772 Topology, Geometry, and Dynamics V. A. Rokhlin-Memorial Conference on Topology, Geometry, and Dynamics: V. A. Rokhlin-100 August 19–23, 2019 The Euler International Mathematical Institute, St. Petersburg, Russia and Steklov Institute of Mathematics, St. Petersburg, Russia Anatoly M. Vershik Victor M. Buchstaber Andrey V. Malyutin Editors Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms Topology, Geometry, and Dynamics V. A. Rokhlin-Memorial Conference on Topology, Geometry, and Dynamics: V. A. Rokhlin-100 August 19–23, 2019 The Euler International Mathematical Institute, St. Petersburg, Russia and Steklov Institute of Mathematics, St. Petersburg, Russia Anatoly M. Vershik Victor M. Buchstaber Andrey V. Malyutin Editors Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms 772 Topology, Geometry, and Dynamics V. A. Rokhlin-Memorial Conference on Topology, Geometry, and Dynamics: V. A. Rokhlin-100 August 19–23, 2019 The Euler International Mathematical Institute, St. Petersburg, Russia and Steklov Institute of Mathematics, St. Petersburg, Russia Anatoly M. Vershik Victor M. Buchstaber Andrey V. Malyutin Editors Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms EDITORIAL COMMITTEE Dennis DeTurck, Managing Editor Michael Loss Kailash Misra Catherine Yan 2020 Mathematics Subject Classification. Primary 14A20, 28D05, 52A22, 55N15, 55N22, 57K10, 57K41. Library of Congress Cataloging-in-Publication Data Names: Conference on Topology, Geometry, and Dynamics: V.A. Rokhlin-100 (2019 : Saint Petersburg,Russia),author. |Vershik,A.M.(Anatol˘ıiMoiseevich),1933–editor. |Buchstaber,V. M.,editor. |Malyutin,AndreyV.,1975–editor. |Rokhlin,V.A.,honouree. |EulerInternational MathematicalInstitute,organizer. |Matematichesk˘ıiinstitutim. V.A.Steklova,organizer. Title: Topology, geometry, and dynamics : V. A. Rokhlin-memorial : Conference on Topology, Geometry, and Dynamics: V. A. Rokhlin-100, August 19-23, 2019, The Euler International MathematicalInstitute,St. Petersburg,Russia,SteklovInstituteofMathematics,St. Peters- burg,Russia/AnatolyM.Vershik,VictorM.Buchstaber,AndreyV.Malyutin,editors. Other titles: Correcttitle: Topology,geometry,anddynamics Description: Providence, Rhode Island : American Mathematical Society, [2021] | Series: Con- temporarymathematics,0271-4132;volume772|Includesbibliographicalreferences. Identifiers: LCCN2020045197|ISBN9781470456641(paperback)|ISBN9781470464516(ebook) Subjects: LCSH: Rokhlin, V. A.–Congresses. | Geometry–Congresses. | Topology–Congresses. | Dynamics–Congresses. | AMS: Algebraic geometry – Foundations – Generalizations (algebraic spaces, stacks). | Measure and integration {For analysis on manifolds, see 58-XX.} – Measure- theoreticergodictheory[Seealso11K50,11K55,22D40,37Axx,47A35,54H20,60Fxx,60G10]– Measure-preservingtransformation|Convexanddiscretegeometry–Generalconvexity–Random convex sets and integral geometry [See also 53C65, 60D05]. | Algebraic topology – Homology and cohomology theories [See also 57Txx] – K-theory [See also 19Lxx] For algebraic K-theory, see18F25,19-XX.|Algebraictopology–Homologyandcohomologytheories[Seealso57Txx]– Bordismandcobordismtheories,formalgrouplaws[Seealso14L05,19L41,57R75,57R77,57R85, 57R90]. Classification: LCCQA440.C6542021|DDC516.3/5–dc23 LCrecordavailableathttps://lccn.loc.gov/2020045197 ContemporaryMathematicsISSN:0271-4132(print);ISSN:1098-3627(online) DOI:https://doi.org/10.1090/conm/772 Colorgraphicpolicy. Anygraphicscreatedincolorwillberenderedingrayscalefortheprinted versionunlesscolorprintingisauthorizedbythePublisher. Ingeneral,colorgraphicswillappear incolorintheonlineversion. Copying and reprinting. Individual readersofthispublication,andnonprofit librariesacting for them, are permitted to make fair use of the material, such as to copy select pages for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews,providedthecustomaryacknowledgmentofthesourceisgiven. Republication,systematiccopying,ormultiplereproductionofanymaterialinthispublication ispermittedonlyunderlicensefromtheAmericanMathematicalSociety. Requestsforpermission toreuseportionsofAMSpublicationcontentarehandledbytheCopyrightClearanceCenter. For moreinformation,pleasevisitwww.ams.org/publications/pubpermissions. Sendrequestsfortranslationrightsandlicensedreprintstoreprint-permission@ams.org. (cid:2)c 2021bytheAmericanMathematicalSociety. Allrightsreserved. TheAmericanMathematicalSocietyretainsallrights exceptthosegrantedtotheUnitedStatesGovernment. PrintedintheUnitedStatesofAmerica. (cid:2)∞ Thepaperusedinthisbookisacid-freeandfallswithintheguidelines establishedtoensurepermanenceanddurability. VisittheAMShomepageathttps://www.ams.org/ 10987654321 262524232221 Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms Contents Preface vii V. A. Rokhlin (23 August 1919 – 3 December 1984), materials for the biography A. M. Vershik 1 Teaching mathematics to non-mathematicians V. A. Rokhlin 19 Vladimir Abramovich Rokhlin and algebraic topology