哈里·弗斯腾伯格Hillel Furstenberg

哈里·弗斯腾伯格的朋友和同事都叫他Harry。他是Solly 弗斯腾伯格（1900-1940）和Berta Gryzb（1908-1987）的儿子。Harry有一个姐姐Ruth Charlotte Furstenberg，生于1932年5月27日。他出生在一个居住在德国的犹太家庭，当时希特勒刚刚上台，纳粹党已经通过了反犹立法。在弗斯腾伯格生命的最初几年里，犹太人的处境变得越来越艰难，而且因为他的祖父母来自波兰和俄罗斯，全家在1938年11月被勒令离开德国。他们由Berta的姐妹Anna Haller担保，计划前往英国，在伦敦待一年，然后移民到美国。在他们离开之前，1938年11月9日至10日夜间，犹太人的商铺被砸毁，犹太会堂被摧毁，这一夜后来被称为“水晶之夜”。尽管Harry当时只有三岁，那一夜的记忆却一直留在他心中[6]：——

我对德国为数不多的记忆之一就是水晶之夜。我记得看着我们公寓里破碎的窗户。我们就住在一座犹太会堂旁边，我记得站在那里盯着碎玻璃。

这对一个小男孩来说必定是一次创伤性的经历。在1939年人口普查时，也就是当年秋天第二次世界大战开始前不久，这家人住在英格兰诺福克郡唐纳姆的Everswell大道2号。Solly的职业被登记为家具经销商和抛光工，他的妻子Berta登记的职业为无薪家务。Berta的兄弟Morris Giles在美国新泽西州有一个养鸡场，这家人计划去那里。然而，Solly有一种健康问题，他认为这可能会阻止他进入美国，所以他在伦敦接受了手术。手术没有成功，Solly于1940年2月去世。他被安葬在伦敦纽汉的东哈姆犹太公墓。

1940年10月18日，Berta、Ruth和Harry Furstenberg乘坐Cameronia从苏格兰格拉斯哥驶往纽约。尽管由于担心德国潜艇袭击，许多船只都编队横渡大西洋，但Cameronia没有护航。事实上，这是Cameronia最后一次载客横渡大西洋的航行之一，因为它在1940年12月变成了一艘运兵船。弗斯腾伯格一家于10月28日抵达纽约。在进入美国时，Harry的记录显示：年龄，5岁；身高，4英尺；头发，浅棕色；眼睛颜色，蓝色；肤色，中等；入籍意向，是。这家人按计划去Morris Giles的养鸡场居住。Harry的母亲在农场工作，而Harry开始在东布伦瑞克的一所小学接受教育。然而，他们在那里没有住很久，就搬到了华盛顿高地，吸引他们的是那里有一个庞大的德国犹太社区，但他们在叶史瓦大学附近一个多种族聚居区找到了一所房子。Harry[6]：——

……不久便成为位于第185街的拉比摩西·索洛维伊奇克经学院的学生，在那里一直读到八年级。在那些小学岁月里，他已经早早显露出数学方面的天赋，比他大三岁的姐姐成了他的辅导老师。“我们班还在学加法的时候，她已经在教我乘法了。我总是走在全班前面，”他说。“当你在学校里表现出色时，你往往会变得对事物感兴趣。我不确定那时我对数学是否有什么抱负，或者对自己想成为什么样的人有什么打算。”

弗斯腾伯格一家陷入经济困难，只有弗雷达做缝纫机操作工的一份收入。为了帮助养家，弗雷达决定让哈里学一门手艺，经过考试后，他被布鲁克林技术高中录取，目标是学习电气工程。然而，当地的拉比说服了哈里的母亲，说他应该接受训练成为拉比，于是在技术高中只待了一天之后，他就进入了塔木德学院（现为查尔斯·弗农·博伊斯大学高中），该校位于曼哈顿北部的查尔斯·弗农·博伊斯大学校园内。他说[10]：-

我上高中时，真的很喜欢那里教的欧几里得几何。我想我是喜欢几何习题带来的挑战。你可以按自己的方式去做。你不必遵循固定的规则，关键在于你自己的思考。如果它清晰而有逻辑，你就能得到正确答案。我很喜欢这一点。我上高中时，我们学到了虚数。我想，如果我能证明在使用像√–1这样的虚数时，会导致数学中某种矛盾，我就能在数学上成名。我写满了一页又一页的计算，当然什么也没得到，但仅仅做这些计算就是一次很好的经历。

1950年美国人口普查记录显示，贝尔塔、露丝和Harry Furstenberg住在纽约市。贝尔塔是一家内衣工厂的机器操作工，露丝是一家保险机构的职员，而哈里在上学。在塔木德学院的最后一年，Harry Furstenberg是科学俱乐部副主席，并担任化学俱乐部的咨询物理学家。他在第十届西屋科学人才搜寻竞赛中获得荣誉提名。他在1951年年鉴中拿他的数学老师开玩笑[32]：-

