DOCSLIB.ORG

The Colorado Mathematical Olympiad and Further Explorations

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
The Colorado Mathematical Olympiad and Further Explorations The Colorado Mathematical Olympiad and Further Explorations Alexander Soifer The Colorado Mathematical Olympiad and Further Explorations From the Mountains of Colorado to the Peaks of Mathematics With over 185 Illustrations Forewords by Philip L. Engel Paul Erdos˝ Martin Gardner Branko Grunbaum¨ Peter D. Johnson, Jr. and Cecil Rousseau 123 Alexander Soifer College of Letters, Arts and Sciences University of Colorado at Colorado Springs 1420 Austin Bluffs Parkway Colorado Springs, CO 80918 USA [email protected] ISBN: 978-0-387-75471-0 e-ISBN: 978-0-387-75472-7 DOI: 10.1007/978-0-387-75472-7 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2011925380 c Alexander Soifer 2011 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com) To all those people throughout the world, who create Olympiads for new generations of mathematicians. Forewords to The Colorado Mathematical Olympiad and Further Explorations We live in an age of extreme specialization – in mathematics as well as in all other sciences, in engineering, in medicine. Hence, to say that probably 90% of mathematicians cannot understand 90% of mathematics currently published is, most likely, too optimistic. In contrast, even a pessimist would have to agree that at least 90% of the material in this book is read- ily accessible to, and understandable by 90% of students in middle and high schools. However, this does not mean that the topics are trivial – they are elementary in the sense that they do not require knowledge of lots of previously studied material, but are sophisticated in requiring attention, concen- tration, and thinking that is not fettered by preconceptions. The organization in groups of five problems for each of the “Olympiads”, for which the participants were allowed four hours, hints at the difficulty of finding complete solutions. I am convinced that most professional mathematicians would be hard pressed to solve a set of five problems in two hours, or even four. There are many collections of problems, for “Olympiads” of various levels, as well as problems in a variety of journals. What sets this book apart from the “competition” are sev- eral aspects that deserve to be noted. vi Forewords vii • The serenity and enthusiasm with which the problems, and their solutions, are presented; • The absence of prerequisites for understanding the prob- lems and their solutions; • The mixture of geometric and combinatorial ideas that are required in almost all cases. The detailed exposition of the trials and tribulations en- dured by the author, as well as the support he received, shed light on the variety of influences which the administration of a university exerts on the faculty. As some of the negative ac- tions are very probably a consequence of mathophobia, the spirit of this book may cure at least a few present or future deciders from that affliction. Many mathematicians are certainly able to come up with an interesting elementary problem. But Soifer may be unique in his persistence, over the decades, of inventing worthwhile problems, and providing amusing historical and other com- ments, all accessible to the intended pre-college students. It is my fervent hope that this book will find the wide readership it deserves, and that its readers will feel motivated to look for enjoyment in mathematics. Branko Gr¨unbaum Departmernt of Mathematics University of Washington, Seattle Here is another wonderful book from Alexander Soifer. This one is a more-than-doubling of an earlier book on the first 10 years of the Colorado Mathematical Olympiad, which was founded and nourished to robust young adulthood by– Alexander Soifer. Like The Mathematical Coloring Book, this book is not so much mathematical literature as it is literature built around mathematics, if you will permit the distinction. Yes, there is plenty of mathematics here, and of the most delicious kind. In case you were unaware of, or had forgotten (as I had), the level of skill, nay, art, necessary to pose good olympiad- or Putnam exam-style problems, or the effect that such a prob- lem can have on a young mind, and even on the thoughts of a jaded sophisticate, then what you have been missing can be found here in plenty – at least a year’s supply of great intellec- tual gustation. If you are a mathematics educator looking for activities for a math. club – your search is over! And with the Further Explorations sections, anyone so inclined could spend a lifetime on the mathematics sprouting from this volume. But since there will be no shortage of praise for the math- ematical and pedagogical contributions of From the Mountains ofColorado..., let me leave that aspect of the work and supply viii Forewords ix a few words about the historical account that surrounds and binds the mathematical trove, and makes a story of it all. The Historical Notes read like a wardiary,oranexplorer’slet- ters home: there is a pleasant, mundane rhythm of reportage – who and how many showed up from where, who the spon- sors were, which luminaries visited, who won, what their prizes were – punctuated by turbulence, events ranging from A. Soifer’s scolding of a local newspaper for printing only the names of the top winners, to the difficulties arising from the weather and the shootings at Columbine High School in 1999 (both matters of life and death in Colorado), to the in- explicable attempts of university administrators to impede, restructure, banish, or destroy the Colorado Mathematical Olympiad, in 1985, 1986, 2001 and 2003. It is fascinating stuff. The very few who have the entrepeneurial spirit to attempt the creation of anything like an Olympiad will be forewarned and inspired. The rest of us will be pleasurably horrified and amazed, our sympathies stimulated and our support aroused for the brave ones who bring new life to the communication of mathematics. Peter D. Johnson. Jr. Department of Mathematics and Statistics Auburn University In the common understanding of things, mathematics is dis- passionate. This unfortunate notion is reinforced by modern mathematical prose, which gets good marks for logic and poor ones for engagement. But the mystery and excitement of mathematical discovery cannot be denied. These qualities overflow all preset boundaries. On July 10, 1796, Gauss wrote in his diary EΥPHKA! num = Δ + Δ + Δ He had discovered a proof that every positive integer is the sum of three triangular numbers {0, 1, 3, 6, 10, . n(n + 1)/ 2,...}. This result was something special. It was right to cele- brate the moment with an exclamation of Eureka! In 1926, an intriguing conjecture was making the rounds of European universities. If the set of positive integers is partitioned into two classes, then at least one of the classes contains an n- term arithmetic progression, no matter how large n is taken to be. x Forewords xi The conjecture had been formulated by the Dutch mathematician P. J. H. Baudet, who told it to his friend and mentor Frederik Schuh. B. L. van der Waerden learned the problem in Schuh’s seminar at the University of Amsterdam. While in Hamburg, van der Waerden told the conjecture to Emil Artin and Otto Schreier as the three had lunch. After lunch, they adjourned to Artin’s office at the University of Hamburg to try to find a proof. They were successful, and the result, now known as van der Waerden’s Theorem, is one of the Three Pearls of Number Theory in Khinchine’s book by that name. The story does not end there. In 1971, van der Waerden published a re- markable paper entitled How the Proof of Baudet’s Conjecture was Found.1 In it, he describes how the three mathematicians searched for a proof by drawing diagrams on the blackboard to represent the classes, and how each mathematician had Einf¨alle (sudden ideas) that were crucial to the proof. In this account, the reader is a fourth person in Artin’s office, observ- ing with each Einfall the rising anticipation that the proof is going to work. Even though unspoken, each of the three must have had a “Eureka moment” when success was assured. From the Mountains of Colorado to the Peaks of Mathemat- ics presents the 20-year history of the Colorado Mathemat- ical Olympiad. It is symbolic that this Olympiad is held in Colorado. Colorado is known for its beauty and spaciousness. In the book there is plenty of space for mathematics. There are wonderful problems with ingenious solutions, taken from ge- ometry, combinatorics, number theory, and other areas. But there is much more. There is space to meet the participants, hear their candid comments, learn of their talents, mathemat- ical and otherwise, and in some cases to follow their paths as professionals. There is space for poetry and references to thearts.Thereisspaceforafullstory of the competition – its 1 Studies in Pure Mathematics (Presented to Richard Rado), L. Mirsky, ed., Aca- demic Press, London, 1971, pp 251-260.
Load more

Recommended publications
  • Algebraic Geometry of Topological Spaces I 3
    ALGEBRAIC GEOMETRY OF TOPOLOGICAL SPACES I GUILLERMO CORTINAS˜ AND ANDREAS THOM Abstract. We use techniques from both real and complex algebraic geometry to study K- theoretic and related invariants of the algebra C(X) of continuous complex-valued functions on a compact Hausdorff topological space X. For example, we prove a parametrized version of a theorem of Joseph Gubeladze; we show that if M is a countable, abelian, cancella- tive, torsion-free, seminormal monoid, and X is contractible, then every finitely generated n projective module over C(X)[M] is free. The particular case M = N0 gives a parametrized version of the celebrated theorem proved independently by Daniel Quillen and Andrei Suslin that finitely generated projective modules over a polynomial ring over a field are free. The conjecture of Jonathan Rosenberg which predicts the homotopy invariance of the negative algebraic K-theory of C(X) follows from the particular case M = Zn. We also give alge- braic conditions for a functor from commutative algebras to abelian groups to be homotopy invariant on C∗-algebras, and for a homology theory of commutative algebras to vanish on C∗-algebras. These criteria have numerous applications. For example, the vanishing cri- terion applied to nil-K-theory implies that commutative C∗-algebras are K-regular. As another application, we show that the familiar formulas of Hochschild-Kostant-Rosenberg and Loday-Quillen for the algebraic Hochschild and cyclic homology of the coordinate ring of a smooth algebraic variety remain valid for the algebraic Hochschild and cyclic homology of C(X). Applications to the conjectures of Be˘ılinson-Soul´eand Farrell-Jones are also given.
    [Show full text]
  • Quillen's Work in Algebraic K-Theory
    J. K-Theory 11 (2013), 527–547 ©2013 ISOPP doi:10.1017/is012011011jkt203 Quillen’s work in algebraic K-theory by DANIEL R. GRAYSON Abstract We survey the genesis and development of higher algebraic K-theory by Daniel Quillen. Key Words: higher algebraic K-theory, Quillen. Mathematics Subject Classification 2010: 19D99. Introduction This paper1 is dedicated to the memory of Daniel Quillen. In it, we examine his brilliant discovery of higher algebraic K-theory, including its roots in and genesis from topological K-theory and ideas connected with the proof of the Adams conjecture, and his development of the field into a complete theory in just a few short years. We provide a few references to further developments, including motivic cohomology. Quillen’s main work on algebraic K-theory appears in the following papers: [65, 59, 62, 60, 61, 63, 55, 57, 64]. There are also the papers [34, 36], which are presentations of Quillen’s results based on hand-written notes of his and on communications with him, with perhaps one simplification and several inaccuracies added by the author. Further details of the plus-construction, presented briefly in [57], appear in [7, pp. 84–88] and in [77]. Quillen’s work on Adams operations in higher algebraic K-theory is exposed by Hiller in [45, sections 1-5]. Useful surveys of K-theory and related topics include [81, 46, 38, 39, 67] and any chapter in Handbook of K-theory [27]. I thank Dale Husemoller, Friedhelm Waldhausen, Chuck Weibel, and Lars Hesselholt for useful background information and advice. 1.
    [Show full text]
  • References Is Provisional, and Will Grow (Or Shrink) As the Course Progresses
    Note: This list of references is provisional, and will grow (or shrink) as the course progresses. It will never be comprehensive. See also the references (and the book) in the online version of Weibel's K- Book [40] at http://www.math.rutgers.edu/ weibel/Kbook.html. References [1] Armand Borel and Jean-Pierre Serre. Le th´eor`emede Riemann-Roch. Bull. Soc. Math. France, 86:97{136, 1958. [2] A. K. Bousfield. The localization of spaces with respect to homology. Topology, 14:133{150, 1975. [3] A. K. Bousfield and E. M. Friedlander. Homotopy theory of Γ-spaces, spectra, and bisimplicial sets. In Geometric applications of homotopy theory (Proc. Conf., Evanston, Ill., 1977), II, volume 658 of Lecture Notes in Math., pages 80{130. Springer, Berlin, 1978. [4] Kenneth S. Brown and Stephen M. Gersten. Algebraic K-theory as gen- eralized sheaf cohomology. In Algebraic K-theory, I: Higher K-theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972), pages 266{ 292. Lecture Notes in Math., Vol. 341. Springer, Berlin, 1973. [5] William Dwyer, Eric Friedlander, Victor Snaith, and Robert Thoma- son. Algebraic K-theory eventually surjects onto topological K-theory. Invent. Math., 66(3):481{491, 1982. [6] Eric M. Friedlander and Daniel R. Grayson, editors. Handbook of K- theory. Vol. 1, 2. Springer-Verlag, Berlin, 2005. [7] Ofer Gabber. K-theory of Henselian local rings and Henselian pairs. In Algebraic K-theory, commutative algebra, and algebraic geometry (Santa Margherita Ligure, 1989), volume 126 of Contemp. Math., pages 59{70. Amer. Math. Soc., Providence, RI, 1992. [8] Henri Gillet and Daniel R.
