The Abel Prize Ceremony May 22, 2012
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FIELDS MEDAL for Mathematical Efforts R
Recognizing the Real and the Potential: FIELDS MEDAL for Mathematical Efforts R Fields Medal recipients since inception Year Winners 1936 Lars Valerian Ahlfors (Harvard University) (April 18, 1907 – October 11, 1996) Jesse Douglas (Massachusetts Institute of Technology) (July 3, 1897 – September 7, 1965) 1950 Atle Selberg (Institute for Advanced Study, Princeton) (June 14, 1917 – August 6, 2007) 1954 Kunihiko Kodaira (Princeton University) (March 16, 1915 – July 26, 1997) 1962 John Willard Milnor (Princeton University) (born February 20, 1931) The Fields Medal 1966 Paul Joseph Cohen (Stanford University) (April 2, 1934 – March 23, 2007) Stephen Smale (University of California, Berkeley) (born July 15, 1930) is awarded 1970 Heisuke Hironaka (Harvard University) (born April 9, 1931) every four years 1974 David Bryant Mumford (Harvard University) (born June 11, 1937) 1978 Charles Louis Fefferman (Princeton University) (born April 18, 1949) on the occasion of the Daniel G. Quillen (Massachusetts Institute of Technology) (June 22, 1940 – April 30, 2011) International Congress 1982 William P. Thurston (Princeton University) (October 30, 1946 – August 21, 2012) Shing-Tung Yau (Institute for Advanced Study, Princeton) (born April 4, 1949) of Mathematicians 1986 Gerd Faltings (Princeton University) (born July 28, 1954) to recognize Michael Freedman (University of California, San Diego) (born April 21, 1951) 1990 Vaughan Jones (University of California, Berkeley) (born December 31, 1952) outstanding Edward Witten (Institute for Advanced Study, -
A Unified Approach to Three Themes in Harmonic Analysis ($1^{St} $ Part)
A UNIFIED APPROACH TO THREE THEMES IN HARMONIC ANALYSIS (I & II) (I) THE LINEAR HILBERT TRANSFORM AND MAXIMAL OPERATOR ALONG VARIABLE CURVES (II) CARLESON TYPE OPERATORS IN THE PRESENCE OF CURVATURE (III) THE BILINEAR HILBERT TRANSFORM AND MAXIMAL OPERATOR ALONG VARIABLE CURVES VICTOR LIE In loving memory of Elias Stein, whose deep mathematical breadth and vision for unified theories and fundamental concepts have shaped the field of harmonic analysis for more than half a century. arXiv:1902.03807v2 [math.AP] 2 Oct 2020 Date: October 5, 2020. Key words and phrases. Wave-packet analysis, Hilbert transform and Maximal oper- ator along curves, Carleson-type operators in the presence of curvature, bilinear Hilbert transform and maximal operators along curves, Zygmund’s differentiation conjecture, Car- leson’s Theorem, shifted square functions, almost orthogonality. The author was supported by the National Science Foundation under Grant No. DMS- 1500958. The most recent revision of the paper was performed while the author was supported by the National Science Foundation under Grant No. DMS-1900801. 1 2 VICTOR LIE Abstract. In the present paper and its sequel [63], we address three rich historical themes in harmonic analysis that rely fundamentally on the concept of non-zero curvature. Namely, we focus on the boundedness properties of (I) the linear Hilbert transform and maximal operator along variable curves, (II) Carleson-type operators in the presence of curvature, and (III) the bilinear Hilbert transform and maximal operator along variable curves. Our Main Theorem states that, given a general variable curve γ(x,t) in the plane that is assumed only to be measurable in x and to satisfy suitable non-zero curvature (in t) and non-degeneracy conditions, all of the above itemized operators defined along the curve γ are Lp-bounded for 1 <p< ∞. -
Ergodic Theory Plays a Key Role in Multiple Fields Steven Ashley Science Writer
CORE CONCEPTS Core Concept: Ergodic theory plays a key role in multiple fields Steven Ashley Science Writer Statistical mechanics is a powerful set of professor Tom Ward, reached a key milestone mathematical tools that uses probability the- in the early 1930s when American mathema- ory to bridge the enormous gap between the tician George D. Birkhoff and Austrian-Hun- unknowable behaviors of individual atoms garian (and later, American) mathematician and molecules and those of large aggregate sys- and physicist John von Neumann separately tems of them—a volume of gas, for example. reconsidered and reformulated Boltzmann’ser- Fundamental to statistical mechanics is godic hypothesis, leading to the pointwise and ergodic theory, which offers a mathematical mean ergodic theories, respectively (see ref. 1). means to study the long-term average behavior These results consider a dynamical sys- of complex systems, such as the behavior of tem—whetheranidealgasorothercomplex molecules in a gas or the interactions of vi- systems—in which some transformation func- brating atoms in a crystal. The landmark con- tion maps the phase state of the system into cepts and methods of ergodic theory continue its state one unit of time later. “Given a mea- to play an important role in statistical mechan- sure-preserving system, a probability space ics, physics, mathematics, and other fields. that is acted on by the transformation in Ergodicity was first introduced by the a way that models physical conservation laws, Austrian physicist Ludwig Boltzmann laid Austrian physicist Ludwig Boltzmann in the what properties might it have?” asks Ward, 1870s, following on the originator of statisti- who is managing editor of the journal Ergodic the foundation for modern-day ergodic the- cal mechanics, physicist James Clark Max- Theory and Dynamical Systems.Themeasure ory. -