在把自己知道的数学知识全都教给利希滕贝格先生之后，哈里现在又大胆地去教格雷策先生初等算术基础了。

Joseph Lichtenberg，文学学士，哥伦比亚大学，1920年；文学硕士，1921年，在Talmudical Academy教授数学，同时皮埃尔·萨米埃尔 L Greitzer，文学硕士，哥伦比亚大学，1936年，教授数学和物理。

弗斯腾伯格 1951年从该科学院毕业，他在叶史瓦学院学习数学和神学。弗斯腾伯格 回忆了他在大学的时光 [12]：-

对我来说，正如无疑对20世纪50年代初就读于叶史瓦学院的许多人一样，数学这门学科与一位杰出的人物——Jekuthiel Ginsburg教授——联系在一起。……在课堂上，他向学生传达了抽象数学思想的内在美。……很难想象有哪个职业生涯比我的职业生涯更归功于一个人和一个机构，我的职业生涯归功于Jekuthiel Ginsburg和叶史瓦大学。除了我学到的数学之外，我还体验到了数学之爱与人类仁慈的融合，这种体验我只能希望自己能为他人复制。

你可以在THIS LINK读到更多弗斯腾伯格对Jekuthiel Ginsburg的回忆。

弗斯腾伯格有幸能够参加顶尖数学家开设的讲座 [12]：-

……还在本科阶段时，我就接触了一系列由来自多所机构的著名教授访问[叶史瓦学院]时所作的高水平高级主题讲座。这些教授包括来自哥伦比亚大学的塞缪尔·艾伦伯格和艾力斯·科尔钦，来自城市学院的杰西·道格拉斯，以及从雪城大学远道而来的阿贝·吉尔巴特。

弗斯腾伯格于1955年担任叶史瓦学院数学俱乐部主席 [24]：-

在会长Harry Furstenberg和副会长Isaac Sadowsky的领导下，数学俱乐部致力于以简化的形式呈现高等数学主题。这些主题以讲座形式准备，并完全由俱乐部的学生成员讲授，包括：“非欧几何”、“几何的变换群”、“拓扑学专题”、“四色定理”以及“Gilel定理的基本例子”[原文如此]。

1955年，弗斯腾伯格从叶史瓦学院毕业，获得了学士和硕士学位。他已经与Note on one type of indeterminate form（1953年）和On the infinitude of primes（1955年）发表了一些论文，均发表在American Mathematical Monthly上。关于素数的论文给出了一个拓扑学证明，证明素数有无穷多个。同样在1955年，即他获得第一个学位的那一年，他在Proceedings of the American Mathematical Society上发表了The inverse operation in groups。这是一篇精彩的论文，给出的结果可以纳入群论课程。Bill Boone评论了这篇论文[4]：-

作者以单个二元运算给出了群的一组优雅公设，这种运算在群论分析中经常出现，$ab^{-1}$。
设G是一个带有运算a*b的系统，使得
(1) 对于G中的任意a, b，a*b在G中，
(2) 对于G中的任意a, b, c，(a*c)*(b*c)=a*b，
(3) 对于G中的任意a，a*G = G。
那么可以推出，存在G中的e，使得对于所有G中的a，a*a = e，G在运算ab=a*(e*b)下是一个群，并且$a^{*}b = ab^{-1}$。如果此外对于所有G中的a, b, c，(c*b)*(c*a) = a*b，那么G是阿贝尔群。类比半群，一个“半群”是满足(1)和(2)的系统G。（并非每个半群都是群。）展示了半群的一个结构定理。

弗斯腾伯格前往普林斯顿大学攻读博士学位，由萨洛蒙·博赫纳指导。此时萨洛蒙·博赫纳对概率论感兴趣，已于1955年出版了他的经典著作Harmonic Analysis and the Theory of Probability，而弗斯腾伯格正是在这一年开始了研究。在1958年提交学位论文Prediction Theory后，弗斯腾伯格获得了博士学位。这篇学位论文于1960年作为Stationary processes and prediction theory出版。P Masani在评论[23]中写道：-

在这项工作中，首先讨论了随机过程的经典预测理论的局限性。根据这一讨论，为单一时间序列制定了一种新的预测理论。在这一发展过程中揭示的思想被证明在预测理论本身之外具有有趣的分支。……这部作品作为一篇关于一个非常困难主题的一流且极具原创性的学位论文而存在。

