2024 邵逸夫数学科学奖授予南非数学家数论大家彼得·萨纳克

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2024 邵逸夫数学科学奖授予南非数学家数论大家彼得·萨纳克

撰文  流风不冻 2024-05-24 12:30 陕西


邵逸夫数学科学奖颁予彼得·萨纳克 Peter Sarnak 以表彰他将数论、分析学、组合学、动力学、几何学和谱理论结合起来，发展出薄群的算术理论和仿射筛法。

彼得·萨纳克简介：

彼得·萨纳克 (Peter Sarnak) 在 1953 年于南非约翰内斯堡出生，现为美国普林斯顿高等研究院数学戈帕·普拉萨德讲座教授及美国普林斯顿大学数学尤金·希金斯讲座教授。他在 1975 年于南非威特沃特斯兰德大学获得数学学士学位，并在 1980 年于美国史丹福大学取得数学博士学位。他曾担任美国纽约大学助理教授 (1980–1983) 和副教授 (1983)。之后，他加入史丹福大学，先后担任副教授 (1984–1987) 和教授 (1987–1991)。自 1991 年起，他转到普林斯顿 大学任职，并被委任为法恩讲座教授 (1995–1996) 和数学系主任 (1996–1999)。他曾是普林斯顿高等研究院的成员 (1999–2002 和 2005–2007)，并自 2007 年起担任该研究院教授。他也是纽约大学库朗数学科学研究所教授 (2001–2005) 和普林斯顿大学数学尤金·希金斯讲座教授 (2002–)。他是美国国家科学院和英国伦敦皇家 学会院士。2019 年获得西尔维斯特奖（英国皇家学会）。2014 年获得沃尔夫奖。

获奖理由：



获奖原文：

The Shaw Prize in Mathematical Sciences 2024 is awarded to Peter Sarnak, Gopal Prasad Professor, School of Mathematics, Institute for Advanced Study and Eugene Higgins Professor of Mathematics, Princeton University, USA, for his development of the arithmetic theory of thin groups and the affine sieve, by bringing together number theory, analysis, combinatorics, dynamics, geometry and spectral theory.

A natural number is called a prime number if it is larger than 1 and is not the product of two strictly smaller natural numbers which themselves are larger than 1. For example, 2 is a prime number, but 4 = 2 × 2 is not. Euclid’s theorem (circa 300 BCE) asserts that any natural number other than 0 and 1 is the product of prime numbers, and that there are infinitely many prime numbers. The study of the distribution of the prime numbers is a core topic in Number Theory.

The search for prime numbers has been a central theme in number theory since the ancient Greeks. One looks for polynomial functions f(x) such that f(x) is prime for infinitely many integers x. Euclid’s theorem says that f(x) = x is one such function. One may enlarge the problem by requiring that f(x) be almost prime valued, that is, the product of a bounded number of primes for infinitely many integers x. For example, the Twin Prime Conjecture is equivalent to the statement that f(x) = x(x+2) is a product of two primes for infinitely many integers x. The Chinese mathematician Jingrun Chen (1973), using Brun’s combinatorial sieve, showed that this function has at most 3 prime factors for infinitely many integers x. One may also restrict the set of x considered by requiring them to lie in a sparse subset of the integers. A similar problem can be posed for any polynomial with integer coefficients in several variables.

Sarnak pioneered the search for almost prime values of polynomials in sparse subsets arising as the orbit of a thin group. A thin group is a subgroup of an arithmetic group with a Goldilocks property: it is neither too large (being of infinite index) nor too small (having the same Zariski closure as the arithmetic group). Thin groups arise very naturally in pure and applied mathematics. For example, the symmetry group of integral Apollonian circle packings is a thin group. In addition, there is an abundance of Kleinian groups, or more generally monodromy groups of differential equations, that are thin groups.

Expanders are highly connected sparse graphs widely used in computer science. Foreseeing how the expander property of finite quotients of a thin group could be used to produce almost primes, Sarnak developed the affine sieve. Sarnak, together with Bourgain and Gamburd, produced expanders out of some thin groups. The construction relies on earlier foundational work by Sarnak and Xue in which they showed a relation between the minimal dimension of representations of finite linear groups and expanders.

Sarnak, together with Bourgain and Gamburd, obtained a precise counting and equidistribution result for integral vectors on an orbit of a thin group which take almost prime values when one applies a given polynomial function to them.

Sarnak, together with Golsefidy, established that, under some natural hypotheses, an integral polynomial function produces almost primes in a Zariski dense subset of a thin orbit.

Sarnak's introduction of combinatorial and ergodic theoretical methods to Diophantine problems has had a profound impact. His original and deep vision has launched a vast research programme that brings together number theory, combinatorics, analysis, dynamics, geometry and spectral theory.

Mathematical Sciences Selection Committee

The Shaw Prize

21 May 2024, Hong Kong

关于邵逸夫数学科学奖：

邵逸夫数学科学奖是 2002 年由香港著名的电影制作人邵逸夫爵士创立的奖项，首届的颁奖礼在 2004 年举行。

邵逸夫奖基金会每年选出世界上在数学、医学及天文学三方面有成就的科学家，颁授一百万美元奖金以作表扬。并设有天文学奖、生命科学与医学奖、数学科学奖，共三个奖项；它是个国际性奖项，形式模仿诺贝尔奖，由邵逸夫奖基金会有限公司作管理。

数学是一切自然科学和现代技术的基础语言。数学在二十世纪发展精进，既开创了新领域，亦解决了重大且棘手的旧难题，影响遍及每一门创造性的科学和技术，社会科学亦受其惠。

因为计算器科学、信息科技与统计学在二十世纪的发展，数学在二十一世纪对人类将会更加重要。

历届获奖人员：





参考链接：

1. https://www.shawprize.org/laurea ... /?type=Contribution

2. https://baike.baidu.com/item/%E9 ... /14077613?fr=ge_ala

3. https://mp.weixin.qq.com/s/bNREH7QWpOL6Vu459srgWg

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