Springer 刚刚推出了 2010 年出版的 “The Abel Prize 2003–2007” 的续集 “The Abel Prize 2008–2012“.
The Abel Prize 2008-2012
Covering the years 2008-2012, this book profiles the life and work of recent winners of the Abel Prize: John G. Thompson and Jacques Tits, 2008; Mikhail Gromov, 2009; John T. Tate Jr., 2010; John W. Milnor, 2011; Endre Szemerédi, 2012.
The profiles feature autobiographical information as well as a description of each mathematician’s work. In addition, each profile contains a complete bibliography, a curriculum vitae, as well as photos — old and new. As an added feature, interviews with the Laureates can be streamed from the Abel Prize web site.
The book also presents a history of the Abel Prize written by the historian Kim Helsvig, and includes a facsimile of a letter from Niels Henrik Abel, which is transcribed, translated into English, and placed into historical perspective by Christian Skau.
The 2014 Wolf Prize in Mathematics is awarded to Peter Sarnak, for his deep contributions in analysis, number theory, geometry, and combinatorics.
Peter Sarnak is on the permanent faculty at the School of Mathematics of the Institute for Advanced Study, Princeton, NJ, USA.
Peter Clive Sarnak (born December 18, 1953) graduated University of the Witwatersrand (B.Sc. 1975) and Stanford University (Ph.D. 1980), under the direction of Paul Cohen.
Prof. Sarnak is a mathematician of an extremely broad spectrum with a far-reaching vision. He has impacted the development of several mathematical fields, often by uncovering deep and unsuspected connections. In analysis, he investigated eigenfunctions of quantum mechanical Hamiltonians which correspond to chaotic classical dynamical systems in a series of fundamental papers. He formulated and supported the “Quantum Unique Ergodicity Conjecture” asserting that all eigenfunctions of the Laplacian on negatively curved manifolds are uniformly distributed in phase space. Sarnak’s introduction of tools from number theory into this domain allowed him to obtain results which had seemed out of reach and paved the way for much further progress, in particular the recent works of E. Lindenstrauss and N. Anantharaman. In his work on L-functions (jointly with Z. Rudnick) the relationship of contemporary research on automorphic forms to random matrix theory and the Riemann hypothesis is brought to a new level by the computation of higher correlation functions of the Riemann zeros. This is a major step forward in the exploration of the link between random matrix theory and the statistical properties of zeros of the Riemann zeta function going back to H. Montgomery and A. Odlyzko. In 1999 it culminates in the fundamental work, jointly with N. Katz, on the statistical properties of low-lying zeros of families of L-functions. Sarnak’s work (with A. Lubotzky and R. Philips) on Ramanujan graphs had a huge impact on combinatorics and computer science. Here again he used deep results in number theory to make surprising and important advances in another discipline.
By his insights and his readiness to share ideas he has inspired the work of students and fellow researchers in many areas of mathematics.
Professor Gerd Faltings, winner of the prize in science, is the Director at the Max-Planck Institute for Mathematics in Bonn. He has made groundbreaking contributions to algebraic geometry and number theory. His work combines ingenuity, vision and technical power. He has introduced stunning new tools and techniques which are now constantly used in modern mathematics.
Faltings’ deep insights into the p-adic cohomology of algebraic varieties have been crucial to modern developments in number theory. His work on moduli spaces of abelian varieties has had great influence on arithmetic algebraic geometry. He has introduced new geometric ideas and techniques in the theory of Diophantine approximation, leading to his proof of Lang’s conjecture on rational points of abelian varieties and to a far-reaching generalization of the subspace theorem. Professor Faltings has also made important contributions to the theory of vector bundles on algebraic curves with his proof of the Verlinde formula.
问题 6 的答案也是 R. L. Graham 提供的: Graham published a proof in 1963 as “A Theorem on Partitions”, Journal of the Australian Mathematical Society 3 (1963), pp. 435-441.
If \(n\) is an integer exceeding \(77\) then there exist positive integers \(k\), \(a_1\), \(a_2\), \(\dotsc\), \(a_k\) such that:
His proof is constructive and fairly short, but it does require a long table of decompositions for relatively small values of \(n\). It would be interesting to see a non-constructive proof that doesn’t require such a long list.
问题 7 也不简单.
Granville and Selfridge, Product of integers in an interval, modulo squares: “We prove a conjecture of Irving Kaplansky which asserts that between any pair of consecutive positive squares there is a set of distinct integers whose product is twice a square.”
The details are Electronic Journal of Combinatorics, Volume 8(1), 2001.
有比问题 8 更普遍的结果. More precisely, let \(\zeta_1\), \(\dotsc\), \(\zeta_k\) be \(n\)-th roots of unity. If
\[|\sum_{i=1}^k n_i\zeta_i|= 1,\]
where \(n_i\in\mathbb Z\), then \(\sum\limits_{i=1}^k n_i \zeta_i\) is also an \(n\)-th root of unit.
美国总统奥巴马在白宫发表电视讲话悼念曼德拉时说: “自己对曼德拉的逝世感到悲痛, 指曼德拉一生追求理想, 并付诸行动, 追求建立正义自由的社会. 他的离世是一大损失, 世界失去一位典范. 奥巴马赞扬曼德拉是时代人物, 无人能及. 现在, 让我们暂时停下来, 感谢曼德拉曾经存在过. 他是一个用自己双手书写历史的伟人, 他将整个精神世界拉向了正义的一边. ” 奥巴马还说, 他和无数人一样, 曼德拉的一生给了他积极的启发. 他现在不再属于我们, 他属于这个时代. 历史永远会记住他.” “Obama: “Nelson Mandela lived for that ideal, and he made it real. He achieved more than could be expected of any man. Today, he has gone home. And we have lost one of the most influential, courageous, and profoundly good human beings that any of us will share time with on this Earth. a man who took history in his hands, and bent the arc of the moral universe toward justice. He no longer belongs to us — he belongs to the ages.”
“The Washington Post(华盛顿邮报)”发表社论作出的评价是: “罗斯福, 斯大林, 丘吉尔, 希特勒……这些是决定 20世纪上半叶的名字; 甘地, 马丁路德·金, 曼德拉, 则是对战后世界有着重大影响的人物.”
自1989 年开始, Ostrowski 基金会每两年发一次奖, 奖金 \(100,000\) Swiss francs(瑞士法郎), 奖励 for recent outstanding achievements in pure mathematics and in the foundations of numerical mathematics. 该奖 1997 年的获得者是 Andrew J. Wiles; 2001 年由 Henryk Iwaniec, Peter Sarnak, Richard L. Taylor 三人分享荣誉; 2005年的奖励授予 Ben Green 和 Terence Tao.
Ostrowski 基金的目的是为了促进数学科学. The Foundation A. M. Ostrowski for an international prize in higher mathematics was created by Alexander Markovich Ostrowski (1893-1986) who from 1928 to 1958 was professor of mathematics at the University of Basel.