Roth's Theorem in the Primes
Annals of Mathematics, 161 (2005), 1609–1636 Roth’s theorem in the primes By Ben Green* Abstract We show that any set containing a positive proportion of the primes con- tains a 3-term arithmetic progression. An important ingredient is a proof that the primes enjoy the so-called Hardy-Littlewood majorant property. We de- rive this by giving a new proof of a rather more general result of Bourgain which, because of a close analogy with a classical argument of Tomas and Stein from Euclidean harmonic analysis, might be called a restriction theorem for the primes. 1. Introduction Arguably the second most famous result of Klaus Roth is his 1953 upper bound [21] on r3(N), defined 17 years previously by Erd˝os and Tur´an to be the cardinality of the largest set A [N] containing no nontrivial 3-term arithmetic ⊆ progression (3AP). Roth was the first person to show that r3(N) = o(N). In fact, he proved the following quantitative version of this statement. Proposition 1.1 (Roth). r (N) N/ log log N. 3 " There was no improvement on this bound for nearly 40 years, until Heath- Brown [15] and Szemer´edi [22] proved that r N(log N) c for some small 3 " − positive constant c. Recently Bourgain [6] provided the best bound currently known. Proposition 1.2 (Bourgain). r (N) N (log log N/ log N)1/2. 3 " *The author is supported by a Fellowship of Trinity College, and for some of the pe- riod during which this work was carried out enjoyed the hospitality of Microsoft Research, Redmond WA and the Alfr´ed R´enyi Institute of the Hungarian Academy of Sciences, Bu- dapest. -
All That Math Portraits of Mathematicians As Young Researchers
Downloaded from orbit.dtu.dk on: Oct 06, 2021 All that Math Portraits of mathematicians as young researchers Hansen, Vagn Lundsgaard Published in: EMS Newsletter Publication date: 2012 Document Version Publisher's PDF, also known as Version of record Link back to DTU Orbit Citation (APA): Hansen, V. L. (2012). All that Math: Portraits of mathematicians as young researchers. EMS Newsletter, (85), 61-62. General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. Users may download and print one copy of any publication from the public portal for the purpose of private study or research. You may not further distribute the material or use it for any profit-making activity or commercial gain You may freely distribute the URL identifying the publication in the public portal If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim. NEWSLETTER OF THE EUROPEAN MATHEMATICAL SOCIETY Editorial Obituary Feature Interview 6ecm Marco Brunella Alan Turing’s Centenary Endre Szemerédi p. 4 p. 29 p. 32 p. 39 September 2012 Issue 85 ISSN 1027-488X S E European M M Mathematical E S Society Applied Mathematics Journals from Cambridge journals.cambridge.org/pem journals.cambridge.org/ejm journals.cambridge.org/psp journals.cambridge.org/flm journals.cambridge.org/anz journals.cambridge.org/pes journals.cambridge.org/prm journals.cambridge.org/anu journals.cambridge.org/mtk Receive a free trial to the latest issue of each of our mathematics journals at journals.cambridge.org/maths Cambridge Press Applied Maths Advert_AW.indd 1 30/07/2012 12:11 Contents Editorial Team Editors-in-Chief Jorge Buescu (2009–2012) European (Book Reviews) Vicente Muñoz (2005–2012) Dep. -
The History of the Abel Prize and the Honorary Abel Prize the History of the Abel Prize
The History of the Abel Prize and the Honorary Abel Prize The History of the Abel Prize Arild Stubhaug On the bicentennial of Niels Henrik Abel’s birth in 2002, the Norwegian Govern- ment decided to establish a memorial fund of NOK 200 million. The chief purpose of the fund was to lay the financial groundwork for an annual international prize of NOK 6 million to one or more mathematicians for outstanding scientific work. The prize was awarded for the first time in 2003. That is the history in brief of the Abel Prize as we know it today. Behind this government decision to commemorate and honor the country’s great mathematician, however, lies a more than hundred year old wish and a short and intense period of activity. Volumes of Abel’s collected works were published in 1839 and 1881. The first was edited by Bernt Michael Holmboe (Abel’s teacher), the second by Sophus Lie and Ludvig Sylow. Both editions were paid for with public funds and published to honor the famous scientist. The first time that there was a discussion in a broader context about honoring Niels Henrik Abel’s memory, was at the meeting of Scan- dinavian natural scientists in Norway’s capital in 1886. These meetings of natural scientists, which were held alternately in each of the Scandinavian capitals (with the exception of the very first meeting in 1839, which took place in Gothenburg, Swe- den), were the most important fora for Scandinavian natural scientists. The meeting in 1886 in Oslo (called Christiania at the time) was the 13th in the series. -