    [Show full text]
  • Prize Is Awarded Every Three Years at the Joint Mathematics Meetings
    AMERICAN MATHEMATICAL SOCIETY LEVI L. CONANT PRIZE This prize was established in 2000 in honor of Levi L. Conant to recognize the best expository paper published in either the Notices of the AMS or the Bulletin of the AMS in the preceding fi ve years. Levi L. Conant (1857–1916) was a math- ematician who taught at Dakota School of Mines for three years and at Worcester Polytechnic Institute for twenty-fi ve years. His will included a bequest to the AMS effective upon his wife’s death, which occurred sixty years after his own demise. Citation Persi Diaconis The Levi L. Conant Prize for 2012 is awarded to Persi Diaconis for his article, “The Markov chain Monte Carlo revolution” (Bulletin Amer. Math. Soc. 46 (2009), no. 2, 179–205). This wonderful article is a lively and engaging overview of modern methods in probability and statistics, and their applications. It opens with a fascinating real- life example: a prison psychologist turns up at Stanford University with encoded messages written by prisoners, and Marc Coram uses the Metropolis algorithm to decrypt them. From there, the article gets even more compelling! After a highly accessible description of Markov chains from fi rst principles, Diaconis colorfully illustrates many of the applications and venues of these ideas. Along the way, he points to some very interesting mathematics and some fascinating open questions, especially about the running time in concrete situ- ations of the Metropolis algorithm, which is a specifi c Monte Carlo method for constructing Markov chains. The article also highlights the use of spectral methods to deduce estimates for the length of the chain needed to achieve mixing.
    [Show full text]
  • A Journal of the K-Theory Foundation(Ktheoryfoundation.Org)
    ANNALSOF K-THEORY Paul Balmer Spencer Bloch no. 1 vol. 1 2016 Alain Connes Guillermo Cortiñas Eric Friedlander Max Karoubi Gennadi Kasparov Alexander Merkurjev Amnon Neeman Jonathan Rosenberg Marco Schlichting Andrei Suslin Vladimir Voevodsky Charles Weibel Guoliang Yu msp A JOURNAL OF THE K-THEORY FOUNDATION ANNALS OF K-THEORY msp.org/akt EDITORIAL BOARD Paul Balmer University of California, Los Angeles, USA [email protected] Spencer Bloch University of Chicago, USA [email protected] Alain Connes Collège de France; Institut des Hautes Études Scientifiques; Ohio State University [email protected] Guillermo Cortiñas Universidad de Buenos Aires and CONICET, Argentina [email protected] Eric Friedlander University of Southern California, USA [email protected] Max Karoubi Institut de Mathématiques de Jussieu – Paris Rive Gauche, France [email protected] Gennadi Kasparov Vanderbilt University, USA [email protected] Alexander Merkurjev University of California, Los Angeles, USA [email protected] Amnon Neeman amnon.Australian National University [email protected] Jonathan Rosenberg (Managing Editor) University of Maryland, USA [email protected] Marco Schlichting University of Warwick, UK [email protected] Andrei Suslin Northwestern University, USA [email protected] Vladimir Voevodsky Institute for Advanced Studies, USA [email protected] Charles Weibel (Managing Editor) Rutgers University, USA [email protected] Guoliang Yu Texas A&M University, USA [email protected] PRODUCTION Silvio Levy (Scientific Editor) [email protected] Annals of K-Theory is a journal of the K-Theory Foundation(ktheoryfoundation.org). The K-Theory Foundation acknowledges the precious support of Foundation Compositio Mathematica, whose help has been instrumental in the launch of the Annals of K-Theory.
