Letters to the Editor

Total Page:16

File Type:pdf, Size:1020Kb

Letters to the Editor letters.qxp 4/22/98 9:02 AM Page 405 Letters to the Editor Corrections on Mac Lane’s In this context it is not worth men- (iii) Max Dehn was never in Berlin Article tioning the missing accents on Brün- as a professor, but was “habilitated” Saunders Mac Lane’s memories of ing and Pólya or the curious spelling in Münster, and then was successively mathematical life in Göttingen imme- “Bertold” for Bertolt (Brecht). in Kiel, Breslau, and Frankfurt (where diately after January 30, 1933 (Notices, Mac Lane also gets some historical he succeeded Bieberbach in 1921). He vol. 42, October 1995, 1134–1138), things wrong. fled in 1939, first to Denmark, and in when he was a foreign student there, (i) Moritz Geiger did not serve in the a very arduous way finally came to the are inevitably interesting, as he is one First World War—he was dismissed on United States. of the few surviving mathematicians September 26, 1933 (one month after (iv) Richard Brauer was not in from that time. However, either Pro- Mac Lane’s departure). Berlin; his elder brother, Alfred (a well- fessor Mac Lane, or more likely the (ii) Mac Lane must have misunder- known number theorist, though not as Notices editor responsible, should have stood Martin Kneser about his father- famous as his younger brother), was. When Richard emigrated in October done a better job of editing these rem- in-law. It was a great-great-grand- 1933, he was a professor at Königs- iniscences. To wit: mother of Hasse who was Jewish (her berg. Mac Lane may have been con- “Richard Pohl” should be Robert name was Itzig). I was shown this doc- fused by the fact that both Alfred and Pohl. umentation by Martin Kneser in 1988. Richard did doctoral dissertations in “Hans Freudental” should be Hans The “non-Aryan rules” for being a Freudenthal. Berlin. member of the NSDAP were far “Edward Tornier” should be Erhard (v) I believe Hans Freudenthal left stricter than for anything else and re- Tornier. Berlin in 1931. He certainly did only quired no non-Aryan ancestor alive in “Karl Ludwig Siegel” should be Carl narrowly escape the Nazis, but that 1800. Hasse’s great-great-grand- Ludwig Siegel. was in Holland. Under the picture (p. 1137) “Schw- mother in question was born in 1775. (vi) Hanna Neumann was not Jew- ertfager” should be Schwerdtfeger (his The story of Hasse’s application for ish (her maiden name was vonCaem- first name was Hans). Nazi Party membership is very com- merer). In January 1933 she met Bern- “Erna Barrow” should presumably plicated. Suffice it to say here that he hard Neumann, who was Jewish and be Erna Bannow, who, incidentally, had a brother, Albrecht Hasse, living who emigrated in August of that year later followed Witt to Hamburg, be- in Berlin who had been allowed to to England. She became secretly en- came his first doctoral student (de- join the Party, and so Helmut Hasse gaged to him in 1934; she passed the gree awarded in 1939), and married had reason to believe he might be Staatsexamen while still at Berlin and him in 1940. able to do so as well. in 1937 actually started work on a APRIL 1996 NOTICES OF THE AMS 405 letters.qxp 4/22/98 9:02 AM Page 406 Letters to the Editor doctorate at Göttingen, to be super- that his “good friend” Gerhard I propose that the AMS take a vised by Hasse. She did not leave Ger- Gentzen “disappeared” when the Rus- stronger, more visible role in the dis- many until July 1938 and married sians arrived in Prague. In fact cussion of the problems of teaching Bernhard secretly later that year. For Gentzen refused to voluntarily give up mathematics. I suggest the following more detail, her obituary in volume his university position in early April action: 17 of the Journal of the Australian 1945; arrested in May, he was placed 1. The Notices editorial board Mathematical Society should be con- in a detention camp, where appar- should invite a series of feature arti- sulted. ently he was murdered by Czech (not cles highlighting the kinds of change (vii) Concerning the story about Russian) soldiers. which are being implemented. There Pólya, since various versions of it cir- are many serious mathematicians who culate (e.g., see also p. 26 of the recent Sanford L. Segal have taken a deep interest in this ac- book George Pólya, Master of Discov- University of Rochester tivity in recent years. Why have they ery, by Harold and Loretta Taylor), the done that? Why do they continue to do (Received December 13, 1995) truth is perhaps worth presenting. On it? Is it, as Professor Krantz suggests, P.S. In Mac Lane’s recent piece in January 18, 1921, Pólya wrote Bieber- because that’s