正是在普林斯顿大学期间，他遇到了他将要娶的女孩。这个故事在[6]中讲述：-

他在普林斯顿大学的室友——当时弗斯腾伯格正在该校攻读数学博士学位——曾在地铁列车上看到一位年轻女子在读一本哲学书，这在1957年可不是常见的景象。Rochelle Cohen从家乡芝加哥来到纽约市度过这一年，她在Boro Park租了一个房间。几个月后，当欢庆妥拉节到来时，她前往当地的一座犹太会堂观看男子们跳舞。恰巧，弗斯腾伯格的室友也在同一座犹太会堂庆祝节日，他认出了地铁上的那个女孩，想到自己单身的室友，便走向Rochelle说：“我知道有个人正适合你。”诚然，一位数学家和一位哲学爱好者未必像是天作之合。“但我常说，她嫁给我的原因之一，”弗斯腾伯格讲述道，“是因为我说服了她，数学中存在美。美来自隐藏之物，而非显露之物。”

在1958-59年担任麻省理工学院讲师一年后，弗斯腾伯格在明尼苏达大学科学、文学与艺术学院数学系工作。在这个系里，他是一个研究概率论的强大团队中的成员。1963年，明尼苏达大学的两个数学系合并为理工学院数学学院，次年弗斯腾伯格被任命为正教授。1965年，他与妻子Rochelle一同前往以色列，被任命为耶路撒冷希伯来大学数学教授。Rochelle是一位作家和杂志编辑，专攻艺术与当代文化。Harry和Rochelle弗斯腾伯格有五个孩子。弗斯腾伯格一直留在希伯来大学，直到2003年退休。他还曾在巴伊兰大学任教。

弗斯腾伯格的许多重要成果都呈现在他的经典专著Recurrence in ergodic theory and combinatorial number theory（1981）中。以下是Michael Keane的书评[22]的摘录：——

这本非常易读的书讨论了动力系统和遍历理论在组合学和数论中的一些近期应用，这些应用主要归功于作者本人。全书分为三部分。第一部分题为“紧空间中的回复与一致回复”，作者介绍了拓扑动力系统中的回复，然后证明了多重乔治·戴维·伯克霍夫回复定理……从这一定理推导出巴特尔·伦德特·范德瓦尔登关于等差数列的定理的多维版本，并给出了在丢番图不等式中的应用。第二部分标题为“保测系统中的回复”。在简要介绍测度论遍历理论的相关部分之后，这一节致力于证明多重回复定理……从这一结果，作者推导出塞迈雷迪·安德烈关于正密度整数序列中存在任意长等差数列的定理的多维版本。第三部分称为“动力学与整数大集”，研究了拓扑动力学中的回复与关于整数有限划分的组合结果（例如Hindman定理、蒂博尔·拉多定理）之间的联系。在这里，邻近性的概念起着核心作用。在阅读这本书时，书评人发现第一部分激发了他的想象力，使他想要继续读下去，第二部分提供了大量工作并考验了他的技术能力，而最后一部分则引导他想象未来研究的可能性。一部优秀的著作！

让我们来看看弗斯腾伯格获得的一些奖项，以便提及他最伟大的数学成就。以色列奖是以色列国颁发的奖项，被视为国家的最高荣誉，于1993年授予弗斯腾伯格。同年，弗斯腾伯格获得了Harvey奖，该奖每年由以色列海法的Technion颁发，表彰[17]：——

……在遍历理论和概率论、索菲斯·李群和拓扑动力学方面的开创性工作。

2004年，他获得了EMET奖，这是一个年度奖项，授予在学术和专业成就方面表现卓越、具有深远影响并对社会做出重大贡献的人。该奖项由以色列科学、艺术与文化促进基金会代表以色列总理颁发。以下是该奖项引文的摘录[27]：-

弗斯腾伯格移民以色列对该国数学领域产生了巨大影响，并帮助以色列转变为遍历论乃至整个数学领域的重要国际中心。在耶路撒冷——他学术活动的中心——他继续创作了一系列不朽的数学著作。1975年，他与Benjamin Weiss教授一起在耶路撒冷开创了遍历论研究年。那一年至今仍被铭记为彻底改变了该领域研究面貌的一年。多年来，他担任世界各地许多大学的客座讲师，包括斯坦福大学、耶鲁大学等。弗斯腾伯格教授并指导了许多在他研究领域及其他更广泛领域攻读高级学位的学生——从而引领了新一代数学家，他们今天在以色列和国外的许多高等教育机构担任教授。