V. M. Buchstaber 33 Amenability of groupoids and asymptotic invariance of convolution powers T. Bu¨hler and V. A. Kaimanovich 69 Slopes of links and signature formulas A. Degtyarev, V. Florens, and A. G. Lecuona 93 B-rigidity of the property to be an almost Pogorelov polytope N. Yu. Erokhovets 107 The first homology of a real cubic is generated by lines S. Finashin and V. Kharlamov 123 Circular orders, ultra-homogeneous order structures, and their automorphism groups E. Glasner and M. Megrelishvili 133 Convergence of equilibrium measures corresponding to finite subgraphs of infinite graphs: New examples B. M. Gurevich 155 Anti-symplectic involutions on rational symplectic 4-manifolds V. Kharlamov and V. Shevchishin 165 Dolbeault cohomology of complex manifolds with torus action R. Krutowski and T. Panov 173 Poincar´e polynomials of generic torus orbit closures in Schubert varieties E. Lee, M. Masuda, S. Park, and J. Song 189 Higher order Massey products and applications I. Limonchenko and D. Millionshchikov 209 Discreteness of deformations of cocompact discrete subgroups G. A. Margulis and G. A. Soifer 241 v Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms vi CONTENTS Topological isotopy and Cochran’s derived invariants S. A. Melikhov 249 Geometric description of the Hochschild cohomology of group algebras A. S. Mishchenko 267 A user’s guide to basic knot and link theory A. Skopenkov 281 Group actions: Entropy, mixing, spectra, and generic properties A. Stepin and S. Tikhonov 311 Rokhlin’s theorem, a problem and a conjecture D. Sullivan 325 Maximally inflected real trigonal curves on Hirzebruch surfaces V. I. Zvonilov 331 Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms Preface This volume in the Contemporary Mathematics series of the American Math- ematical Society is dedicated to the memory of the outstanding Russian mathe- matician Vladimir Abramovich Rokhlin (23 August 1919 – 3 December 1984) on the occasion of the 100th anniversary of his birth. To celebrate this anniversary, SteklovInstituteofMathematicsanditsSt.PetersburgDepartment, togetherwith the Euler International Mathematical Institute, organized an international confer- ence Topology, Geometry, Dynamics: Rokhlin-100, which was held in St. Peters- burg, Russia, August 19–23, 2019. The volume contains papers based on keynote talks of this conference. Their subjects, in accordance with the title of the volume, cover modern topics in topology, the theory of dynamical systems, geometry, and other areas of mathematics related to Rokhlin’s broad research interests. Further information on the conference (program, list of participants, etc.) can be found at http://www.pdmi.ras.ru/EIMI/2019/tgd/index.html. Some of the talks and papers in this volume directly discuss the seminal role of Rokhlin’s work in the subsequent development of mathematics. His talent for teachingisreflectedinhisbrilliantlecture,publishedinthevolume,aboutteaching mathematics to non-mathematicians. We also include a paper about Rokhlin’s astounding biography, which seems to absorb all hardships and tragedies of the 20th century. Unfortunately, limited space prevented us from presenting a fuller picture of the modern areas of mathematics influenced by Rokhlin. We hope to do this in other publications. AsignificantcontributiontotheworkonthisvolumewasmadebyV.M.Buch- staber, A. V. Malyutin, T. E. Panov, N. V. Tsilevich, P. B. Zatitskii, F. V. Petrov, P. N. Nikitin, N. N. Vasiliev. In addition to the articles by D. Sullivan and V. Buchstaber published in this volume,IwouldliketomentionotherimportantpublicationsdiscussingV.A.Rokh- lin’s remarkable contribution to mathematics; see the references below. A. M. Vershik July 15, 2020 References [1] V. A. Rokhlin, Selected Works, 2nd edition. Ed.: A. M. Vershik. MC- CME, 2010, 572 p., ISBN 978-5-94057-518-4, in Russian. [2] Topology, Ergodic Theory, Real Algebraic Geometry: Rokhlin’s Memorial. Eds.: V. Turaev, A. Vershik. Steklov Mathematical Institute, St. Pe- tersburg, Russia. American Mathematical Society Translations, Series 2, Volume 202, 2001, 286 p. vii Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms viii PREFACE [3] J. Milnor, Topology through the centuries: Low dimensional manifolds. Bulletin of the American Mathematical Society, Volume 52, Number 4, 2015, pp. 545–584. [4] B.Weiss,On theworkof V. A. Rokhlininergodictheory. ErgodicTheory and Dynamical Systems, Volume 9, Issue 4, 1989, pp. 619–627. [5] O. Ya. Viro, V. M. Kharlamov, V. A. Rokhlin’s work in topology. Alge- bra i Analiz, Volume 2, Issue 2, 1990, pp. 242–248, in Russian. English translation: Leningrad Mathematical Journal, Volume 2, Issue 2, 1991, pp. 433–438. [6] A. M. Vershik, V. A. Rokhlin’s work in ergodic theory. Algebra i Analiz, Volume 2, Issue 2, 1990, pp. 236–241, in Russian. English translation: Leningrad Mathematical Journal, Volume 2, Issue 2, 1991, pp. 427–431. Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms