Alfred Rosenblatt (1880–1947). Polish–Peruvian Mathematician
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Recollections and Notes, Vol. 1 (1887–1945) Translated by Abe
Vita Mathematica 18 Hugo Steinhaus Mathematician for All Seasons Recollections and Notes, Vol. 1 (1887–1945) Translated by Abe Shenitzer Edited by Robert G. Burns, Irena Szymaniec and Aleksander Weron Vita Mathematica Volume 18 Edited by Martin MattmullerR More information about this series at http://www.springer.com/series/4834 Hugo Steinhaus Mathematician for All Seasons Recollections and Notes, Vol. 1 (1887–1945) Translated by Abe Shenitzer Edited by Robert G. Burns, Irena Szymaniec and Aleksander Weron Author Hugo Steinhaus (1887–1972) Translator Abe Shenitzer Brookline, MA, USA Editors Robert G. Burns York University Dept. Mathematics & Statistics Toronto, ON, Canada Irena Szymaniec Wrocław, Poland Aleksander Weron The Hugo Steinhaus Center Wrocław University of Technology Wrocław, Poland Vita Mathematica ISBN 978-3-319-21983-7 ISBN 978-3-319-21984-4 (eBook) DOI 10.1007/978-3-319-21984-4 Library of Congress Control Number: 2015954183 Springer Cham Heidelberg New York Dordrecht London © Springer International Publishing Switzerland 2015 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. -
My Dear Colleague! [1/2] Thank You for the Cards and Congratulations
Ithaca, N.Y. 6.I.1940 [1/1] My Dear Colleague! [1/2] Thank you for the cards and congratulations. I [1/3] have not written you in for a long while because I did not receive [1/4] any answers to two letters [1/5] and I did not know what that meant. To Bloch [1/6] I wrote often and without results. I even had [1/7] the impression that he went back to Poland. [1/8] With me everything is in the best order. [1/9] I have familiarized myself with the regulations [1/10] and shortly I will begin applying for a non-quota visa. [1/11] I was told by the National Refugee Service that the matter is [1/12] easy. I only have to sign a few papers, [1/13] but that usually takes a while. [1/14] My parents sent me news that they are alive [1/15] and healthy. They are in the Russian portion. In Warsaw and [1/16] in the German part in general there are supposedly [1/17] some horrific things happening. [1/18] I have nothing from Steinhaus. Zygmund1 will probably [1/19] arrive at M.I.T. for next [1/20] semester. I was told2 that Marcinkiewicz and Kaczmarz3 [1/21] were shot because they were officers.4 [2/1] All of it is monstrous and sometimes5 I just don’t want to [2/2] think about it. [2/3] I hope that you are doing well [2/4] and that you are satisfied like I am. -
Quantitative Tauberian Theorems
Quantitative Tauberian theorems omas Powell University of Bath Mathematical Logic: Proof Theory, Constructive Mathematics MFO (delivered online) 11 November 2020 ese slides are available at https://t-powell.github.io/talks Introduction Abel’s theorem Let fang be a sequence of reals, and suppose that the power series 1 X i F(x) := aix i=0 converges on jxj < 1. en whenever 1 X ai = s i=0 it follows that F(x) ! s as x % 1: is is a classical result in elementary analysis called Abel’s theorem (N.b. it also holds in the complex setting). You can use it to e.g. prove that 1 X (−1)i = log(2): i + 1 i=0 Does the converse of Abel’s theorem hold? NO. For a counterexample, define F :(−1; 1) ! R by 1 1 X F(x) = = (−1)ixi 1 + x i=0 en 1 F(x) ! as x % 1 2 but 1 X (−1)i does not converge i=0 Tauber’s theorem fixes this Let fang be a sequence of reals, and suppose that the power series 1 X i F(x) := aix i=0 converges on jxj < 1. en whenever F(x) ! s as x % 1 AND jnanj ! 0 it follows that 1 X ai = s i=0 is is Tauber’s theorem, proven in 1897 by Austrian mathematician Alfred Tauber (1866 - 1942). Tauberian theorems e basic structure of Tauber’s theorem is: 1 X i Let F(x) = aix i=0 en if we know (A) Something about the behaviour of F(x) as x % 1 (B) Something about the growth of fang as n ! 1 en we can conclude P1 (C) Something about the convergence of i=0 ai. -
