (1915–1918). Cham: Birkhäuser. ISBN 978-3-032-17105-4/hbk; 978-3-032-17108-5/pbk; 978-3-032-17106-1/ebook. x, 512 pp.
Ryabogin, Dmitry 2022: A negative answer to Ulam’s problem 19 from the Scottish Book. Annals of Mathematics (2) 195(3), pp. 1111–1150.
Ryabogin, Dmitry 2023: On bodies floating in equilibrium in every orientation. Geometriae Dedicata 217(4), Paper No. 70, 17 pp.
Schauder, Juliusz 1930: Der Fixpunktsatz in Funktionalräumen. Studia Mathematica 2, pp. 171–180.
Sierpiński, Wacław 1929: Sur les images biunivoques de l’ensemble de tous les nombres irrationels. Matematica 1, pp. 18–20.
Stawiska, Małgorzata 2015: Matematyka polska na Joint Mathematics Meetings w San Antonio (Texas), USA, styczeń 2015. Antiquitates Mathematicae 9(1), pp. 181–192.
Steinhaus, Hugo 1929: Anwendungen der Funktionalanalysis auf einige Fragen der reellen Funktionentheorie. Studia Mathematica 1, pp. 51–81.
Steinhaus, Hugo 1992/2015: Wspomnienia i zapiski. Londyn: Wydawnictwo Aneks. English translation: 2015: Mathematician for All Seasons – Recollections and Notes, 1 (1887–1945). With a foreword by Kazimierz Dziewanowski. Translated from the Polish by A. Shenitzer. Edited and with an introduction by R.G. Burns, I. Szymaniec and A. Weron, Cham: Birkhäuser–Springer. ISBN 978-3-319-21983-7 (“Vita Mathematica” 18), xiii + 491 pp.
Storozh, Oleh G.; Chuyko, Galina I.; Kudryk, Taras S.; Mykytyuk, Yaroslav V.; Sushchyk, Natan. S. 2021: Vladyslav Elijovych Lyantse (in honour of the 100th anniversary of birth). Matematychni Studii 55(2), pp. 214–222.
Szpilrajn-Marczewski, Edward 1945: Sur deux propriétés des classes d’ensembles. Fundamenta Mathematicae 33(1), pp. 303–307.
Telgársky, Rastislav 1987: Topological games: on the 50th anniversary of the Banach-Mazur game, Rocky Mountain Journal of Mathematics 17, pp. 227–276.
Teschke, Olaf 2021: The “Jahrbuch über die Fortschritte der Mathematik” as a part of zbMATH Open. European Mathematical Society Magazine 122, pp. 62–64.
Toruńczyk, Henryk 1981: Characterizing Hilbert space topology. Fundamenta Mathematicae 111(3), pp. 247–262.
Turowicz, Andrzej 1948: Sur les fonctionnelles continues et multiplicatives. Annales Societatis Mathematicae Polonae 20, pp. 135–156.
Turowicz, Andrzej 1989: Tape recording. Tyniec. Courtesy of Z. Pawlikowska-Brożek, transcribed by S. Domoradzki.
Tychonoff, Andriej 1935: Ein Fixpunktsatz. Mathematische Annalen 3, pp. 767–776.
Ulam, Stanisław Marcin 1976: Adventures of a mathematician. New York: Charles Scribner’s Sons, xi+317 pp. (25 plates). Polish edition: 1996: Przygody matematyka, przełożyła Agnieszka Górnicka. Warszawa: Prószyński i S-ka, ISBN 83-86868-25-2.
Ulam, Stanislaw 2015: An Anecdotal History of the Scottish Book. Chapter 1. [In:] Richard Daniel Mauldin (ed.) 2015: The Scottish Book. Mathematics from The Scottish Café. With selected problems from The New Scottish Book. 2nd updated and enlarged edition. Cham: Birkhäuser/Springer, ISBN 978-3-319-22896-9, pp. 3–12.
Vasilyev, Ioann 2025: On the zero sets of harmonic polynomials. DOI: 10.48550/arXiv.2506.24116.
Wortal Stefana Banacha 2025: The Scottish Book (multiple versions and authors). Copies of the manuscript and the typescript in Polish and English. URL: http://kielich.amu.ed u.pl/Stefan_Banach/archiwalia.html (accessed June 15, 2026).
Zahorski, Zygmunt 1947: Sur l’ensemble des points singuliers d’une fonction d’une variable réelle admettant les derivées de tous les orders. Fundamenta Mathematicae 34, pp. 183–245.
Zahorski, Zygmunt 1986: Zarys biografii. Zeszyty Naukowe Politechniki Śla̧skiej, Matematyka–Fizyka 48, pp. 7–25. ISSN 0072-420X.
Zahorski, Zygmunt (1992); answers to the survey by the Historical Commission of the Polish Mathematical Society sent to dr. Zofia Pawlikowska-Brożek (unpublished), in the possession of Stanisław Domoradzki.
Zawalski, Bartłomiej 2025: On the homothety conjecture for the body of flotation and the body of buoyancy on a plane. DOI: 10.48550/arXiv.2507.11850.
Zieliński, Krzysztof 2019: Goście i gospodarze Kawiarni Szkockiej [Guests and hosts of the Scottish Café]. Wiadomości Matematyczne 55(1), pp. 197–215.
Downloads
Published
Issue
Section
How to Cite