where the big bucks the Mathematical Intelligencer (vol. 16, bach (then in Frankfurt) a lengthy let- are? I don’t think so. Let’s find out. no. 3 (1994), 9–10) an error relevant ter from Zürich (a copy is in my pos- 2. The Society should schedule me- to this letter also occurs. The “math- session) in which the incident alluded diated discussions at AMS meetings on ematical anti-Semite” who said to by both Mac Lane and the Taylors the issues of change. The participants “Princeton ist ein kleines Negerdorf” (following a taped interview with should represent various positions on was not in Berlin and was not in- Pólya’s nephew) is described. Inci- the questions of change, and these significant. He was the great geome- dentally, the context of this letter is should not simply be panel discus- ter Wilhelm Blaschke, who was in that Pólya might be considered (the sions. They should also not degener- Hamburg. chance, in fact, was very remote) as ate into debates, since debate implies Bieberbach’s successor when he went confrontation, and in debate, victory to Berlin. Presumably Pólya in 1921 is tends to go to the glib participant. rather more accurate about something Innovations in Mathematics The goal should be information for that happened in 1913 than other peo- Education the membership. I suggest discussion ple’s secondhand memories many The AMS has been presented with an mediated by persons who are trained years later. So far as I know this is the opportunity for positive action and for that activity. first time this story has become pub- change by Steven Krantz. He has pre- 3. The Committee on Education lic. Christmas 1913, Pólya was travel- sented readers of AMS publications should better highlight the Society’s ling from Zürich to Frankfurt and had with a collection of singularly unpro- activities related to education at its an exchange of words with the young fessional diatribes opposing change in Web site, and the Committee on Edu- man sitting opposite him in the train the way we teach mathematics and cation should attempt liaison with compartment over Pólya’s trunk, attacking those who promote change. other organizations interested in the which had fallen down. Pólya, who was The most recent, in “Math for Sale”, same problems. in an “overly irritated state”, chal- the editorial in the October 1995 issue Seven years ago Jerry Uhl and Ho- lenged the fellow to a duel. He refused, of the Notices, carries the authority of racio Porta decided to try using new where upon Pólya punched him. It the Society, since it was written by an tools and ideas in teaching calculus to turned out that his unwilling oppo- editor and published by the Society. undergraduates. The Calculus&Math- nent was the son of an important man Do you know that the Society has ematica project was born. I came along and a student in Göttingen. Pólya had a Committee on Education? Do you shortly after they started. We tried to leave Göttingen as a consequence. know what it is doing? You can find these things and were very excited by Pólya says, “privately the story is also out by going to the AMS home page what we saw: first-year undergrads not worth a defense.” Thus, there were on the World Wide Web. Do you know (and not just honors students) talking no anti-Semitic remarks as in the Tay- that the Society has a listserv for dis- about mathematics in ways we hadn’t lors’ version, and Pólya was the one cussion of calculus reform issues? seen before, students taking charge of who demanded a duel rather than the Will you find reference to it on the their own explorations into mathe- other way around. Pólya blames him- AMS home page? Not as this is being matics, and students who could and self completely for the incident. Inci- written. Is the Society serious about did explain in their own terms what dentally, Pólya does mention anti- addressing problems in the teaching the calculus does and is. We were Semitism elsewhere in the letter, so of mathematics? hooked. We have continued to invent there is every reason to believe this ver- I believe that the Society and its and pursue change, and we are still ex- sion. Of course, neither Mac Lane nor membership have a vital interest in cited by what we see happening. Martin Kneser could have known about questions about teaching mathemat- Changing the way we teach math- this. ics. The questions are difficult, and ematics is a serious enterprise and In this connection, one must also points of view vary, but there is strong should not be demeaned by an opin- mention Mac Lane’s letter in the same evidence to suggest that general im- ionated, uninformed member of the issue of the Notices, when he hints provement is possible.
