威廉·瓦兰斯·道格拉斯·霍奇W. V. D. Hodge

威廉·瓦兰斯·道格拉斯·霍奇的父母是Janet Vallance（生于1875年），她的父亲霍奇 Vallance在苏格兰爱丁堡拥有一家糖果店，以及Archibald James Hodge（1869-1938），他在房地产市场工作，是杰西·道格拉斯 and Company公司的合伙人。Archibald Hodge和Janet Vallance于1900年结婚（[5]或[6]）：-

起初，Vallance家族强烈反对这桩婚事。尽管此时两个家庭都生活宽裕，但Vallance家族由于精心积累储蓄，明显更为富有，他们怀疑年轻的Archibald是贪图他们女儿的钱（这完全不属实）。此外，Archibald作为富裕父母的独子，过着放荡的生活，并养成了饮酒的嗜好。

霍奇出生于苏格兰爱丁堡的1 Church Hill Place。他是父母三个孩子中的第二个，有一个哥哥Archibald Vallance 霍奇和一个妹妹Janet Horsburgh 霍奇。他四岁开始上学，在当地幼儿园度过了两年。

离开幼儿园后，霍奇的全部教育都在爱丁堡的George Watson's 查尔斯·弗农·博伊斯 College，他于1909年入学，在那里学习直到1920年。他在学校的各门学科都表现出色，这对他很重要，因为虽然他的父亲和兄弟都是优秀的高尔夫球手，霍奇在高尔夫球场上却明显缺乏能力。因此，他的学业成功对他是一种补偿，他决心充分利用这一点。霍奇在第一次世界大战爆发时十一岁，接下来的四年战争岁月十分艰难。在这几年的中间，即1916年，他的叔叔霍奇去世，这导致他的哥哥Archibald Vallance 霍奇离开学校，成为Vallance糖果企业的学徒。家族决定Archibald在培训后将接管企业。这对霍奇产生了相当大的影响，因为现在他必须在George Watson's自己闯荡，这使他与学业上上进的学生成为朋友。从那时起，他的表现明显改善，但由于无人指导，他在科目选择上做出了一些糟糕的决定。他选择选修拉丁语和希腊语课程，而不是科学课程。这个决定，他在后来的生活中觉得是一个重大错误。

1918年战争结束，住房需求急剧上升，但由于道格拉斯公司的许多员工仍在军队中，家里决定霍奇应该离开学校，加入公司。1918年复活节，他离开乔治·沃森学院，在道格拉斯公司担任文员。霍奇胜任这项工作，但觉得这种相当琐碎无聊的例行工作对他毫无挑战。做了六个月文员工作后，家里决定让他回到乔治·沃森学院，他又在那里度过了两年。在这两年里，霍奇的家人就他未来的职业进行了许多讨论，最终决定他应该以进入文官队伍为目标。这将要求他拥有大学学位，因此作为计划的一部分，他准备参加爱丁堡大学助学金竞赛。他最好的学校科目是数学，所以他的目标是攻读数学和哲学学位。他的数学才能是由他的老师Peter Ramsay激发出来的，这位老师（[5]或[6]）：

……具有一种天赋，能够以霍奇所描述的那种讽刺与善意奇妙结合的方式，激发学生身上任何可能存在的数学才能。

1920年6月，霍奇参加了助学金竞赛。他表现相当不错，在名单上排名相当靠前，但看起来并不突出。然而，有一份试卷旨在为弗瑞兹·约翰威尔士数学助学金寻找最优秀的数学学生，在这份特殊的数学试卷中，他确实名列榜首。赢得这项数学助学金使他在经济上有了保障，并使他得以进入爱丁堡大学学习。他在特殊试卷中表现的另一个非常积极的方面是，甚至在他开始学习之前，埃德蒙·泰勒·惠特克教授就已经知道了他这个有才华的学生。他于1920年10月进入爱丁堡大学，在那里由埃德蒙·泰勒·惠特克授课，并成为埃德蒙·泰勒·惠特克的儿子、才华横溢的约翰·麦克纳滕·惠特克的同学。霍奇在学校表现不错，但在大多数班级中仅能排在前三分之一，却发现在大学的数学课上轻松地名列第一。除了数学，他还选修了英语、经济学和物理学课程，但在这些科目上他的表现不那么令人印象深刻。在这个阶段，他仍然打算毕业后进入公务员系统，但由于他的成就似乎仅仅依赖于他的数学能力，他开始觉得仅凭数学专长进入公务员系统会限制他的职业生涯。在埃德蒙·泰勒·惠特克的大力鼓励下，他开始考虑学术生涯，埃德蒙·泰勒·惠特克建议他继续在剑桥学习。1923年从爱丁堡大学以数学一等荣誉毕业后，他于同年10月进入剑桥大学圣约翰学院。

