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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. -
Of the American Mathematical Society August 2017 Volume 64, Number 7
ISSN 0002-9920 (print) ISSN 1088-9477 (online) of the American Mathematical Society August 2017 Volume 64, Number 7 The Mathematics of Gravitational Waves: A Two-Part Feature page 684 The Travel Ban: Affected Mathematicians Tell Their Stories page 678 The Global Math Project: Uplifting Mathematics for All page 712 2015–2016 Doctoral Degrees Conferred page 727 Gravitational waves are produced by black holes spiraling inward (see page 674). American Mathematical Society LEARNING ® MEDIA MATHSCINET ONLINE RESOURCES MATHEMATICS WASHINGTON, DC CONFERENCES MATHEMATICAL INCLUSION REVIEWS STUDENTS MENTORING PROFESSION GRAD PUBLISHING STUDENTS OUTREACH TOOLS EMPLOYMENT MATH VISUALIZATIONS EXCLUSION TEACHING CAREERS MATH STEM ART REVIEWS MEETINGS FUNDING WORKSHOPS BOOKS EDUCATION MATH ADVOCACY NETWORKING DIVERSITY blogs.ams.org Notices of the American Mathematical Society August 2017 FEATURED 684684 718 26 678 Gravitational Waves The Graduate Student The Travel Ban: Affected Introduction Section Mathematicians Tell Their by Christina Sormani Karen E. Smith Interview Stories How the Green Light was Given for by Laure Flapan Gravitational Wave Research by Alexander Diaz-Lopez, Allyn by C. Denson Hill and Paweł Nurowski WHAT IS...a CR Submanifold? Jackson, and Stephen Kennedy by Phillip S. Harrington and Andrew Gravitational Waves and Their Raich Mathematics by Lydia Bieri, David Garfinkle, and Nicolás Yunes This season of the Perseid meteor shower August 12 and the third sighting in June make our cover feature on the discovery of gravitational waves -
Hilbert Constance Reid
Hilbert Constance Reid Hilbert c COPERNICUS AN IMPRINT OF SPRINGER-VERLAG © 1996 Springer Science+Business Media New York Originally published by Springer-Verlag New York, Inc in 1996 All rights reserved. No part ofthis publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publisher. Library ofCongress Cataloging·in-Publication Data Reid, Constance. Hilbert/Constance Reid. p. Ctn. Originally published: Berlin; New York: Springer-Verlag, 1970. Includes bibliographical references and index. ISBN 978-0-387-94674-0 ISBN 978-1-4612-0739-9 (eBook) DOI 10.1007/978-1-4612-0739-9 I. Hilbert, David, 1862-1943. 2. Mathematicians-Germany Biography. 1. Title. QA29.HsR4 1996 SIO'.92-dc20 [B] 96-33753 Manufactured in the United States of America. Printed on acid-free paper. 9 8 7 6 543 2 1 ISBN 978-0-387-94674-0 SPIN 10524543 Questions upon Rereading Hilbert By 1965 I had written several popular books, such as From liro to Infinity and A Long Way from Euclid, in which I had attempted to explain certain easily grasped, although often quite sophisticated, mathematical ideas for readers very much like myself-interested in mathematics but essentially untrained. At this point, with almost no mathematical training and never having done any bio graphical writing, I became determined to write the life of David Hilbert, whom many considered the profoundest mathematician of the early part of the 20th century. Now, thirty years later, rereading Hilbert, certain questions come to my mind. -
Early History of the Riemann Hypothesis in Positive Characteristic
The Legacy of Bernhard Riemann c Higher Education Press After One Hundred and Fifty Years and International Press ALM 35, pp. 595–631 Beijing–Boston Early History of the Riemann Hypothesis in Positive Characteristic Frans Oort∗ , Norbert Schappacher† Abstract The classical Riemann Hypothesis RH is among the most prominent unsolved prob- lems in modern mathematics. The development of Number Theory in the 19th century spawned an arithmetic theory of polynomials over finite fields in which an analogue of the Riemann Hypothesis suggested itself. We describe the history of this topic essentially between 1920 and 1940. This includes the proof of the ana- logue of the Riemann Hyothesis for elliptic curves over a finite field, and various ideas about how to generalize this to curves of higher genus. The 1930ies were also a period of conflicting views about the right method to approach this problem. The later history, from the proof by Weil of the Riemann Hypothesis in charac- teristic p for all algebraic curves over a finite field, to the Weil conjectures, proofs by Grothendieck, Deligne and many others, as well as developments up to now are described in the second part of this diptych: [44]. 2000 Mathematics Subject Classification: 14G15, 11M99, 14H52. Keywords and Phrases: Riemann Hypothesis, rational points over a finite field. Contents 1 From Richard Dedekind to Emil Artin 597 2 Some formulas for zeta functions. The Riemann Hypothesis in characteristic p 600 3 F.K. Schmidt 603 ∗Department of Mathematics, Utrecht University, Princetonplein 5, 3584 CC -