Recommended publications
  • Tōhoku Rick Jardine
    INFERENCE / Vol. 1, No. 3 Tōhoku Rick Jardine he publication of Alexander Grothendieck’s learning led to great advances: the axiomatic description paper, “Sur quelques points d’algèbre homo- of homology theory, the theory of adjoint functors, and, of logique” (Some Aspects of Homological Algebra), course, the concepts introduced in Tōhoku.5 Tin the 1957 number of the Tōhoku Mathematical Journal, This great paper has elicited much by way of commen- was a turning point in homological algebra, algebraic tary, but Grothendieck’s motivations in writing it remain topology and algebraic geometry.1 The paper introduced obscure. In a letter to Serre, he wrote that he was making a ideas that are now fundamental; its language has with- systematic review of his thoughts on homological algebra.6 stood the test of time. It is still widely read today for the He did not say why, but the context suggests that he was clarity of its ideas and proofs. Mathematicians refer to it thinking about sheaf cohomology. He may have been think- simply as the Tōhoku paper. ing as he did, because he could. This is how many research One word is almost always enough—Tōhoku. projects in mathematics begin. The radical change in Gro- Grothendieck’s doctoral thesis was, by way of contrast, thendieck’s interests was best explained by Colin McLarty, on functional analysis.2 The thesis contained important who suggested that in 1953 or so, Serre inveigled Gro- results on the tensor products of topological vector spaces, thendieck into working on the Weil conjectures.7 The Weil and introduced mathematicians to the theory of nuclear conjectures were certainly well known within the Paris spaces.
    [Show full text]
  • Roots of L-Functions of Characters Over Function Fields, Generic
    Roots of L-functions of characters over function fields, generic linear independence and biases Corentin Perret-Gentil Abstract. We first show joint uniform distribution of values of Kloost- erman sums or Birch sums among all extensions of a finite field Fq, for ˆ almost all couples of arguments in Fq , as well as lower bounds on dif- ferences. Using similar ideas, we then study the biases in the distribu- tion of generalized angles of Gaussian primes over function fields and primes in short intervals over function fields, following recent works of Rudnick–Waxman and Keating–Rudnick, building on cohomological in- terpretations and determinations of monodromy groups by Katz. Our results are based on generic linear independence of Frobenius eigenval- ues of ℓ-adic representations, that we obtain from integral monodromy information via the strategy of Kowalski, which combines his large sieve for Frobenius with a method of Girstmair. An extension of the large sieve is given to handle wild ramification of sheaves on varieties. Contents 1. Introduction and statement of the results 1 2. Kloosterman sums and Birch sums 12 3. Angles of Gaussian primes 14 4. Prime polynomials in short intervals 20 5. An extension of the large sieve for Frobenius 22 6. Generic maximality of splitting fields and linear independence 33 7. Proof of the generic linear independence theorems 36 References 37 1. Introduction and statement of the results arXiv:1903.05491v2 [math.NT] 17 Dec 2019 Throughout, p will denote a prime larger than 5 and q a power of p. ˆ 1.1. Kloosterman and Birch sums.
    [Show full text]
  • Publications of Members, 1930-1954
    THE INSTITUTE FOR ADVANCED STUDY PUBLICATIONS OF MEMBERS 1930 • 1954 PRINCETON, NEW JERSEY . 1955 COPYRIGHT 1955, BY THE INSTITUTE FOR ADVANCED STUDY MANUFACTURED IN THE UNITED STATES OF AMERICA BY PRINCETON UNIVERSITY PRESS, PRINCETON, N.J. CONTENTS FOREWORD 3 BIBLIOGRAPHY 9 DIRECTORY OF INSTITUTE MEMBERS, 1930-1954 205 MEMBERS WITH APPOINTMENTS OF LONG TERM 265 TRUSTEES 269 buH FOREWORD FOREWORD Publication of this bibliography marks the 25th Anniversary of the foundation of the Institute for Advanced Study. The certificate of incorporation of the Institute was signed on the 20th day of May, 1930. The first academic appointments, naming Albert Einstein and Oswald Veblen as Professors at the Institute, were approved two and one- half years later, in initiation of academic work. The Institute for Advanced Study is devoted to the encouragement, support and patronage of learning—of science, in the old, broad, undifferentiated sense of the word. The Institute partakes of the character both of a university and of a research institute j but it also differs in significant ways from both. It is unlike a university, for instance, in its small size—its academic membership at any one time numbers only a little over a hundred. It is unlike a university in that it has no formal curriculum, no scheduled courses of instruction, no commitment that all branches of learning be rep- resented in its faculty and members. It is unlike a research institute in that its purposes are broader, that it supports many separate fields of study, that, with one exception, it maintains no laboratories; and above all in that it welcomes temporary members, whose intellectual development and growth are one of its principal purposes.