他在剑桥的学习由他从爱丁堡大学赢得的范邓禄普助学金和圣约翰学院的60英镑奖学金资助。剑桥大学对年轻的霍奇来说有点令人震惊，因为他在爱丁堡学的是相当老式的数学课程。剑桥的数学荣誉学位考试刚刚经历了一次重大修订，霍奇由利特尔伍德和F P怀特（讲授射影几何）授课。他还由亨利·弗雷德里克·贝克授课，而霍奇由于对这些材料准备不足，要跟上这些要求很高的课程十分吃力。然而，正是怀特的课程最激励霍奇（[5]或[6]）：

到达剑桥后不久，霍奇就不得不决定他打算在B计划中专攻什么学科。结果这个决定变得极其容易。从他加入F P White的射影几何课程那一刻起，他就再没想过任何其他主题。这似乎令人惊讶，因为White虽然是一位热情的几何学家和迷人的人，却并未自称是杰出的讲师。解释在于爱丁堡和剑桥对射影几何的处理方式之间的对比。在爱丁堡，它是欧几里得几何的附属物，而White的课程全面阐述了基于关联公理的合成方法及其与坐标几何的关系。这为霍奇打开了一个全新的世界，用他自己的话说，他很快就被“迷住了”。

1921年，Alan Broadbent已注册入约翰学院，一旦霍奇来到同一学院，两人便成了朋友。他们年龄几乎完全相同，霍奇比Broadbent小约两周。在1925年数学荣誉学位考试中取得优异成绩后，霍奇继续赢得史密斯奖，并靠弗格森奖学金在剑桥进行了一年的研究。他的朋友Broadbent此时也在利特尔伍德的指导下进行研究，但霍奇似乎无意花超过一年时间进行研究，因此他只非正式地从亨利·弗雷德里克·贝克那里获得建议。在某些方面，这一年并不十分成功，因为霍奇在一年的大部分时间里没有找到任何能取得进展的问题。然而，在其他方面，这一年是有用的，因为在紧张的荣誉学位考试岁月之后，他不得不付出巨大努力以取得顶尖成绩，现在压力减轻了。1925-26年，他与Broadbent住在约翰学院的房间里，两人将在余生中保持亲密友谊。事实上，Broadbent于1930年娶了霍奇的妹妹珍妮特（妮塔）霍奇。

霍奇于1926年被任命为布里斯托大学的助理讲师，并在那里度过了五年。这对他是一段非常有益的时期，意识到他的研究潜力后，系主任Henry Ronald Hassé给了他轻松的教学负担，以便他能专注于研究。他在布里斯托期间开始发表论文，即Linear systems of algebraic curves on a plane and on a cone（1928）、The isolated singularities of an algebraic surface（1929）、On multiple integrals attached to an algebraic variety（1930）和Further properties of abelian integrals attached to algebraic varieties（1931）。正是1930年的论文为他带来了国际关注。在其中，他应用了所罗门·莱夫谢茨1929年论文中的拓扑思想，该论文研究了曲线上的积分。霍奇能够使用所罗门·莱夫谢茨的拓扑方法解决弗朗切斯科·塞维里提出的关于曲面积分的问题。同年，他当选为剑桥圣约翰学院的会士。

前一年，即1929年7月27日，他与凯瑟琳·安妮·卡梅伦结婚，她是牛津大学出版社爱丁堡分社经理罗伯特·史蒂文森·卡梅伦的女儿。他们有一个儿子和一个女儿。1931年，在赢得1851年展览学生奖学金后，他前往普林斯顿，以便与他极为钦佩的所罗门·莱夫谢茨一起工作。他与所罗门·莱夫谢茨的初次接触很困难，因为所罗门·莱夫谢茨拒绝相信霍奇在 On multiple integrals attached to an algebraic variety中的结果是正确的[5]：-

起初，所罗门·莱夫谢茨公开坚称霍奇错了，并写信要求他撤回论文。最终，所罗门·莱夫谢茨和霍奇于1931年5月在剑桥马克斯·纽曼的房间里会面。经过长时间讨论，达成了武装中立状态，并邀请霍奇在下一学年前往普林斯顿。

霍奇在普林斯顿的第一个月里，一直试图说服所罗门·莱夫谢茨相信他是对的。一旦被说服，所罗门·莱夫谢茨公开承认了自己的错误，并非常慷慨地赞扬了霍奇的工作。在美国期间，霍奇在约翰威廉·霍普金斯大学与奥斯卡·扎里斯基一起学习了两个月。这是所罗门·莱夫谢茨的建议，他说霍奇将从奥斯卡·扎里斯基那里获得关于代数几何的许多有用见解。

访问美国后，霍奇于1932年返回英国剑桥。次年，他被任命为大学讲师，并于1935年当选为剑桥彭布罗克学院的研究员。他的教学负担非常重，但尽管如此，在此期间他发展了几何、分析和拓扑学之间的关系，并在调和积分理论方面产生了一些他最令人难忘的工作。由于这些贡献，霍奇于1937年赢得了约翰·柯西·亚当斯奖，赫尔曼·外尔将这一贡献描述为：-

