4VG H e r i t a g e o f E u r o p e a n M a t h e m a t i c s 0iö t Yos e t Va a o r M l st ei o t rt fra Cag o - Gösta Mittag-Leffler and L r r e e f fl s Vito Volterra p e o r n a 40 Years of Correspondence d n e d n c e ERELF Edited by domar is ué tosmrd rsanaa Sentéric Frédéric Jaëck TaM J l alaa Laurent Mazliak zezë znlc iot iak lDk Emma Sallent Del Colombo e l C Rossana Tazzioli o l o m b o ISBN 978-3-03719-199-6 www.ems-ph.org Jaëck et al. | Heritage | RB 31 mm Heritage of European Mathematics Advisory Board Ciro Ciliberto, Roma Ildar A. Ibragimov, St. Petersburg Władysław Narkiewicz, Wroclaw Peter M. Neumann, Oxford Samuel J. Patterson, Göttingen Previously published Andrzej Schinzel, Selecta, Volume I: Diophantine Problems and Polynomials; Volume II: Elementary, Analytic and Geometric Number Theory, Henryk Iwaniec, Władysław Narkiewicz, and Jerzy Urbanowicz (Eds.) Thomas Harriot’s Doctrine of Triangular Numbers: the ‘Magisteria Magna’, Janet Beery and Jacqueline Stedall (Eds.) Hans Freudenthal, Selecta, Tony A. Springer and Dirk van Dalen (Eds.) Nikolai I. Lobachevsky, Pangeometry, Athanase Papadopoulos (Transl. and Ed.) Jacqueline Stedall, From Cardano’s great art to Lagrange’s reflections: filling a gap in the history of algebra Peter M. Neumann, The mathematical writings of Évariste Galois Peter Roquette, Contributions to the History of Number Theory in the 20th Century Jacques Tits, Œuvres – Collected Works, Volumes I–IV, Francis Buekenhout, Bernhard Mühlherr, Jean-Pierre Tignol and Hendrik Van Maldeghem (Eds.) Della Dumbaugh and Joachim Schwermer, Emil Artin and Beyond – Class Field Theory and L-Functions Martina Becˇváˇrová and Ivan Netuka, Karl Löwner and His Student Lipman Bers – Pre-war Prague Mathematicians Henri-Paul de Saint-Gervais, Uniformization of Riemann Surfaces. Revisiting a hundred-year-old theorem Gösta Mittag-Leffler and Vito Volterra 40 Years of Correspondence Edited by Frédéric Jaëck Laurent Mazliak Emma Sallent Del Colombo Rossana Tazzioli Editors: Frédéric Jaëck Laurent Mazliak Département de Mathématiques et Applications Sorbonne Université LPSM École Normale Supérieure Case courrier 188, 4 place Jussieu 45 Rue d’Ulm 75252 Paris CEDEX 05 75230 Paris CEDEX 05 France France [email protected] [email protected] Rossana Tazzioli Emma Sallent Del Colombo Département de Mathématiques Independent researcher Université de Lille Carrer de València 69 Cité Scientifique, Avenue Gauss 08015 Barcelona 59655 Villeneuve-d’Ascq CEDEX Spain France [email protected] [email protected] 2010 Mathematics Subject Classification (primary; secondary): 01-XX Key words: Mittag-Leffler, Volterra, correspondence, mathematics, history of mathematics ISBN 978-3-03719-199-6 The Swiss National Library lists this publication in The Swiss Book, the Swiss national bibliography, and the detailed bibliographic data are available on the Internet at http://www.helveticat.ch. This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broad- casting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use permission of the copyright owner must be obtained. © 2019 European Mathematical Society Contact address: European Mathematical Society Publishing House Technische Universität Berlin Mathematikgebäude Strasse des 17. Juni 136 10623 Berlin Germany Homepage: www.ems-ph.org Typeset using the editors’ TEX files: Nicole Bloye, Cardiff, UK Printing and binding: Beltz Bad Langensalza GmbH, Bad Langensalza, Germany ∞ Printed on acid free paper 9 8 7 6 5 4 3 2 1 Preface It is a great pleasure to see the current volume on the correspondence between Gösta Mittag-Leffler and Vito Volterra published by the European Mathematical Society. The exchangeofideasbetweenthetwoandtheircommonpassionformathematicscertainly helpedMittag-Lefflerformhisvisionoftheresearchinstitutethathefoundedbydonating his magnificent villa with its library to the Royal Swedish Academy of Sciences, to transform it into a center dedicated to mathematical research. The villa is located in beautifulDjursholmincloseproximitytoStockholmwheretheinstitute,whichisinfact theoldestmathematicsresearchcenterintheworld,stilloperatestoday. FollowingtheideasandwishesofGöstaMittag-Leffler,theinstituteisaninternational meetingplaceforresearchandpostdoctoraltraininginthemathematicalsciences. Itfound itscurrentformin1969underthevisionaryleadershipofLennartCarleson. Itisrununder theauspicesoftheRoyalSwedishAcademyofSciencesandisgovernedbyaboardwith representativesfromallNordiccountries. Itwelcomesvisitorsallyearforshortorlonger stays. Thepremisesoftheinstituteencompassseveralbuildings: themainbuildingincluding alibrary,officeanddiscussionspacesforresearchersandstaff,aseminarroombuilding, and five other buildings with housing facilities for visiting researchers. The mission of InstitutMittag-Leffleristosupportinternationaltop-levelresearchinmathematics,with special attention to the development in the Nordic countries. The institute is a hub for theinternationalmathematicalresearchcommunityandformathematiciansintheNordic countries. Themainactivitiesincluderesearchprograms,conferences,workshops,seminarsand