    [Show full text]
  • License Or Copyright Restrictions May Apply to Redistribution; See Https
    License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use EMIL ARTIN BY RICHARD BRAUER Emil Artin died of a heart attack on December 20, 1962 at the age of 64. His unexpected death came as a tremendous shock to all who knew him. There had not been any danger signals. It was hard to realize that a person of such strong vitality was gone, that such a great mind had been extinguished by a physical failure of the body. Artin was born in Vienna on March 3,1898. He grew up in Reichen- berg, now Tschechoslovakia, then still part of the Austrian empire. His childhood seems to have been lonely. Among the happiest periods was a school year which he spent in France. What he liked best to remember was his enveloping interest in chemistry during his high school days. In his own view, his inclination towards mathematics did not show before his sixteenth year, while earlier no trace of mathe­ matical aptitude had been apparent.1 I have often wondered what kind of experience it must have been for a high school teacher to have a student such as Artin in his class. During the first world war, he was drafted into the Austrian Army. After the war, he studied at the University of Leipzig from which he received his Ph.D. in 1921. He became "Privatdozent" at the Univer­ sity of Hamburg in 1923.
    [Show full text]
  • Emil Artin in America
    MATHEMATICAL PERSPECTIVES BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 50, Number 2, April 2013, Pages 321–330 S 0273-0979(2012)01398-8 Article electronically published on December 18, 2012 CREATING A LIFE: EMIL ARTIN IN AMERICA DELLA DUMBAUGH AND JOACHIM SCHWERMER 1. Introduction In January 1933, Adolf Hitler and the Nazi party assumed control of Germany. On 7 April of that year the Nazis created the notion of “non-Aryan descent”.1 “It was only a question of time”, Richard Brauer would later describe it, “until [Emil] Artin, with his feeling for individual freedom, his sense of justice, his abhorrence of physical violence would leave Germany” [5, p. 28]. By the time Hitler issued the edict on 26 January 1937, which removed any employee married to a Jew from their position as of 1 July 1937,2 Artin had already begun to make plans to leave Germany. Artin had married his former student, Natalie Jasny, in 1929, and, since she had at least one Jewish grandparent, the Nazis classified her as Jewish. On 1 October 1937, Artin and his family arrived in America [19, p. 80]. The surprising combination of a Roman Catholic university and a celebrated American mathematician known for his gnarly personality played a critical role in Artin’s emigration to America. Solomon Lefschetz had just served as AMS president from 1935–1936 when Artin came to his attention: “A few days ago I returned from a meeting of the American Mathematical Society where as President, I was particularly well placed to know what was going on”, Lefschetz wrote to the president of Notre Dame on 12 January 1937, exactly two weeks prior to the announcement of the Hitler edict that would influence Artin directly.