Klaus Friedrich Roth, 1925–2015 the Godfrey Argent Studio C
e Bull. London Math. Soc. 50 (2018) 529–560 C 2017 Authors doi:10.1112/blms.12143 OBITUARY Klaus Friedrich Roth, 1925–2015 The Godfrey Argent Studio C Klaus Friedrich Roth, who died in Inverness on 10 November 2015 aged 90, made fundamental contributions to different areas of number theory, including diophantine approximation, the large sieve, irregularities of distribution and what is nowadays known as arithmetic combinatorics. He was the first British winner of the Fields Medal, awarded in 1958 for his solution in 1955 of the famous Siegel conjecture concerning approximation of algebraic numbers by rationals. He was elected a member of the London Mathematical Society on 17 May 1951, and received its De Morgan Medal in 1983. 1. Life and career Klaus Roth, son of Franz and Matilde (n´ee Liebrecht), was born on 29 October 1925, in the German city of Breslau, in Lower Silesia, Prussia, now Wroclaw in Poland. To escape from Nazism, he and his parents moved to England in 1933 and settled in London. He would recall that the flight from Berlin to London took eight hours and landed in Croydon. Franz, a solicitor by training, had suffered from gas poisoning during the First World War, and died a few years after their arrival in England. Roth studied at St Paul’s School between 1937 and 1943, during which time the school was relocated to Easthampstead Park, near Crowthorne in Berkshire, as part of the wartime evac- uation of London. There he excelled in mathematics and chess, and one master, Mr Dowswell, observed interestingly that he possessed complete intellectual honesty. -
On Some Recent Developments in the Theory of Buildings
On some recent developments in the theory of buildings Bertrand REMY∗ Abstract. Buildings are cell complexes with so remarkable symmetry properties that many groups from important families act on them. We present some examples of results in Lie theory and geometric group theory obtained thanks to these highly transitive actions. The chosen examples are related to classical and less classical (often non-linear) group-theoretic situations. Mathematics Subject Classification (2010). 51E24, 20E42, 20E32, 20F65, 22E65, 14G22, 20F20. Keywords. Algebraic, discrete, profinite group, rigidity, linearity, simplicity, building, Bruhat-Tits' theory, Kac-Moody theory. Introduction Buildings are cell complexes with distinguished subcomplexes, called apartments, requested to satisfy strong incidence properties. The notion was invented by J. Tits about 50 years ago and quickly became useful in many group-theoretic situations [75]. By their very definition, buildings are expected to have many symmetries, and this is indeed the case quite often. Buildings are relevant to Lie theory since the geometry of apartments is described by means of Coxeter groups: apartments are so to speak generalized tilings, where a usual (spherical, Euclidean or hyper- bolic) reflection group may be replaced by a more general Coxeter group. One consequence of the existence of sufficiently large automorphism groups is the fact that many buildings admit group actions with very strong transitivity properties, leading to a better understanding of the groups under consideration. The beginning of the development of the theory is closely related to the theory of algebraic groups, more precisely to Borel-Tits' theory of isotropic reductive groups over arbitrary fields and to Bruhat-Tits' theory of reductive groups over non-archimedean valued fields. -
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]. -
EMS Newsletter September 2012 1 EMS Agenda EMS Executive Committee EMS Agenda
NEWSLETTER OF THE EUROPEAN MATHEMATICAL SOCIETY Editorial Obituary Feature Interview 6ecm Marco Brunella Alan Turing’s Centenary Endre Szemerédi p. 4 p. 29 p. 32 p. 39 September 2012 Issue 85 ISSN 1027-488X S E European M M Mathematical E S Society Applied Mathematics Journals from Cambridge journals.cambridge.org/pem journals.cambridge.org/ejm journals.cambridge.org/psp journals.cambridge.org/flm journals.cambridge.org/anz journals.cambridge.org/pes journals.cambridge.org/prm journals.cambridge.org/anu journals.cambridge.org/mtk Receive a free trial to the latest issue of each of our mathematics journals at journals.cambridge.org/maths Cambridge Press Applied Maths Advert_AW.indd 1 30/07/2012 12:11 Contents Editorial Team Editors-in-Chief Jorge Buescu (2009–2012) European (Book Reviews) Vicente Muñoz (2005–2012) Dep. Matemática, Faculdade