The Littlewood Tauberian Theorem
Cambridge University Press 978-1-107-05945-0 - Twelve Landmarks of Twentieth-Century Analysis D. Choimet and H. Queffélec Excerpt More information 1 The Littlewood Tauberian theorem 1.1 Introduction In 1897, the Austrian mathematician Alfred Tauber published a short article on the convergence of numerical series [173], which can be summarised as follows. ∞ = Let an be a convergent series of complex numbers, with n=0 an .A theorem of Abel [1] states that ∞ n f (x) = an x → as x 1. (1.1) n=0 A theorem of Kronecker [116] states that n 1 → . n kak 0 (1.2) k=1 The converse of these two theorems is false: neither of the two conditions (1) nor (2) is sufficient to imply the convergence of the series an. However, if both conditions are satisfied simultaneously, the series an converges, giving the following theorem. 1.1.1 Theorem A necessary and sufficient condition for an to converge (with sum ) is that: ∞ (1) f (x) = a xn → as x 1, n=0 n 1 n → (2) n k=1 kak 0. The proof of Theorem 1.1.1 follows that of the following special case. 1 © in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-1-107-05945-0 - Twelve Landmarks of Twentieth-Century Analysis D. Choimet and H. Queffélec Excerpt More information 2 The Littlewood Tauberian theorem ( ) → → ∞ = 1.1.2 Theorem f x as x 1, and nan 0 implies n=0 an . A few remarks on this article: the theorem of Abel cited above gave rise to Abelian theorems [113], that is, theorems of the form If an is a convergent complex series with sum , and (bn,x ) an infinite rectangular matrix indexed by N × X, where X is a set with an associated point at infinity satisfying bn,x −→ 1 forevery n ∈ N, x→∞ then ∞ f (x) = anbn,x n=0 is defined for x ∈ X and f (x) → as x →∞. -
An Application of Tauber Theory: Proving the Prime Number Theorem
faculty of science mathematics and applied and engineering mathematics An application of Tauber theory: proving the Prime Number Theorem Bachelor’s Project Mathematics February 2018 Student: J. Koolstra Supervisors: Prof.dr. J. Top and Dr. A. Sterk Abstract We look at the origins of Tauber theory, and apply it to prove the prime number theorem (PNT). Specifically, we prove a weak version of the Wiener-Ikehara Tauberian theorem due to Newman. Its appli- cation requires us to establish some properties of the Riemann zeta function. Most notably with regard to its meromorphic continuation, and the distribution of its zeros. 2 Contents Introduction 5 Discrete Tauber theory 7 Summability methods . .7 The theorems of Tauber and Hardy-Littlewood . 13 The Wiener-Ikehara theorem 19 Some complex analytic technicalities . 19 A proof of the Wiener-Ikehara theorem . 19 The Riemann zeta function 28 The meromorphic continuation of the zeta function . 28 The zeta function is nonzero on the line Re z =1.......... 32 The prime number theorem 37 The linear bound on the Tchebychef -function . 37 Equivalent formulations of PNT . 39 Appendix A: the prime polynomial theorem 42 References 48 3 Even before I had begun my more detailed investigations into higher arithmetic, one of my first projects was to turn my attention to the decreasing frequency of primes, to which end I counted primes in several chiliads. I soon recognized that behind all of its fluctuations, this frequency is on average inversely proportional to the logarithm. Gauss to Encke, 1849 It is not knowledge, but the act of learning, not possession but the act of getting there, which grants the greatest enjoyment. -
Geometry of Algebraic Curves