    [Show full text]
  • A Century of Mathematics in America, Peter Duren Et Ai., (Eds.), Vol
    Garrett Birkhoff has had a lifelong connection with Harvard mathematics. He was an infant when his father, the famous mathematician G. D. Birkhoff, joined the Harvard faculty. He has had a long academic career at Harvard: A.B. in 1932, Society of Fellows in 1933-1936, and a faculty appointmentfrom 1936 until his retirement in 1981. His research has ranged widely through alge­ bra, lattice theory, hydrodynamics, differential equations, scientific computing, and history of mathematics. Among his many publications are books on lattice theory and hydrodynamics, and the pioneering textbook A Survey of Modern Algebra, written jointly with S. Mac Lane. He has served as president ofSIAM and is a member of the National Academy of Sciences. Mathematics at Harvard, 1836-1944 GARRETT BIRKHOFF O. OUTLINE As my contribution to the history of mathematics in America, I decided to write a connected account of mathematical activity at Harvard from 1836 (Harvard's bicentennial) to the present day. During that time, many mathe­ maticians at Harvard have tried to respond constructively to the challenges and opportunities confronting them in a rapidly changing world. This essay reviews what might be called the indigenous period, lasting through World War II, during which most members of the Harvard mathe­ matical faculty had also studied there. Indeed, as will be explained in §§ 1-3 below, mathematical activity at Harvard was dominated by Benjamin Peirce and his students in the first half of this period. Then, from 1890 until around 1920, while our country was becoming a great power economically, basic mathematical research of high quality, mostly in traditional areas of analysis and theoretical celestial mechanics, was carried on by several faculty members.
    [Show full text]
  • National Academy of Sciences July 1, 1979 Officers
    NATIONAL ACADEMY OF SCIENCES JULY 1, 1979 OFFICERS Term expires President-PHILIP HANDLER June 30, 1981 Vice-President-SAUNDERS MAC LANE June 30, 1981 Home Secretary-BRYCE CRAWFORD,JR. June 30, 1983 Foreign Secretary-THOMAS F. MALONE June 30, 1982 Treasurer-E. R. PIORE June 30, 1980 Executive Officer Comptroller Robert M. White David Williams COUNCIL Abelson, Philip H. (1981) Markert,C. L. (1980) Berg, Paul (1982) Nierenberg,William A. (1982) Berliner, Robert W. (1981) Piore, E. R. (1980) Bing, R. H. (1980) Ranney, H. M. (1980) Crawford,Bryce, Jr. (1983) Simon, Herbert A. (1981) Friedman, Herbert (1982) Solow, R. M. (1980) Handler, Philip (1981) Thomas, Lewis (1982) Mac Lane, Saunders (1981) Townes, Charles H. (1981) Malone, Thomas F. (1982) Downloaded by guest on September 30, 2021 SECTIONS The Academyis divided into the followingSections, to which membersare assigned at their own choice: (11) Mathematics (31) Engineering (12) Astronomy (32) Applied Biology (13) Physics (33) Applied Physical and (14) Chemistry Mathematical Sciences (15) Geology (41) Medical Genetics Hema- (16) Geophysics tology, and Oncology (21) Biochemistry (42) Medical Physiology, En- (22) Cellularand Develop- docrinology,and Me- mental Biology tabolism (23) Physiological and Phar- (43) Medical Microbiology macologicalSciences and Immunology (24) Neurobiology (51) Anthropology (25) Botany (52) Psychology (26) Genetics (53) Social and Political Sci- (27) Population Biology, Evo- ences lution, and Ecology (54) Economic Sciences In the alphabetical list of members,the numbersin parentheses, followingyear of election, indicate the respective Class and Section of the member. CLASSES The members of Sections are grouped in the following Classes: I. Physical and Mathematical Sciences (Sections 11, 12, 13, 14, 15, 16).