    [Show full text]
  • Theoretic Timeline Converging on Motivic Cohomology, Then Briefly Discuss Algebraic 퐾-Theory and Its More Concrete Cousin Milnor 퐾-Theory
    AN OVERVIEW OF MOTIVIC COHOMOLOGY PETER J. HAINE Abstract. In this talk we give an overview of some of the motivations behind and applica- tions of motivic cohomology. We first present a 퐾-theoretic timeline converging on motivic cohomology, then briefly discuss algebraic 퐾-theory and its more concrete cousin Milnor 퐾-theory. We then present a geometric definition of motivic cohomology via Bloch’s higher Chow groups and outline the main features of motivic cohomology, namely, its relation to Milnor 퐾-theory. In the last part of the talk we discuss the role of motivic cohomology in Voevodsky’s proof of the Bloch–Kato conjecture. The statement of the Bloch–Kato conjec- ture is elementary and predates motivic cohomolgy, but its proof heavily relies on motivic cohomology and motivic homotopy theory. Contents 1. A Timeline Converging to Motivic Cohomology 1 2. A Taste of Algebraic 퐾-Theory 3 3. Milnor 퐾-theory 4 4. Bloch’s Higher Chow Groups 4 5. Properties of Motivic Cohomology 6 6. The Bloch–Kato Conjecture 7 6.1. The Kummer Sequence 7 References 10 1. A Timeline Converging to Motivic Cohomology In this section we give a timeline of events converging on motivic cohomology. We first say a few words about what motivic cohomology is. 푝 op • Motivic cohomology is a bigraded cohomology theory H (−; 퐙(푞))∶ Sm/푘 → Ab. • Voevodsky won the fields medal because “he defined and developed motivic coho- mology and the 퐀1-homotopy theory of algebraic varieties; he proved the Milnor conjectures on the 퐾-theory of fields.” Moreover, Voevodsky’s proof of the Milnor conjecture makes extensive use of motivic cohomology.
    [Show full text]
  • The Mathematics of Andrei Suslin
    B U L L E T I N ( N e w Series) O F T H E A M E R I C A N MATHEMATICAL SOCIETY Vo lume 57, N u mb e r 1, J a n u a r y 2020, P a g e s 1–22 https://doi.org/10.1090/bull/1680 Art icle electronically publis hed o n S e p t e mbe r 24, 2019 THE MATHEMATICS OF ANDREI SUSLIN ERIC M. FRIEDLANDER AND ALEXANDER S. MERKURJEV C o n t e n t s 1. Introduction 1 2. Projective modules 2 3. K 2 of fields and the Brauer group 3 4. Milnor K-theory versus algebraic K-theory 7 5. K-theory and cohomology theories 9 6. Motivic cohomology and K-theories 12 7. Modular representation theory 16 Acknowledgments 18 About the authors 18 References 18 0. Introduction Andrei Suslin (1950–2018) was both friend and mentor to us. This article dis- cusses some of his many mathematical achievements, focusing on the role he played in shaping aspects of algebra and algebraic geometry. We mention some of the many important results Andrei proved in his career, proceeding more or less chronolog- ically beginning with Serre’s Conjecture proved by Andrei in 1976 (and simul- taneously by Daniel Quillen). As the reader will quickly ascertain, this article does not do justice to the many mathematicians who contributed to algebraic K- theory and related subjects in recent decades. In particular, work of Hyman Bass, Alexander Be˘ılinson, Spencer Bloch, Alexander Grothendieck, Daniel Quillen, Jean- Pierre Serre, and Christophe Soul`e strongly influenced Andrei’s mathematics and the mathematical developments we discuss.