授予弗斯腾伯格的极负盛名的奖项之一是2007年沃尔夫奖[20]（另见[13]）：-

……因他对遍历论、概率论、拓扑动力系统、对称空间上的分析以及齐性流做出的深刻贡献。

引文更详细地介绍了弗斯腾伯格在这些领域的贡献，这些贡献使他获得了该奖项[20]（另见[13]）：-

弗斯腾伯格是概率论、遍历论和拓扑动力学的伟大大师之一。他的贡献包括：将遍历论思想应用于数论和组合数学，以及将概率思想应用于索菲斯·李群及其离散子群的理论。在概率论中，他是研究随机矩阵乘积的先驱，并展示了其极限行为如何与索菲斯·李群中的深刻结构定理紧密相关。这一结果对该领域后续的所有工作产生了重大影响——该领域不仅已成为概率论的主要分支，也成为了统计物理及其他领域的主要分支。在拓扑动力学中，弗斯腾伯格对极小远距流的结构定理的证明引入了全新的技术，彻底改变了该领域。他关于常负曲率曲面上的极限圆流是唯一遍历的定理，已成为索菲斯·李群作用动力学理论的主要组成部分。在研究齐性空间上的随机过程时，他引入了平稳方法，对这些方法的研究使他定义了现在所谓的群的弗斯腾伯格边界。他对群上随机游走渐近行为的分析，对该领域的后续工作产生了持久影响，包括对索菲斯·李群中的格以及群作用的上循环的研究。在遍历论中，弗斯腾伯格发展了动力嵌入的基本概念。这使他在组合数学中取得了惊人的应用，包括对等差数列的塞迈雷迪·安德烈定理的新证明及其深远推广。

除了这些荣誉之外，弗斯腾伯格还当选为以色列科学院（1974年）、美国国家科学院（1989年）和美国艺术与科学院（1995年）的成员。科学院对他的成员身份记录如下[18]：-

弗斯腾伯格是耶路撒冷希伯来大学阿尔伯特·爱因斯坦数学研究所的数学荣休教授。他的研究聚焦于随机现象和遍历理论与数学其他分支的交互作用；特别是群论和数论。他研究了在索菲斯·李群上的随机游走和调和函数理论背景下矩阵随机乘积的行为。他探讨了遍历理论中的回归现象与组合数论的交互作用。目前他正在研究遍历理论在分形几何中的应用。

2003年，在弗斯腾伯格退休之际，以色列科学基金会为他组织了一次数学中的概率研究研讨会。他在2006年作了图兰·帕尔纪念讲座。三次讲座的主题是[24]：-

Lecture 1. Number Theory, Combinatorics and Recurrence in Dynamical Systems; the Correspondence Principle.
Lecture 2. Ergodicity, Mixing, Conventional and non-Conventional Ergodic Theorems.
Lecture 3. The Long Term Memory of Dynamical Systems and the Strange Role of Nilpotent Groups and Nilflows.

2008年，他在乔治·伯克利作了第二十八届年度Bowen讲座。他给出了以下摘要[1]：-

这些讲座将聚焦于遍历理论在分形几何中的作用。我们将考察这样的动力系统，其中时间的推进对应于欧几里得空间中分形的逐步放大。从这个观点来看，特殊分形的自相似现象可以视为对应于动力系统中轨道的周期性。更一般的几乎周期性和回归的动力现象在分形几何中也有其对应物，我们的讨论将大部分致力于阐明这一点。处理“分形测度”，即支撑在分形集上的测度，将是方便的，因为遍历理论的工具将可用。这些想法将在涉及费利克斯·豪斯多夫维数的问题中找到应用，但我们将看到“遍历分形测度”是具有独立兴趣的对象。

我们还注意到，弗斯腾伯格是1984年在布里斯托尔举行的英国数学学术讨论会的全体会议演讲者，当时他作了Ergodic theory and Diophantine problems讲座。

尼尔斯·阿贝尔奖被公认为授予数学家的最高奖项。2020年，该奖联合授予了弗斯腾伯格和格列戈里·马尔古利斯 [33]：-

……因开创性地将概率和动力学方法用于群论、数论和组合数学。

新闻稿称 [34]：-

弗斯腾伯格和格列戈里·马尔古利斯发明了随机游走技术来研究群和图等数学对象，并由此引入概率方法，解决了群论、数论、组合数学和图论中的许多未决问题。随机游走是由一系列随机步骤组成的路径，随机游走的研究是概率论的一个核心分支。“弗斯腾伯格和格列戈里·马尔古利斯的工作展示了跨越不同数学学科界限的有效性，并推倒了纯数学与应用数学之间的传统壁垒，”尼尔斯·阿贝尔委员会主席Hans Munthe-Kaas说。他继续道：“弗斯腾伯格和格列戈里·马尔古利斯巧妙地运用概率方法和随机游走，解决了数学不同领域的深刻问题，震惊了数学界。这带来了大量新结果，例如素数存在长等差数列、理解索菲斯·李群中格的结构，以及构造出应用于通信技术和计算机科学的扩展图，仅举几例。”

弗斯腾伯格1935年生于柏林。他的家庭是犹太人，1939年设法逃离纳粹德国前往美国。不幸的是，他的父亲未能挺过这段旅程，弗斯腾伯格与母亲和姐姐在纽约的一个正统派社区长大。当他发表早期论文之一时，有谣言流传说他不是一个人，而是一群数学家的笔名。那篇论文包含来自如此多不同领域的想法，肯定不可能是一个人的作品吧？在美国几所大学从事数学职业后，他于1965年离开美国前往耶路撒冷希伯来大学，并在那里一直待到2003年退休。他的大部分职业生涯都在以色列度过，帮助该国成为世界数学中心。弗斯腾伯格曾获以色列奖和沃尔夫奖。