Fundamental Algebraic Geometry
http://dx.doi.org/10.1090/surv/123 hematical Surveys and onographs olume 123 Fundamental Algebraic Geometry Grothendieck's FGA Explained Barbara Fantechi Lothar Gottsche Luc lllusie Steven L. Kleiman Nitin Nitsure AngeloVistoli American Mathematical Society U^VDED^ EDITORIAL COMMITTEE Jerry L. Bona Peter S. Landweber Michael G. Eastwood Michael P. Loss J. T. Stafford, Chair 2000 Mathematics Subject Classification. Primary 14-01, 14C20, 13D10, 14D15, 14K30, 18F10, 18D30. For additional information and updates on this book, visit www.ams.org/bookpages/surv-123 Library of Congress Cataloging-in-Publication Data Fundamental algebraic geometry : Grothendieck's FGA explained / Barbara Fantechi p. cm. — (Mathematical surveys and monographs, ISSN 0076-5376 ; v. 123) Includes bibliographical references and index. ISBN 0-8218-3541-6 (pbk. : acid-free paper) ISBN 0-8218-4245-5 (soft cover : acid-free paper) 1. Geometry, Algebraic. 2. Grothendieck groups. 3. Grothendieck categories. I Barbara, 1966- II. Mathematical surveys and monographs ; no. 123. QA564.F86 2005 516.3'5—dc22 2005053614 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA. -
Mathematicians Fleeing from Nazi Germany
Mathematicians Fleeing from Nazi Germany Mathematicians Fleeing from Nazi Germany Individual Fates and Global Impact Reinhard Siegmund-Schultze princeton university press princeton and oxford Copyright 2009 © by Princeton University Press Published by Princeton University Press, 41 William Street, Princeton, New Jersey 08540 In the United Kingdom: Princeton University Press, 6 Oxford Street, Woodstock, Oxfordshire OX20 1TW All Rights Reserved Library of Congress Cataloging-in-Publication Data Siegmund-Schultze, R. (Reinhard) Mathematicians fleeing from Nazi Germany: individual fates and global impact / Reinhard Siegmund-Schultze. p. cm. Includes bibliographical references and index. ISBN 978-0-691-12593-0 (cloth) — ISBN 978-0-691-14041-4 (pbk.) 1. Mathematicians—Germany—History—20th century. 2. Mathematicians— United States—History—20th century. 3. Mathematicians—Germany—Biography. 4. Mathematicians—United States—Biography. 5. World War, 1939–1945— Refuges—Germany. 6. Germany—Emigration and immigration—History—1933–1945. 7. Germans—United States—History—20th century. 8. Immigrants—United States—History—20th century. 9. Mathematics—Germany—History—20th century. 10. Mathematics—United States—History—20th century. I. Title. QA27.G4S53 2008 510.09'04—dc22 2008048855 British Library Cataloging-in-Publication Data is available This book has been composed in Sabon Printed on acid-free paper. ∞ press.princeton.edu Printed in the United States of America 10 987654321 Contents List of Figures and Tables xiii Preface xvii Chapter 1 The Terms “German-Speaking Mathematician,” “Forced,” and“Voluntary Emigration” 1 Chapter 2 The Notion of “Mathematician” Plus Quantitative Figures on Persecution 13 Chapter 3 Early Emigration 30 3.1. The Push-Factor 32 3.2. The Pull-Factor 36 3.D. -
Academic Genealogy of the Oakland University Department Of
Basilios Bessarion Mystras 1436 Guarino da Verona Johannes Argyropoulos 1408 Università di Padova 1444 Academic Genealogy of the Oakland University Vittorino da Feltre Marsilio Ficino Cristoforo Landino Università di Padova 1416 Università di Firenze 1462 Theodoros Gazes Ognibene (Omnibonus Leonicenus) Bonisoli da Lonigo Angelo Poliziano Florens Florentius Radwyn Radewyns Geert Gerardus Magnus Groote Università di Mantova 1433 Università di Mantova Università di Firenze 1477 Constantinople 1433 DepartmentThe Mathematics Genealogy Project of is a serviceMathematics of North Dakota State University and and the American Statistics Mathematical Society. Demetrios Chalcocondyles http://www.mathgenealogy.org/ Heinrich von Langenstein Gaetano da Thiene Sigismondo Polcastro Leo Outers Moses Perez Scipione Fortiguerra Rudolf Agricola Thomas von Kempen à Kempis Jacob ben Jehiel Loans Accademia Romana 1452 Université de Paris 1363, 1375 Université Catholique de Louvain 1485 Università di Firenze 1493 Università degli Studi di Ferrara 1478 Mystras 1452 Jan Standonck Johann (Johannes Kapnion) Reuchlin Johannes von Gmunden Nicoletto Vernia Pietro Roccabonella Pelope Maarten (Martinus Dorpius) van Dorp Jean Tagault François Dubois Janus Lascaris Girolamo (Hieronymus Aleander) Aleandro Matthaeus Adrianus Alexander Hegius Johannes Stöffler Collège Sainte-Barbe 1474 Universität Basel 1477 Universität Wien 1406 Università di Padova Università di Padova Université Catholique de Louvain 1504, 1515 Université de Paris 1516 Università di Padova 1472 Università -