    [Show full text]
  • Techniques, Computations, and Conjectures for Semi-Topological K-Theory
    TECHNIQUES, COMPUTATIONS, AND CONJECTURES FOR SEMI-TOPOLOGICAL K-THEORY ERIC M. FRIEDLANDER, CHRISTIAN HAESEMEYER, AND MARK E. WALKER Abstract. We establish the existence of an \Atiyah-Hirzebruch-like" spectral sequence relating the morphic cohomology groups of a smooth, quasi-projective complex variety to its semi-topological K-groups. This spectral sequence is compatible with (and, indeed, is built from) the motivic spectral sequence that connects the motivic cohomology and algebraic K-theory of varieties, and it is also compatible with the classical Atiyah-Hirzebruch spectral sequence in algebraic topology. In the second part of this paper, we use this spectral sequence in conjunction with another computational tool that we introduce | namely, a variation on the integral weight filtration of the Borel-Moore (singular) cohomology of complex varieties introduced by H. Gillet and C. Soul´e | to compute the semi-topological K-theory of a large class of varieties. In particular, we prove that for curves, surfaces, toric varieties, projective rational three-folds, and related varieties, the semi-topological K-groups and topological K-groups are isomorphic in all degrees permitted by cohomological considerations. We also formulate integral conjectures relating semi-topological K-theory to topological K-theory analogous to more familiar conjectures (namely, the Quillen-Lichtenbaum and Beilinson-Lichtenbaum Conjectures) concerning mod- n algebraic K-theory and motivic cohomology. In particular, we prove a local vanishing result for morphic cohomology which enables us to formulate pre- cisely a conjectural identification of morphic cohomology by A. Suslin. Our computations verify that these conjectures hold for the list of varieties above. Contents 1.
    [Show full text]
  • Mathematical Society April 2003 Volume 50, Number 4
    ISS N 0002 -9920 of the American Mathematical Society April 2003 Volume 50, Number 4 An Introduction to Analysis on Metric Spaces page 438 Artful Mathematics: The Heritage of M. C. Escher page 446 Filling in Escher's blank space (see page 457) The Open Computer Algebra System ,.;' MuPAD Pro - [DtffEq.mnb] fo:B. MuPAD Pro is a full-featured computer algebra system in an integrated and 0 30 Vtewer - [VCamKiem2.vca] < Edit View Tools Animation Window Help open environment for symbolic and numeric computing . The MuPAD language has a Pascal-like syntax and allows imperative, functional, and object­ oriented programming . Its domains and categories are like object-oriented MuP .AJ) recognized an inhomogenous linear differential equation classes that allow overriding and corresponding homogenous system is expresse~ using a new ide overloading methods and operators, The differential equation y'"'(x) - 5y"(x) + 4 y(x) = 2 cos(x) o inheritance, and generic algorithms. A • solve( ode ( y''''(x)- S*y''(x) + 4*y comfortable notebook interface includes a graphics tool for visualization, an integrated source-level debugger, a profiler, and hypertext help. It is also possible to solve systems of differential equations. Consi Some of the new features in version 2.5 include: f(x) - j(x) + g'(x) + 2 g(x) =l+eA g'(x) + 2 g(x) + h'(x) + h(x) =2 +ex X • High quality 3-D graphics • HTML export Ready • Increased computational capability • Available with Scilab for very fast numerical computations ~MacKichan SOFTWAR E, INC . Tools for Scientific Creativity Since
    [Show full text]
  • Report for the Academic Year 2017–2018
    Institute for Advanced Study Re port for 2 0 1 7–2 0 INSTITUTE FOR ADVANCED STUDY 1 8 EINSTEIN DRIVE PRINCETON, NEW JERSEY 08540 Report for the Academic Year (609) 734-8000 www.ias.edu 2017–2018 Cover: SHATEMA THREADCRAFT, Ralph E. and Doris M. Hansmann Member in the School of Social Science (right), gives a talk moderated by DIDIER FASSIN (left), James D. Wolfensohn Professor, on spectacular black death at Ideas 2017–18. Opposite: Fuld Hall COVER PHOTO: DAN KOMODA Table of Contents DAN KOMODA DAN Reports of the Chair and the Director 4 The Institute for Advanced Study 6 School of Historical Studies 8 School of Mathematics 20 School of Natural Sciences 30 School of Social Science 40 Special Programs and Outreach 48 Record of Events 57 80 Acknowledgments 88 Founders, Trustees, and Officers of the Board and of the Corporation 89 Administration 90 Present and Past Directors and Faculty 91 Independent Auditors’ Report THOMAS CLARKE REPORT OF THE CHAIR The Institute for Advanced Study’s independence and excellence led by Sanjeev Arora, Visiting Professor in the School require the dedication of many benefactors, and in 2017–18, of Mathematics. we celebrated the retirements of our venerable Vice Chairs The Board was delighted to welcome new Trustees Mark Shelby White and Jim Simons, whose extraordinary service has Heising, Founder and Managing Director of the San Francisco enhanced the Institute beyond measure. I am immensely grateful investment firm Medley Partners, and Dutch astronomer and and feel exceptionally privileged to have worked with both chemist Ewine Fleur van Dishoeck, Professor of Molecular Shelby and Jim in shaping and guiding the Institute into the Astrophysics at the University of Leiden.