关于这个尼尔斯·阿贝尔奖的更多信息，请见THIS LINK。

作为最后评论，让我们注意到Ruth Furstenberg，哈利的姐姐，嫁给了工程师Herbert Berger（1931年10月3日出生于纽约市）。他们移民到以色列，Ruth于2001年8月8日在那里去世。

Hillel Furstenberg is known to his friends and colleagues as Harry. He was the son of Solly Furstenberg (1900-1940) and Berta Gryzb (1908-1987). Harry had an older sister, Ruth Charlotte Furstenberg, born 27 May 1932. He was born into a Jewish family living in Germany shortly after Hitler had come to power and the Nazi party had passed anti-Semitic legislation. Problems for Jewish people became increasingly difficult over the first few years of Hillel's life and, because his grandparents were from Poland and Russia, the family were ordered to leave Germany in November 1938. They were sponsored by Berta's sister Anna Haller, and planned to go to England, spend a year in London, then emigrate to the United States. Before they left, on the night of 9-10 November 1938, Jewish businesses were smashed up, synagogues were destroyed on what has become known as Kristallnacht. Although Harry was only three years old, the memory of that night stayed with him [6]:-

One of my few memories from Germany is Kristallnacht. I remember looking at the broken windows in our apartment. We lived right next to a shul and I remember standing there staring at the broken glass.

It must have been a traumatic experience for the young boy. At the time of the 1939 census, shortly before the start of World War II in the autumn of that year, the family was living at 2 Everswell Avenue, Downham, Norfolk, England. Solly's occupation is given as Furniture Dealer and Polisher with his wife Berta records her occupation as Unpaid Domestic Duties. Berta's brother, Morris Giles, had a poultry farm in New Jersey, United States, and the family planned to go there. Solly, however, had a medical condition which he believed might prevent him entering the United States so he underwent surgery in London. The operation was not successful and Solly died in February 1940. He was buried in the East Ham Jewish Cemetery in Newham, London.

On 18 October 1940 Berta, Ruth and Harry Furstenberg sailed from Glasgow, Scotland, to New York on the Cameronia. Although many ships were crossing the Atlantic in convoy because of fear from German submarine attacks, the Cameronia was unescorted. It was, in fact one of the Cameronia's last passenger carrying journeys across the Atlantic since she became a troop carrying ship in December 1940. The Furstenberg family arrived in New York on 28 October. On entering the United States, Harry's record shows: Age, 5; Height, 4 ft; Hair, light brown; Eye colour, blue; Complexion, medium; Citizenship intention, yes. The family went, as intended, to live with on Morris Giles's poultry farm. Harry's mother worked on the farm while Harry began his education in an elementary school in East Brunswick. They did not live there for very long, however, and they moved to Washington Heights, the attraction there being a large German Jewish community, but they found a house in a multi-ethnic area near Yeshiva University. Harry [6]:-

... soon became a student at Yeshiva Rabbi Moses Soloveichik on 185th Street, which he attended through the eighth grade. During those elementary school years, he had already shown early promise in mathematics, and his sister, three years his senior, became his tutor. "She was teaching me multiplication when my class was learning addition. I was always ahead of my class," he says. "When you're doing well in school, you tend to become interested in things. I'm not sure whether I had any kind of ambitions in mathematics at that time or had any plans about what I wanted to be."

The Furstenberg family suffered financial hardships; only having an income from Freda working as a sewing machine operator. To help support the family, Freda decided that Harry should learn a trade and, after taking a test, he was admitted to Brooklyn Technical High School with the aim of studying electrical engineering. The local Rabbi, however, persuaded Harry's mother that he should train to become a Rabbi, so after only one day at the Technical High School he entered the Talmudical Academy (now the Yeshiva University High School for Boys), located on the campus of Yeshiva University in northern Manhattan. He said [10]:-

When I was in high school, I really enjoyed the Euclidean geometry that was taught there. I guess I enjoyed the challenge of geometry exercises. You are able to do things your way. You do not have to follow definite rules, it is about your own thinking. If it is clear and logical, you get to the right answer. I enjoyed that. We learned about imaginary numbers when I was in high school. I thought I could make my name in mathematics if I proved that, when using imaginary numbers like √–1, it was going to lead to some contradiction in mathematics. I filled pages and pages of calculations and of course it didn't get anywhere, but it was a good experience just doing the calculations.

The 1950 US Census records Berta, Ruth and Harry Furstenberg living in New York City. Berta is a machine operator in an underwear factory, Ruth is a clerk in an insurance agency while Harry is at school. In his final year at Talmudical Academy, Harry Furstenberg was vice-president of the Science Club and consulting physicist to the Chemistry Club. He won an honourable mention in the Tenth Annual Westinghouse Science Talent Search Contest. He poked fun at his mathematics teachers in the 1951 yearbook [32]:-

After having taught Mr Lichtenberg all he knows about Math, Harry is now venturing forth to teach Mr Greitzer the Fundamentals of Elementary Arithmetic.