Doing Mathematics in Emmy Noether's Time
Doing mathematics in Emmy Noether’s time: A survey on the study and work situation of women mathematicians in Germany between the two World Wars Prof.Dr.habil. Ilka Agricola Philipps-Universit¨at Marburg Madrid, April 2019 — a tribute to the 100th anniversary of Noether’s habilitation — 1 What this talk is about. • History of science, because science is done by scientists, so their working and living conditions influence scientific progress • To be seen in historical context: Industrialisation and wars lead to deep changes in society, the formation of a working class, the need for engineers and other professionals, and sometimes a shortage of male employees • Partially based on a case study carried out in Marburg (2017/18) about the first generation of female math students in Marburg (see refs.) • Applies mainly to Germany and, to some extent, neighbouring countries; very different from situation in South Europe and elsewhere 1 Marburg: An old traditional university • Founded in 1527 by Philip I, Landgrave of Hesse • oldest protestant university in the world • Of relevance for us: As a consequence of the Austro-Prussian War (1866), Marburg and G¨ottingen became part of Prussia (the Electorate of Hesse and and the Kingdom of Hannover disappeared) • Hence, Marburg is typical for the development of universities in Germany’s largest state – to be honest, they developed much better after 1866. 1 Das ‘Mathematische Seminar’ • Mathematics was taught in Marburg since the foundation of the university 1527 (Papin, Wolff. ) • 1817: Foundation of the ‘Mathematisch- physikalisches Institut’ by Christian Gerling (student of C.F. Gauß) • 1885: Foudation of the Faculty by H. -
Kronecker Classes of Algebraic Number Fields WOLFRAM
JOURNAL OF NUMBER THFDRY 9, 279-320 (1977) Kronecker Classes of Algebraic Number Fields WOLFRAM JEHNE Mathematisches Institut Universitiit Kiiln, 5 K&I 41, Weyertal 86-90, Germany Communicated by P. Roquette Received December 15, 1974 DEDICATED TO HELMUT HASSE INTRODUCTION In a rather programmatic paper of 1880 Kronecker gave the impetus to a long and fruitful development in algebraic number theory, although no explicit program is formulated in his paper [17]. As mentioned in Hasse’s Zahlbericht [8], he was the first who tried to characterize finite algebraic number fields by the decomposition behavior of the prime divisors of the ground field. Besides a fundamental density relation, Kronecker’s main result states that two finite extensions (of prime degree) have the same Galois hull if every prime divisor of the ground field possesses in both extensions the same number of prime divisors of first relative degree (see, for instance, [8, Part II, Sects. 24, 251). In 1916, Bauer [3] showed in particular that, among Galois extensions, a Galois extension K 1 k is already characterized by the set of all prime divisors of k which decompose completely in K. More generally, following Kronecker, Bauer, and Hasse, for an arbitrary finite extension K I k we consider the set D(K 1 k) of all prime divisors of k having a prime divisor of first relative degree in K; we call this set the Kronecker set of K 1 k. Obviously, in the case of a Galois extension, the Kronecker set coincides with the set just mentioned. In a counterexample Gassmann [4] showed in 1926 that finite extensions. -
Cartan and Complex Analytic Geometry