summer schools, which all aim to conduct and develop current mathematical research. The institute also publishes two mathematical journals, Acta Mathematica (founded by Mittag-Leffler in 1882) and Arkiv för matematik (founded in 1903), all volumes of both journalsbeingfreelyavailableonline. Inthiswaytheinstituteoperatestothisdayinthe spiritofitsfounder. Itdevelopsmathematicsthroughinteractionsandmeetingsbetween mathematicians from all over the world and offers a research environment where new mathematicalideasareborn,discussed,andtestedbyworldleadingresearchers. ItisremarkabletonotethatmanysubjectsoftheexchangebetweenMittag-Lefflerand hisItaliancolleague—forexamplemathematicalphysicsandpartialdifferentialequations as well as various aspects of Analysis — are connected to important research areas still well represented at the institute, for example with recent programs in Fractal Geometry andDynamics(2016)andSpectralMethodsinMathematicalPhysics(2019). vi The present book gives unique and direct access to the line of thought of two mathematiciansthatinmanyways, acenturyago, helpedshapethemathematicalworld of today. A particularly useful aspect is the exposition of ‘mathematics in the raw’. In the letters, and the accompanying contextual introduction that analyzes their contents, we meet profound ideas and problems studied directly, without formal definitions and established methods (that were eventually formulated only later). Altogether, this set of lettersdeepensourunderstandingbothofmathematicsasasubjectandoftheintellectual environmentoutofwhichInstitutMittag-Lefflerwasborn. TobiasEkholm DirectorInstitutMittag-Leffler Acknowledgments Many people contributed to the realization of this work in different contexts and ways, and we would like to express our gratitude to them. We are grateful to first mention the two main institutions, which allowed us to publish the present correspondence — the Accademia dei Lincei in Rome and the Mittag-Leffler Institute in Djursholm near Stockholm. Moreover,theInstitutewelcomedusfortwoperiodsofstudyintheSummers 2017 and 2018 in order to complete the book, and we were thus able to take advantage of the exceptional working conditions it provides to its visitors. We want especially to warmlythankMikaelRågstedt,thelibrarianoftheInstitute,whohasfacilitatedourstay attheInstituteinallpossibleways. WeexpressourthankstotheNobelFoundation,the KungligaBibliothekofStockholm, andtheArchiviodellaScuolaNormaleSuperiorein Pisaforprovidinguswithdocumentsandletters,andtotheVolterrafamilyforpermission to publish some photographs of their property. We thank our colleagues Ciro Ciliberto andManfredKarbefortheirlongstandinginterestintheproject. Ofcourse, thesupport of the European Mathematical Society and its proposal to welcome the book among its publications were essential to encourage us to lead the project to the end. Thomas Hintermann, who oversees the policy of the European Mathematical Society Publishing House, has been always efficient and responsive, and it is a great pleasure for us to mention his role in this story. It also our pleasure to thank Nicole Bloye for her huge work in editing the English language. We want also to mention the help provided by HansEllegren,AlessandraFiocca,ÅseFrid,andJesperLützeninvariousways. Lastbut notleast,wemustexpressourmostheartfeltgratitudetoourcolleagueandfriendPietro Nastasiwhohadbeenyearsagotheinitiatorofthisenterprise,towhichhedevolvedgreat enthusiasm and energy. His original drafts gave us a good starting impulse and we are gladtoassociatehimwiththepresentbook. Barcelona–Lille–Paris,April2019 FrédéricJaëck,LaurentMazliak,EmmaSallentDelColombo,RossanaTazzioli Contents Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii I Introductiontothecorrespondence . . . . . . . . . . . . . . . . . . . . . . . 1 A Presentationofthecorrespondence . . . . . . . . . . . . . . . . . . . . . 3 B GöstaMittag-LefflerbeforehisfirstmeetingwithVolterra . . . . . . . . 4 C ThefirstmeetingbetweenMittag-LefflerandVolterra . . . . . . . . . . . 8 D VolterrainDini’sfootsteps . . . . . . . . . . . . . . . . . . . . . . . . . 9 E FoundationofthejournalActaMathematica . . . . . . . . . . . . . . . . 13 F FromDinitoBetti,1882–1887 . . . . . . . . . . . . . . . . . . . . . . . 16 G Functionalanalysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 H Thesecondmeeting. Guccia’sroleandVolterra’sinternationaldébuts . . 20 I 1888–1902: Abel’smanuscripts . . . . . . . . . . . . . . . . . . . . . . 26 J 1891: SonyaKovalevskaya’serror . . . . . . . . . . . . . . . . . . . . . 28 K 1896: Inversionofintegrals . . . . . . . . . . . . . . . . . . . . . . . . . 31 L 1898: Mittag-Leffler’sstaranditsprehistory . . . . . . . . . . . . . . . . 36 M Mittag-Leffler’sengagementfortheNobelPrizeinPhysics . . . . . . . . 40 N 1906: Volterra’slecturesinStockholm . . . . . . . . . . . . . . . . . . . 44 O 1910: Integro-differentialequations . . . . . . . . . . . . . . . . . . . . 46 P WorldWarIanditsconsequences,1914–1920 . . . . . . . . . . . . . . . 55 II Letters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423