Joseph Lichtenberg, B.A., Columbia University, 1920; M.A., 1921, taught mathematics at Talmudical Academy while Samuel L Greitzer, M.A., Columbia University, 1936, taught both mathematics and physics.

After Furstenberg graduated from the Academy in 1951 he studied mathematics and divinity at Yeshiva College. Furstenberg recalled his time at university [12]:-

To me, as undoubtedly to many who attended Yeshiva College in the early 1950s, the subject of mathematics was identified with one remarkable individual, Professor Jekuthiel Ginsburg. ... In the classroom, he communicated to his students the innate beauty of abstract mathematical ideas. ... It is hard to imagine a professional career that owes more to one individual and to one institution than my own career owes to Jekuthiel Ginsburg and Yeshiva University. Over and beyond the mathematics I learned, I experienced the love of mathematics blended with human-kindness, an experience I can only wish I could replicate for others.

You can read more of Furstenberg's memories of Jekuthiel Ginsburg at THIS LINK.

At Yeshiva College, Furstenberg was fortunate to be able to attend lectures given by leading mathematicians [12]:-

... while still an undergraduate, I was exposed to a series of high-level lectures in advanced topics given by prominent professors who visited [Yeshiva College] from a number of institutions. These included Samuel Eilenberg and Ellis Kolchin from Columbia University, Jesse Douglas from City College, and Abe Gelbart who travelled from Syracuse University.

Furstenberg was president of the Math Club at Yeshiva College in 1955 [24]:-

The Math Club, under the leadership of Harry Furstenberg, president, and Isaac Sadowsky, vice-president, endeavours to present advanced mathematical topics in simplified form. The topics prepared in lecture form and given entirely by student members of the club included: "Non-Euclidean Geometry", "Groups of Transformation of Geometry", "Topics in Topology", "Four Colour Theorem", and "Elemental Examples of Gilel's Theorem"[sic].

In 1955 Furstenberg graduated from Yeshiva College having been awarded both a B.A. and an M.Sc. He had already published a number of papers with Note on one type of indeterminate form (1953) and On the infinitude of primes (1955) both appearing in the American Mathematical Monthly. The paper on primes gives a topological proof that there are infinitely many primes. Also in 1955, the year he gained his first degree, he published The inverse operation in groups in the Proceedings of the American Mathematical Society. This is a lovely paper, giving results which could be incorporated into a group theory course. Bill Boone reviewed the paper [4]:-

The author gives an elegant set of postulates for groups in terms of a single binary operation which occurs quite frequently in group theoretic analyses, $ab^{-1}$.
Let G be a system with an operation a*b such that
   (1) a*b in G for any a, b in G,
   (2) (a*c)*(b*c)=a*b for any a, b, c in G,
   (3) a*G = G for any a in G.
Then it follows that there is an e in G such that a*a = e for all a in G, that G is a group under the operation ab=a*(e*b), and that $a^{*}b = ab^{-1}$. If in addition (c*b)*(c*a) = a*b for all a, b, c in G, then G is abelian. In analogy with semi-groups, a "half-group" is a system G satisfying (1) and (2). (Not every half-group is a group.) A structure theorem for half-groups is demonstrated.

Furstenberg went to Princeton University to study for his doctorate, supervised by Salomon Bochner. At this time Bochner was interested in probability, having published his classic text Harmonic Analysis and the Theory of Probability in 1955, the year in which Furstenberg began research. After submitting his thesis Prediction Theory in 1958, Furstenberg was awarded his doctorate. This thesis was published as Stationary processes and prediction theory in 1960. P Masani writes in the review [23]:-

In this work the limitations of the classical prediction theory of stochastic processes are first discussed. In the light of this discussion a new prediction theory for single time-sequences is formulated. The ideas uncovered in the course of this development are shown to have interesting ramifications outside prediction theory proper. ... the work stands as a first-rate and highly original dissertation on a very difficult subject.

It was while he was at Princeton University that he met the girl he was to marry. The story is related in [6]:-

His roommate from Princeton University, where Furstenberg was then enrolled in the mathematics doctoral program, had seen a young woman on a subway train reading a philosophy book - not the most common sight in 1957. Rochelle Cohen had come from her hometown of Chicago to spend the year in New York City, where she was renting a room in Boro Park. A few months later, when Simchat Torah arrived, she made her way to a local shul to watch the men dance. As it happens, Furstenberg's roommate was celebrating Yom Tov at that same shul, and recognising the girl on the subway and thinking of his bachelor roommate, he approached Rochelle and said: "I know just the guy for you." True, a mathematician and a philosophy aficionada don't necessarily seem like a match made in Heaven. "But I often say that one of the reasons she married me," relates Furstenberg, "is because I convinced her that there's beauty in mathematics. Beauty comes from the hidden, not the revealed."