    [Show full text]
  • Cohomological Hasse Principle for Schemes Over Valuation Rings Of
    Cohomological Hasse principle for schemes over valuation rings of higher dimensional local fields Patrick Forré ˚ Abstract K. Kato’s conjecture about the cohomological Hasse principle for regular connected schemes X which are flat and proper over the com- plete discrete valuation rings ON of higher local fields FN is proven. This generalizes the work of M. Kerz, S. Saito and U. Jannsen for finite fields to the case of all higher local fields. For that purpose a p-alteration theorem for the local uniformization of schemes over valu- ation rings of arbitrary finite rank and a corresponding Bertini theorem is developed extending the results of O. Gabber, J. deJong, L. Illusie, M. Temkin, S. Saito, U. Jannsen to the non-noetherian world. As an application it is shown that certain motivic cohomology groups of va- rieties over higher local fields are finite. This is one of the rare cases where such a result could be shown for schemes without finite or sep- arably closed residue fields. Furthermore, it will be derived that the X : SK ab kernels of the reciprocity map ρ N pXqÑ π1 pXq and norm map : SK M 1 NX|F N pXq Ñ KN pFN q modulo maximal p -divisible subgroups are finite for regular X which are proper over a higher local field FN with final residue characteristic p. This generalizes results of S. Bloch, K. Kato, U. Jannsen, S. Saito from varieties over finite and local fields to varieties over higher local fields, both of arbitrary dimensions. 2010 Mathematics Subject Classifications: 11G45, 14E15, 14F42, 11G25 Keywords: Kato conjecture, cohomological Hasse
    [Show full text]
  • The Work of Vladimir Voevodsky 101
    ICM 2002 Vol. I 99–103 · · The Work of Vladimir Voevodsky Christophe Soul´e∗ Vladimir Voevodsky was born in 1966. He studied at Moscow State University and Harvard university. He is now Professor at the Institute for Advanced Study in Princeton. Among his main achievements are the following: he defined and developed motivic cohomology and the A1-homotopy theory of algebraic varieties; he proved the Milnor conjectures on the K-theory of fields. Let us state the first Milnor conjecture. Let F be a field and n a positive M integer. The Milnor K-group of F is the abelian group Kn (F ) defined by the following generators and relations. The generators are sequences a1,...,an of n ∗ { } units ai F . The relations are ∈ a1,...,ak−1,xy,ak+1,...,an { } = a1,...,ak−1,x,ak+1,...,an + a1,...,ak−1,y,ak+1,...,an { } { } ∗ for all ai,x,y F , 1 k n, and the Steinberg relation ∈ ≤ ≤ a1,...,x,..., 1 x,...,an =0 { − } ∗ for all ai F and x F 0, 1 . ∈ ∈ −{ } On the other hand, let F be an algebraic closure of F and G = Gal(F /F ) the absolute Galois group of F , with its profinite topology. The Galois cohomology of F with Z/2 coefficients is, by definition, n n H (F, Z/2) = Hcontinuous(G, Z/2) . arXiv:math/0212418v1 [math.AG] 1 Dec 2002 Theorem 1. (Voevodsky 1996 [5]) Assume 1/2 F and n 1. The Galois symbol ∈ ≥ M M n hn : K (F )/2K (F ) H (F, Z/2) n n → is an isomorphism.
    [Show full text]