    [Show full text]
  • Mathematical Genealogy of the Wellesley College Department Of
    Nilos Kabasilas Mathematical Genealogy of the Wellesley College Department of Mathematics Elissaeus Judaeus Demetrios Kydones The Mathematics Genealogy Project is a service of North Dakota State University and the American Mathematical Society. http://www.genealogy.math.ndsu.nodak.edu/ Georgios Plethon Gemistos Manuel Chrysoloras 1380, 1393 Basilios Bessarion 1436 Mystras Johannes Argyropoulos Guarino da Verona 1444 Università di Padova 1408 Cristoforo Landino Marsilio Ficino Vittorino da Feltre 1462 Università di Firenze 1416 Università di Padova Angelo Poliziano Theodoros Gazes Ognibene (Omnibonus Leonicenus) Bonisoli da Lonigo 1477 Università di Firenze 1433 Constantinople / Università di Mantova Università di Mantova Leo Outers Moses Perez Scipione Fortiguerra Demetrios Chalcocondyles Jacob ben Jehiel Loans Thomas à Kempis Rudolf Agricola Alessandro Sermoneta Gaetano da Thiene Heinrich von Langenstein 1485 Université Catholique de Louvain 1493 Università di Firenze 1452 Mystras / Accademia Romana 1478 Università degli Studi di Ferrara 1363, 1375 Université de Paris Maarten (Martinus Dorpius) van Dorp Girolamo (Hieronymus Aleander) Aleandro François Dubois Jean Tagault Janus Lascaris Matthaeus Adrianus Pelope Johann (Johannes Kapnion) Reuchlin Jan Standonck Alexander Hegius Pietro Roccabonella Nicoletto Vernia Johannes von Gmunden 1504, 1515 Université Catholique de Louvain 1499, 1508 Università di Padova 1516 Université de Paris 1472 Università di Padova 1477, 1481 Universität Basel / Université de Poitiers 1474, 1490 Collège Sainte-Barbe
    [Show full text]
  • Math 126 Lecture 4. Basic Facts in Representation Theory
    Math 126 Lecture 4. Basic facts in representation theory. Notice. Definition of a representation of a group. The theory of group representations is the creation of Frobenius: Georg Frobenius lived from 1849 to 1917 Frobenius combined results from the theory of algebraic equations, geometry, and number theory, which led him to the study of abstract groups, the representation theory of groups and the character theory of groups. Find out more at: http://www-history.mcs.st-andrews.ac.uk/history/ Mathematicians/Frobenius.html Matrix form of a representation. Equivalence of two representations. Invariant subspaces. Irreducible representations. One dimensional representations. Representations of cyclic groups. Direct sums. Tensor product. Unitary representations. Averaging over the group. Maschke’s theorem. Heinrich Maschke 1853 - 1908 Schur’s lemma. Issai Schur Biography of Schur. Issai Schur Born: 10 Jan 1875 in Mogilyov, Mogilyov province, Russian Empire (now Belarus) Died: 10 Jan 1941 in Tel Aviv, Palestine (now Israel) Although Issai Schur was born in Mogilyov on the Dnieper, he spoke German without a trace of an accent, and nobody even guessed that it was not his first language. He went to Latvia at the age of 13 and there he attended the Gymnasium in Libau, now called Liepaja. In 1894 Schur entered the University of Berlin to read mathematics and physics. Frobenius was one of his teachers and he was to greatly influence Schur and later to direct his doctoral studies. Frobenius and Burnside had been the two main founders of the theory of representations of groups as groups of matrices. This theory proved a very powerful tool in the study of groups and Schur was to learn the foundations of this subject from Frobenius.
    [Show full text]
  • Mathematisches Forschungsinstitut Oberwolfach Emigration Of
    Mathematisches Forschungsinstitut Oberwolfach Report No. 51/2011 DOI: 10.4171/OWR/2011/51 Emigration of Mathematicians and Transmission of Mathematics: Historical Lessons and Consequences of the Third Reich Organised by June Barrow-Green, Milton-Keynes Della Fenster, Richmond Joachim Schwermer, Wien Reinhard Siegmund-Schultze, Kristiansand October 30th – November 5th, 2011 Abstract. This conference provided a focused venue to explore the intellec- tual migration of mathematicians and mathematics spurred by the Nazis and still influential today. The week of talks and discussions (both formal and informal) created a rich opportunity for the cross-fertilization of ideas among almost 50 mathematicians, historians of mathematics, general historians, and curators. Mathematics Subject Classification (2000): 01A60. Introduction by the Organisers The talks at this conference tended to fall into the two categories of lists of sources and historical arguments built from collections of sources. This combi- nation yielded an unexpected richness as new archival materials and new angles of investigation of those archival materials came together to forge a deeper un- derstanding of the migration of mathematicians and mathematics during the Nazi era. The idea of measurement, for example, emerged as a critical idea of the confer- ence. The conference called attention to and, in fact, relied on, the seemingly stan- dard approach to measuring emigration and immigration by counting emigrants and/or immigrants and their host or departing countries. Looking further than this numerical approach, however, the conference participants learned the value of measuring emigration/immigration via other less obvious forms of measurement. 2892 Oberwolfach Report 51/2011 Forms completed by individuals on religious beliefs and other personal attributes provided an interesting cartography of Italian society in the 1930s and early 1940s.