After a year 1958-59 as an Instructor at Massachusetts Institute of Technology, Furstenberg worked at the Mathematics Department in the College of Science, Letters, and Arts of the University of Minnesota. In this Department he was a member of a strong group working on probability theory. In 1963 the two University of Minnesota Departments of Mathematics were merged into the School of Mathematics in the Institute of Technology and in the following year Furstenberg was appointed a full professor. In 1965, along with his wife Rochelle, he went to Israel when he was appointed as Professor of Mathematics at the Hebrew University of Jerusalem. Rochelle is a writer and magazine editor specialising in arts and contemporary culture. Harry and Rochelle Furstenberg have five children. Furstenberg remained at the Hebrew University until he retired in 2003. He has also taught at Bar Ilan University.

Many important results due to Furstenberg are presented in his classic monograph Recurrence in ergodic theory and combinatorial number theory (1981). Here are extracts from a review by Michael Keane [22]:-

This very readable book discusses some recent applications, due principally to the author, of dynamical systems and ergodic theory to combinatorics and number theory. It is divided into three parts. In Part I, entitled "Recurrence and uniform recurrence in compact spaces", the author gives an introduction to recurrence in topological dynamical systems, and then proves the multiple Birkhoff recurrence theorem ... From this theorem a multidimensional version of van der Waerden's theorem on arithmetic progressions is deduced, and applications to Diophantine inequalities are given. Part II carries the title "Recurrence in measure preserving systems". After a short introduction to the relevant part of measure-theoretic ergodic theory, this section is devoted to a proof of the multiple recurrence theorem ... From this result the author deduces a multidimensional version of Szemerédi's theorem on the existence of arbitrarily long arithmetic progressions in sequences of integers with positive density. Part III, called "Dynamics and large sets of integers", investigates the connections between recurrence in topological dynamics and combinatorial results concerning finite partitions of the integers (e.g., Hindman's theorem, Rado's theorem). Here the notion of proximality plays a central role. In reading this book, the reviewer found that the first part tickled his imagination and made him want to continue, the second part provided a good deal of work and tested his technical ability, while the last part led him to imagine the future possibilities for research. An excellent work!

Let us look at some of the awards that Furstenberg has received so that we can mention his greatest mathematical achievements. The Israel Prize, an award made by the State of Israel that is regarded as the state's highest honour, was presented to Furstenberg in 1993. In the same year Furstenberg received the Harvey Prize, awarded annually by the Technion in Haifa, Israel, for [17]:-

... ground-breaking work in ergodic theory and probability, Lie groups and topological dynamics.

In 2004 he received the EMET Prize, an annual award given for excellence in academic and professional achievements that have far reaching influence and significant contribution to society. The prizes are sponsored by the Foundation for the Advancement of Science, Art and Culture in Israel acting for the Prime Minister of Israel. Here is an extract from the citation for the Prize [27]:-

Professor Furstenberg's immigration to Israel had great influence on the field of mathematics in the country, and helped transformed Israel into an important international centre in ergodic theory in particular and in mathematics in general. In Jerusalem, which was the centre of his academic activities, he continued producing a long series of monumental mathematical works. In 1975 he inaugurated, along with Professor Benjamin Weiss, an ergodic theory research year in Jerusalem. That year is still remembered as the year that entirely changed the face of research in the field. Through the years, he has been a guest lecturer at many universities around the world, including Stanford, Yale, and others. Professor Furstenberg has taught and guided many students studying towards advanced degrees in his field of research and in other, wider fields - thus ushering a new generation of mathematicians who today serve as professors at institutions of higher learning in Israel and abroad.

One of the highly prestigious awards given to Furstenberg has been the 2007 Wolf Prize [20] (see also [13]):-

... for his profound contributions to ergodic theory, probability, topological dynamics, analysis on symmetric spaces and homogenous flows.

The citation goes into more details of Furstenberg's contributions to these areas which led to the award [20] (see also [13]):-

Professor Harry Furstenberg is one of the great masters of probability theory, ergodic theory and topological dynamics. Among his contributions: the application of ergodic theoretic ideas to number theory and combinatorics and the application of probabilistic ideas to the theory of Lie groups and their discrete subgroups. In probability theory he was a pioneer in studying products of random matrices and showing how their limiting behaviour was intimately tied to deep structure theorems in Lie groups. This result has had a major influence on all subsequent work in this area - which has emerged as a major branch not only in probability, but also in statistical physics and other fields. In topological dynamics, Furstenberg's proof of the structure theorem for minimal distal flows, introduced radically new techniques and revolutionised the field. His theorem that the horocycle flow on surfaces of constant negative curvature is uniquely ergodic, has become a major part of the dynamical theory of Lie group actions. In his study of stochastic processes on homogenous spaces, he introduced stationary methods whose study led him to define what is now called the Furstenberg Boundary of a group. His analysis of the asymptotic behaviour of random walks on groups, has had a lasting influence on subsequent work in this area, including the study of lattices in Lie groups and co-cycles of group actions. In ergodic theory, Furstenberg developed the fundamental concept of dynamical embedding. This led him to spectacular applications in combinatorics, including a new proof of the Szemerédi Theorem on arithmetical progressions and far-reaching generalisations thereof.