Geometry of Algebraic Curves Fall 2011 Course taught by Joe Harris Notes by Atanas Atanasov One Oxford Street, Cambridge, MA 02138 E-mail address: [email protected] Contents Lecture 1. September 2, 2011 6 Lecture 2. September 7, 2011 10 2.1. Riemann surfaces associated to a polynomial 10 2.2. The degree of KX and Riemann-Hurwitz 13 2.3. Maps into projective space 15 2.4. An amusing fact 16 Lecture 3. September 9, 2011 17 3.1. Embedding Riemann surfaces in projective space 17 3.2. Geometric Riemann-Roch 17 3.3. Adjunction 18 Lecture 4. September 12, 2011 21 4.1. A change of viewpoint 21 4.2. The Brill-Noether problem 21 Lecture 5. September 16, 2011 25 5.1. Remark on a homework problem 25 5.2. Abel's Theorem 25 5.3. Examples and applications 27 Lecture 6. September 21, 2011 30 6.1. The canonical divisor on a smooth plane curve 30 6.2. More general divisors on smooth plane curves 31 6.3. The canonical divisor on a nodal plane curve 32 6.4. More general divisors on nodal plane curves 33 Lecture 7. September 23, 2011 35 7.1. More on divisors 35 7.2. Riemann-Roch, finally 36 7.3. Fun applications 37 7.4. Sheaf cohomology 37 Lecture 8. September 28, 2011 40 8.1. Examples of low genus 40 8.2. Hyperelliptic curves 40 8.3. Low genus examples 42 Lecture 9. September 30, 2011 44 9.1. Automorphisms of genus 0 an 1 curves 44 9.2. -
Math 632: Algebraic Geometry Ii Cohomology on Algebraic Varieties
MATH 632: ALGEBRAIC GEOMETRY II COHOMOLOGY ON ALGEBRAIC VARIETIES LECTURES BY PROF. MIRCEA MUSTA¸TA;˘ NOTES BY ALEKSANDER HORAWA These are notes from Math 632: Algebraic geometry II taught by Professor Mircea Musta¸t˘a in Winter 2018, LATEX'ed by Aleksander Horawa (who is the only person responsible for any mistakes that may be found in them). This version is from May 24, 2018. Check for the latest version of these notes at http://www-personal.umich.edu/~ahorawa/index.html If you find any typos or mistakes, please let me know at [email protected]. The problem sets, homeworks, and official notes can be found on the course website: http://www-personal.umich.edu/~mmustata/632-2018.html This course is a continuation of Math 631: Algebraic Geometry I. We will assume the material of that course and use the results without specific references. For notes from the classes (similar to these), see: http://www-personal.umich.edu/~ahorawa/math_631.pdf and for the official lecture notes, see: http://www-personal.umich.edu/~mmustata/ag-1213-2017.pdf The focus of the previous part of the course was on algebraic varieties and it will continue this course. Algebraic varieties are closer to geometric intuition than schemes and understanding them well should make learning schemes later easy. The focus will be placed on sheaves, technical tools such as cohomology, and their applications. Date: May 24, 2018. 1 2 MIRCEA MUSTA¸TA˘ Contents 1. Sheaves3 1.1. Quasicoherent and coherent sheaves on algebraic varieties3 1.2. Locally free sheaves8 1.3. -
A Century of Complex Tauberian Theory 1
BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 39, Number 4, Pages 475{531 S 0273-0979(02)00951-5 Article electronically published on July 8, 2002 A CENTURY OF COMPLEX TAUBERIAN THEORY J. KOREVAAR Abstract. Complex-analytic and related boundary properties of transforms give information on the behavior of pre-images. The transforms may be power series, Dirichlet series or Laplace-type integrals; the pre-images are series (of numbers) or functions. The chief impulse for complex Tauberian theory came from number the- ory. The first part of the survey emphasizes methods which permit simple derivations of the prime number theorem, associated with the labels Landau- Wiener-Ikehara and Newman. Other important areas in complex Tauberian theory are associated with the names Fatou-Riesz and Ingham. Recent re- finements have been motivated by operator theory and include local H1 and pseudofunction boundary behavior of transforms. Complex information has also led to better remainder estimates in connection with classical Tauberian theorems. Applications include the distribution of zeros and eigenvalues. 1. Introduction Tauberian theorems involve a class of objects S (functions, series, sequences) and a transformation . The transformation is an `averaging' operation. It must have a continuity property:T certain limit behavior of the original S implies related limit behavior of the image S. The aim of a Tauberian is to reverse the averaging: to go from a limit propertyT of S to a limit property of S, or another transform of S. The converses typically requireT an additional condition, a `Tauberian condition', on S and perhaps a condition on the transform S. -