    [Show full text]
  • Lecture Notes C Sarah Rasmussen, 2019
    Part III 3-manifolds Lecture Notes c Sarah Rasmussen, 2019 Contents Lecture 0 (not lectured): Preliminaries2 Lecture 1: Why not ≥ 5?9 Lecture 2: Why 3-manifolds? + Introduction to knots and embeddings 13 Lecture 3: Link diagrams and Alexander polynomial skein relations 17 Lecture 4: Handle decompositions from Morse critical points 20 Lecture 5: Handles as Cells; Morse functions from handle decompositions 24 Lecture 6: Handle-bodies and Heegaard diagrams 28 Lecture 7: Fundamental group presentations from Heegaard diagrams 36 Lecture 8: Alexander polynomials from fundamental groups 39 Lecture 9: Fox calculus 43 Lecture 10: Dehn presentations and Kauffman states 48 Lecture 11: Mapping tori and Mapping Class Groups 54 Lecture 12: Nielsen-Thurston classification for mapping class groups 58 Lecture 13: Dehn filling 61 Lecture 14: Dehn surgery 64 Lecture 15: 3-manifolds from Dehn surgery 68 Lecture 16: Seifert fibered spaces 72 Lecture 17: Hyperbolic manifolds 76 Lecture 18: Embedded surface representatives 80 Lecture 19: Incompressible and essential surfaces 83 Lecture 20: Connected sum 86 Lecture 21: JSJ decomposition and geometrization 89 Lecture 22: Turaev torsion and knot decompositions 92 Lecture 23: Foliations 96 Lecture 24. Taut Foliations 98 Errata: Catalogue of errors/changes/addenda 102 References 106 1 2 Lecture 0 (not lectured): Preliminaries 0. Notation and conventions. Notation. @X { (the manifold given by) the boundary of X, for X a manifold with boundary. th @iX { the i connected component of @X. ν(X) { a tubular (or collared) neighborhood of X in Y , for an embedding X ⊂ Y . ◦ ν(X) { the interior of ν(X). This notation is somewhat redundant, but emphasises openness.
    [Show full text]
  • Fundamental Theorems in Mathematics
    SOME FUNDAMENTAL THEOREMS IN MATHEMATICS OLIVER KNILL Abstract. An expository hitchhikers guide to some theorems in mathematics. Criteria for the current list of 243 theorems are whether the result can be formulated elegantly, whether it is beautiful or useful and whether it could serve as a guide [6] without leading to panic. The order is not a ranking but ordered along a time-line when things were writ- ten down. Since [556] stated “a mathematical theorem only becomes beautiful if presented as a crown jewel within a context" we try sometimes to give some context. Of course, any such list of theorems is a matter of personal preferences, taste and limitations. The num- ber of theorems is arbitrary, the initial obvious goal was 42 but that number got eventually surpassed as it is hard to stop, once started. As a compensation, there are 42 “tweetable" theorems with included proofs. More comments on the choice of the theorems is included in an epilogue. For literature on general mathematics, see [193, 189, 29, 235, 254, 619, 412, 138], for history [217, 625, 376, 73, 46, 208, 379, 365, 690, 113, 618, 79, 259, 341], for popular, beautiful or elegant things [12, 529, 201, 182, 17, 672, 673, 44, 204, 190, 245, 446, 616, 303, 201, 2, 127, 146, 128, 502, 261, 172]. For comprehensive overviews in large parts of math- ematics, [74, 165, 166, 51, 593] or predictions on developments [47]. For reflections about mathematics in general [145, 455, 45, 306, 439, 99, 561]. Encyclopedic source examples are [188, 705, 670, 102, 192, 152, 221, 191, 111, 635].
    [Show full text]