In addition to these honours, Furstenberg has been elected to the Israel Academy of Sciences (1974), the United States National Academy of Sciences (1989), and the American Academy of Arts and Sciences (1995). The Academy recorded his membership as follows [18]:-

Dr Hillel Furstenberg is a Professor Emeritus of Mathematics at the Hebrew University of Jerusalem in the Einstein Institute of Mathematics. His research focuses on the interaction of stochastic phenomena and ergodic theory with other branches of mathematics; particularly group theory and number theory. He has studied the behaviour of random products of matrices in the context of random walks on Lie groups and the theory of harmonic functions. He has investigated the interaction of recurrence phenomena in ergodic theory with combinatorial number theory. Currently he is studying applications of ergodic theory to the geometry of fractals.

In 2003, on the occasion of Furstenberg's retirement, the Israel Science Foundation organised a research workshop Conference on Probability in Mathematics in his honour. He delivered the Paul Turán Memorial Lectures in 2006. The topics of the three lectures were [24]:-

Lecture 1. Number Theory, Combinatorics and Recurrence in Dynamical Systems; the Correspondence Principle.
Lecture 2. Ergodicity, Mixing, Conventional and non-Conventional Ergodic Theorems.
Lecture 3. The Long Term Memory of Dynamical Systems and the Strange Role of Nilpotent Groups and Nilflows.

In 2008 he delivered the Twenty-Eighth Annual Bowen Lectures at Berkeley. He gave the following Abstract [1]:-

These lectures will focus on the role of ergodic theory in the geometry of fractals. We shall be looking at dynamical systems in which progression in time corresponds to progressively increasing magnification of fractals in Euclidean space. From this point of view the phenomenon of self-similarity for special fractals can be regarded as corresponding to that of periodicity of orbits in dynamical systems. The more general dynamical phenomena of almost periodicity and recurrence also have their counterparts in the geometry of fractals, and much of our discussion will be devoted to clarifying this. It will be convenient to deal with "fractal measures", i.e., measures supported on fractal sets, for which tools of ergodic theory will be available. These ideas will find application in questions involving Hausdorff dimension, but we will see that "ergodic fractal measures" are objects of independent interest.

We also note that Furstenberg was a plenary speaker at the British Mathematical Colloquium at Bristol in 1984 when he gave the lecture Ergodic theory and Diophantine problems.

The Abel Prize is recognised as the highest possible award to a mathematician. It was presented jointly to Hillel Furstenberg and Gergory Margulis in 2020 [33]:-

... for pioneering the use of methods from probability and dynamics in group theory, number theory and combinatorics.

The Press Release states [34]:-

Hillel Furstenberg and Gregory Margulis invented random walk techniques to investigate mathematical objects such as groups and graphs, and in so doing introduced probabilistic methods to solve many open problems in group theory, number theory, combinatorics and graph theory. A random walk is a path consisting of a succession of random steps, and the study of random walks is a central branch of probability theory. "The works of Furstenberg and Margulis have demonstrated the effectiveness of crossing boundaries between separate mathematical disciplines and brought down the traditional wall between pure and applied mathematics," says Hans Munthe-Kaas, chair of the Abel committee. He continues: "Furstenberg and Margulis stunned the mathematical world by their ingenious use of probabilistic methods and random walks to solve deep problems in diverse areas of mathematics. This has opened up a wealth of new results, such as the existence of long arithmetic progressions of prime numbers, understanding the structure of lattices in Lie groups, and the construction of expander graphs with applications to communication technology and computer science, to mention a few."

Hillel Furstenberg was born in Berlin in 1935. His family was Jewish and they managed to flee from Nazi Germany to the U.S. in 1939. Sadly, his father did not survive the journey, and Furstenberg grew up with his mother and sister in an orthodox community in New York. When he published one of his early papers, a rumour circulated that he was not an individual but instead a pseudonym for a group of mathematicians. The paper contained ideas from so many different areas, surely it could not possibly be the work of one man? Following a career in mathematics at several universities in the U.S., he left the country in 1965 for the Hebrew University of Jerusalem, where he stayed until his retirement in 2003. Spending most of his career in Israel, he helped establish the country as a world centre for mathematics. Furstenberg has won the Israel Prize and the Wolf Prize.

There is more about this Abel prize at THIS LINK.

As a final comment, let us note that Ruth Furstenberg, Harry's older sister, married the engineer Herbert Berger (born in New York City on 3 October 1931). They emigrated to Israel and Ruth died there on 8 August 2001.