SERRE DUALITY and APPLICATIONS Contents 1
SERRE DUALITY AND APPLICATIONS JUN HOU FUNG Abstract. We carefully develop the theory of Serre duality and dualizing sheaves. We differ from the approach in [12] in that the use of spectral se- quences and the Yoneda pairing are emphasized to put the proofs in a more systematic framework. As applications of the theory, we discuss the Riemann- Roch theorem for curves and Bott's theorem in representation theory (follow- ing [8]) using the algebraic-geometric machinery presented. Contents 1. Prerequisites 1 1.1. A crash course in sheaves and schemes 2 2. Serre duality theory 5 2.1. The cohomology of projective space 6 2.2. Twisted sheaves 9 2.3. The Yoneda pairing 10 2.4. Proof of theorem 2.1 12 2.5. The Grothendieck spectral sequence 13 2.6. Towards Grothendieck duality: dualizing sheaves 16 3. The Riemann-Roch theorem for curves 22 4. Bott's theorem 24 4.1. Statement and proof 24 4.2. Some facts from algebraic geometry 29 4.3. Proof of theorem 4.5 33 Acknowledgments 34 References 35 1. Prerequisites Studying algebraic geometry from the modern perspective now requires a some- what substantial background in commutative and homological algebra, and it would be impractical to go through the many definitions and theorems in a short paper. In any case, I can do no better than the usual treatments of the various subjects, for which references are provided in the bibliography. To a first approximation, this paper can be read in conjunction with chapter III of Hartshorne [12]. Also in the bibliography are a couple of texts on algebraic groups and Lie theory which may be beneficial to understanding the latter parts of this paper. -
Report No. 26/2008
Mathematisches Forschungsinstitut Oberwolfach Report No. 26/2008 Classical Algebraic Geometry Organised by David Eisenbud, Berkeley Joe Harris, Harvard Frank-Olaf Schreyer, Saarbr¨ucken Ravi Vakil, Stanford June 8th – June 14th, 2008 Abstract. Algebraic geometry studies properties of specific algebraic vari- eties, on the one hand, and moduli spaces of all varieties of fixed topological type on the other hand. Of special importance is the moduli space of curves, whose properties are subject of ongoing research. The rationality versus general type question of these spaces is of classical and also very modern interest with recent progress presented in the conference. Certain different birational models of the moduli space of curves have an interpretation as moduli spaces of singular curves. The moduli spaces in a more general set- ting are algebraic stacks. In the conference we learned about a surprisingly simple characterization under which circumstances a stack can be regarded as a scheme. For specific varieties a wide range of questions was addressed, such as normal generation and regularity of ideal sheaves, generalized inequalities of Castelnuovo-de Franchis type, tropical mirror symmetry constructions for Calabi-Yau manifolds, Riemann-Roch theorems for Gromov-Witten theory in the virtual setting, cone of effective cycles and the Hodge conjecture, Frobe- nius splitting, ampleness criteria on holomorphic symplectic manifolds, and more. Mathematics Subject Classification (2000): 14xx. Introduction by the Organisers The Workshop on Classical Algebraic Geometry, organized by David Eisenbud (Berkeley), Joe Harris (Harvard), Frank-Olaf Schreyer (Saarbr¨ucken) and Ravi Vakil (Stanford), was held June 8th to June 14th. It was attended by about 45 participants from USA, Canada, Japan, Norway, Sweden, UK, Italy, France and Germany, among of them a large number of strong young mathematicians. -