Cartan and Complex Analytic Geometry Jean-Pierre Demailly effort in France to recruit teachers, due to the much increased access of pupils and students to On the Mathematical Heritage of Henri higher education, along with a strong research ef- Cartan fort in technology and science. I remember quite Henri Cartan left us on August 13, 2008, at the well that my father had a book with a mysterious age of 104. His influence on generations of math- title: Théorie Élémentaire des Fonctions Analytiques ematicians worldwide has been considerable. In d’une ou Plusieurs Variables Complexes (Hermann, France especially, his role as a 4th edition from 1961) [Ca3], by Henri Cartan, professor at École Normale Su- which contained magical stuff such as contour périeure in Paris between 1940 integrals and residues. I could then, of course, and 1965 led him to supervise not understand much of it, but my father was the Ph.D. theses of Jean-Pierre quite absorbed with the book; I was equally im- Serre (Fields Medal 1954), René pressed by the photograph of Cartan on the cover Thom (Fields Medal 1958), and pages and by the style of the contents, which had many other prominent math- obvious similarity to the “New Math” we started ematicians such as Pierre Cart- being taught at school—namely set theory and ier, Jean Cerf, Adrien Douady, symbols like ∪, ∩, ∈, ⊂,... My father explained Roger Godement, Max Karoubi, to me that Henri Cartan was one of the leading and Jean-Louis Koszul. French mathematicians and that he was one of However, rather than rewrit- the founding members of the somewhat secre- ing history that is well known tive Bourbaki group, which had been the source to many people, I would like of inspiration for the new symbolism and for the here to share lesser known reform of education. -
Prof. Alexander Grothendieck - the Math Genius! - 11-19-2014 by Gonit Sora - Gonit Sora
Prof. Alexander Grothendieck - The math genius! - 11-19-2014 by Gonit Sora - Gonit Sora - https://gonitsora.com Prof. Alexander Grothendieck - The math genius! by Gonit Sora - Wednesday, November 19, 2014 https://gonitsora.com/prof-alexander-grothendieck-math-genius/ I was reading a math-related post in Facebook when suddenly the message icon in the top right corner of the facebook page turned red and the speakers of my system made a beep sound. It was a message from my friend, Manjil, asking me to write an obituary of the famous mathematician Alexander Grothendieck. I immediately replied "Yes of course!". Incidentally, the math-related post I mentioned in the first line was a news report on the recent death of the same math genius. I opened an extra tab on my browser and searched for more details about Prof. Grothendieck. And yes, I was shocked to see a huge bunch of info about him on various sites. So, I realised that writing about him is going to be tough! Ultimately I have come up with a brief write up on the life and works of Prof. Grothendieck. I, on behalf of GonitSora, dedicate this write up to the revolutionary mathematician Prof. Alexander Grothendieck. May his soul rest in peace! Alexander Grothendieck Prof. Grothendeick was a French mathematician and the leading figure in creating modern algebraic geometry. His research program vastly extended the scope of the field, incorporating major elements of commutative algebra, homological algebra, sheaf theory, and category theory into its foundations. This new perspective led to revolutionary advances across many areas of pure mathematics. -
Mathematical Perspectives
MATHEMATICAL PERSPECTIVES BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 46, Number 4, October 2009, Pages 661–666 S 0273-0979(09)01264-6 Article electronically published on May 14, 2009 ANDREWEIL´ ARMAND BOREL The mathematician Andr´e Weil died in Princeton at the age of ninety-two, after a gradual decline in his physical, but not mental, capacities. In January 1999, a conference on his work and its influence took place at the Institute for Advanced AndreWeil´ 6 May 1906–6 August 1998 This article is from the Proceedings of the American Philosophical Society, vol. 145, no. 1, March 2001, pp. 108–114. Reprinted with permission. c 2009 American Mathematical Society 661 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 662 ARMAND BOREL Study, the poster of which gave this capsule description of him: A man of formidable intellectual power, moved by a global view and knowledge of mathematics, of its history and a strong belief in its unity, Andr´e Weil has profoundly influenced the course of mathe- matics by the breadth and depth of his publications, his correspon- dence and his leading contributions to the work of N. Bourbaki, a statement of which this text is basically just an elaboration. In attempting to give an impression of Weil’s work and thought processes, I shall have to navigate between the Charybdis of undefined mathematical jargon and the Scylla of vague, seemingly but not necessarily more understandable, statements. I hope the reader will bear with me! Andr´e Weil was born in Paris in a Jewish family, to an Alsatian physician and his Russian-born wife, of Austrian origin.