Facultad de Matematicas de Ciências, Edifício C6, Universidad Complutense Piso 2 Campo Grande Mathematical de Madrid 1749-006 Lisboa, Portugal e-mail: [email protected] Plaza de Ciencias 3, 28040 Madrid, Spain Eva-Maria Feichtner e-mail: [email protected] (2012–2015) Society Department of Mathematics Lucia Di Vizio (2012–2016) Université de Versailles- University of Bremen St Quentin 28359 Bremen, Germany e-mail: [email protected] Laboratoire de Mathématiques Newsletter No. 85, September 2012 45 avenue des États-Unis Eva Miranda (2010–2013) 78035 Versailles cedex, France Departament de Matemàtica e-mail: [email protected] Aplicada I EMS Agenda .......................................................................................................................................................... 2 EPSEB, Edifici P Editorial – S. Jackowski ........................................................................................................................... 3 Associate Editors Universitat Politècnica de Catalunya Opening Ceremony of the 6ECM – M. -
Roth's Orthogonal Function Method in Discrepancy Theory and Some
Roth’s Orthogonal Function Method in Discrepancy Theory and Some New Connections Dmitriy Bilyk Abstract In this survey we give a comprehensive, but gentle introduction to the cir- cle of questions surrounding the classical problems of discrepancy theory, unified by the same approach originated in the work of Klaus Roth [85] and based on mul- tiparameter Haar (or other orthogonal) function expansions. Traditionally, the most important estimates of the discrepancy function were obtained using variations of this method. However, despite a large amount of work in this direction, the most im- portant questions in the subject remain wide open, even at the level of conjectures. The area, as well as the method, has enjoyed an outburst of activity due to the recent breakthrough improvement of the higher-dimensional discrepancy bounds and the revealed important connections between this subject and harmonic analysis, proba- bility (small deviation of the Brownian motion), and approximation theory (metric entropy of spaces with mixed smoothness). Without assuming any prior knowledge of the subject, we present the history and different manifestations of the method, its applications to related problems in various fields, and a detailed and intuitive outline of the latest higher-dimensional discrepancy estimate. 1 Introduction The subject and the structure of the present chapter is slightly unconventional. In- stead of building the exposition around the results from one area, united by a com- mon topic, we concentrate on problems from different fields which all share a com- mon method. The starting point of our discussion is one of the earliest results in discrepancy theory, Roth’s 1954 L2 bound of the discrepancy function in dimensions d ≥ 2 [85], as well as the tool employed to obtain this bound, which later evolved into a pow- Dmitriy Bilyk Department of Mathematics, University of South Carolina, Columbia, SC 29208 USA e-mail: [email protected] 1 2 Dmitriy Bilyk erful orthogonal function method in discrepancy theory. -
Calculus Redux
THE NEWSLETTER OF THE MATHEMATICAL ASSOCIATION OF AMERICA VOLUME 6 NUMBER 2 MARCH-APRIL 1986 Calculus Redux Paul Zorn hould calculus be taught differently? Can it? Common labus to match, little or no feedback on regular assignments, wisdom says "no"-which topics are taught, and when, and worst of all, a rich and powerful subject reduced to Sare dictated by the logic of the subject and by client mechanical drills. departments. The surprising answer from a four-day Sloan Client department's demands are sometimes blamed for Foundation-sponsored conference on calculus instruction, calculus's overcrowded and rigid syllabus. The conference's chaired by Ronald Douglas, SUNY at Stony Brook, is that first surprise was a general agreement that there is room for significant change is possible, desirable, and necessary. change. What is needed, for further mathematics as well as Meeting at Tulane University in New Orleans in January, a for client disciplines, is a deep and sure understanding of diverse and sometimes contentious group of twenty-five fac the central ideas and uses of calculus. Mac Van Valkenberg, ulty, university and foundation administrators, and scientists Dean of Engineering at the University of Illinois, James Ste from client departments, put aside their differences to call venson, a physicist from Georgia Tech, and Robert van der for a leaner, livelier, more contemporary course, more sharply Vaart, in biomathematics at North Carolina State, all stressed focused on calculus's central ideas and on its role as the that while their departments want to be consulted, they are language of science. less concerned that all the standard topics be covered than That calculus instruction was found to be ailing came as that students learn to use concepts to attack problems in a no surprise.