Samuel Fogelson (1902 – Po 1941) Streszczenie
ANTIQUITATES MATHEMATICAE Vol. 10(1) 2016, p. 19–43 doi: 10.14708/am.v10i0.1549 Lech Maligranda (Lule˚a) Walerian Piotrowski (Warszawa) Samuel Fogelson (1902 – po 1941) Streszczenie. Samuel Fogelson to zapomniany w Polsce matematyk i statystyk warszawski oraz agent wywiadu sowieckiego. Brak o nim wzmianki w Słowniku Biograficznym Matematyków Polskich (2003) i w Polskim Słowniku Biograficznym; jest informacja w Wikipedii [3], ale opisana została tam głównie jego działalność szpiegowska, przed- stawiona w artykułach A. Poczobuta [6]–[8]. My chcemy przybliżyć osobę Fogelsona jako matematyka i statystyka warszawskiego, który opublikował wiele prac, zwłaszcza ze statystyki opisowej. Staraliśmy się dotrzeć do jego publikacji i osiągnięć w matematyce i statystyce. Wspomnimy też krótko o jego działalności szpiegowskiej na rzecz pań- stwa sowieckiego. 2010 Klasyfikacja tematyczna AMS (2010): 01A72; 01A60. Słowa kluczowe: matematyka dwudziestego wieku, matematycy w Eu- ropie, polska matematyka. 1. Biografia Fogelsona. Samuel Fogelson [Fogielson, Fogel- sohn, Fogielsohn], pseudonim „St. Manert”, urodził się 20 paździer- nika 1902 roku w Warszawie w rodzinie Moszka Fogielsona i Chai-Sary z domu Halbrajch. Nauki początkowe otrzymał w domu, a w 1912 roku wstąpił do siedmioklasowej Szkoły Handlowej Zgromadzenia Kup- ców miasta Warszawy („Handlówki”). W 1915 roku otrzymał pro- mocję do czwartej klasy i wraz z rodzicami wyjechał do Rosji, gdzie w 1918 roku ukończył ośmioklasową szkołę handlową w Rostowie nad Donem. W tym samym roku wstąpił jako słuchacz rzeczywisty na Wy- dział Matematyczno-Fizyczny Uniwersytetu Dońskiego (były rosyjski Uniwersytet Warszawski, ewakuowany został w 1915 roku do Rostowa nad Donem). W 1920 roku wraz z rodzicami powrócił do Warszawy i zapisał się na Wydział Przyrodniczo-Matematyczny Wolnej Wszechnicy Polskiej (WWP). -
Presentation of the Austrian Mathematical Society - E-Mail: [email protected] La Rochelle University Lasie, Avenue Michel Crépeau B
NEWSLETTER OF THE EUROPEAN MATHEMATICAL SOCIETY Features S E European A Problem for the 21st/22nd Century M M Mathematical Euler, Stirling and Wallis E S Society History Grothendieck: The Myth of a Break December 2019 Issue 114 Society ISSN 1027-488X The Austrian Mathematical Society Yerevan, venue of the EMS Executive Committee Meeting New books published by the Individual members of the EMS, member S societies or societies with a reciprocity agree- E European ment (such as the American, Australian and M M Mathematical Canadian Mathematical Societies) are entitled to a discount of 20% on any book purchases, if E S Society ordered directly at the EMS Publishing House. Todd Fisher (Brigham Young University, Provo, USA) and Boris Hasselblatt (Tufts University, Medford, USA) Hyperbolic Flows (Zürich Lectures in Advanced Mathematics) ISBN 978-3-03719-200-9. 2019. 737 pages. Softcover. 17 x 24 cm. 78.00 Euro The origins of dynamical systems trace back to flows and differential equations, and this is a modern text and reference on dynamical systems in which continuous-time dynamics is primary. It addresses needs unmet by modern books on dynamical systems, which largely focus on discrete time. Students have lacked a useful introduction to flows, and researchers have difficulty finding references to cite for core results in the theory of flows. Even when these are known substantial diligence and consulta- tion with experts is often needed to find them. This book presents the theory of flows from the topological, smooth, and measurable points of view. The first part introduces the general topological and ergodic theory of flows, and the second part presents the core theory of hyperbolic flows as well as a range of recent developments.