……是本世纪科学史上的伟大里程碑之一。

霍奇于1941年在The theory and applications of harmonic integrals一书中发表了他重要理论的精炼论述。这部著作标志着剑桥几何学派方向上的重要转变，该学派在Baker的领导下，已变得与其他数学领域有些隔绝。

关于霍奇1941年专著评论的一些摘录，见THIS LINK。

1936年3月，霍奇被任命为Lowndean天文学与几何学讲席教授，接替亨利·弗雷德里克·贝克，他在剑桥担任此讲席直至1970年。近20年后，他谈到自己的任命［12］：-

我清楚记得，1936年我成为剑桥教授时，戈弗雷·哈罗德·哈代不遗余力地帮助我，鼓励几何学沿着我认为最好的方向发展，并给了我许多建议，这些建议后来证明具有极大价值。

当他被任命为讲席教授时，他也成为彭布罗克学院的教授级会士，从而继续与该学院保持联系。二战期间他继续在剑桥工作，但承担了额外职责，以弥补因人员参军而造成的短缺；特别是他担任了彭布罗克学院的财务总管。规定不允许教授担任财务总管，但通过将该职位称为彭布罗克学院管事［6］而绕过了这一规定：-

霍奇负责学院事务的内务方面。这是他第一次接触行政管理，他很快惊讶而有趣地发现，自己非常高效地应对着新的职责。这成为霍奇一生的转折点。一旦发现他既喜欢行政管理又擅长行政管理，他就不断被邀请在校内和校外担任各种职务。

战争期间，1941年，霍奇 开始与 Daniel Pedoe 合作，后者曾在剑桥在 亨利·弗雷德里克·贝克 指导下攻读博士学位并从事研究。霍奇 曾是 Pedoe 1937年学位论文 The Exceptional Curves on an Algebraic Surface 的审查人之一。霍奇 与 Pedoe 的合作促成了三卷本著作 Methods of Algebraic Geometry。

关于这三卷评论的一些摘录，见THIS LINK。

霍奇于1950年访问哈佛。他是当年9月在马萨诸塞州剑桥举行的国际数学家大会的受邀全体会议演讲者，他在那里发表了The Topological Invariants of Algebraic Varieties的演讲。然而，他从1950年1月起在哈佛度过了一年，奥斯卡·扎里斯基安排他作为访问讲师。霍奇与家人去了美国，每个星期天他们都在奥斯卡·扎里斯基家吃饭。在这些场合，霍奇希望社交，不与主人讨论数学。

1958年，他被任命为彭布罗克学院院长，担任该职位直到1970年从大学生活退休。

霍奇 是英国数学讨论会的创始人之一，该讨论会每年在不同英国大学举行。他还在1952年建立 国际数学联盟 中发挥了重要作用，并于1954年至1958年被选为副主席。1958年8月，他担任爱丁堡国际数学家大会主席。1959年，London Mathematical Society 授予他 奥古斯塔斯·德摩根 奖章。他于1938年当选为 伦敦皇家学会 会士，1957年获得该学会的皇家奖章，并于1959年至1965年担任副主席。皇家奖章的授予理由是：-

……表彰他在代数几何方面的杰出工作。

1974年，他还被 皇家学会 授予科普利奖章：-

……表彰他在代数几何方面的开创性工作，尤其是他的调和积分理论。

他还获得了许多其他荣誉。他当选为 爱丁堡皇家学会 和 美国国家科学院 的成员。许多大学授予他荣誉学位，包括布里斯托尔（1957年）、爱丁堡（1958年）、莱斯特（1959年）、谢菲尔德（1960年）、埃克塞特（1961年）、威尔士（1961年）和利物浦（1961年）。他于1959年被封为爵士。

在4中，他被描述如下：-

霍奇与人们通常印象中的数学家形象大不相同。他快活、随和、务实，很容易被误认为是一位商人。

迈克尔·阿蒂亚是霍奇的学生之一，他写道，霍奇是：-

……谦逊而不张扬。为人和蔼，性情温和，兼具苏格兰人坚实的常识，他与同事和学生都相处融洽。

最后，让我们引用霍奇自己对几何学地位变化的评价，这一变化发生在他于剑桥12读本科之后：-

过去三十年[1925年至1955年]间，几何学作为数学一个分支的地位有了巨大改善，或者更确切地说，几何学被重新整合进了数学的主要结构之中。事实上，还可以进一步说，随着几何学在数学体系中恢复其应有位置，那种曾对数学造成极大危害的碎片化过程已被逆转，我们有望迎来这样一天：不再有分析学家、代数学家、几何学家等等之分，而只有数学家。数学研究有两个方面，动机与技巧，当后者占据主导时，结果往往就是过度的专门化。几何思想的革命，以及几何学作为主要数学学科之一的恢复，有助于促成数学的统一，我们完全可以公正地将此视为过去二十五年间对该学科的主要贡献之一。

William Hodge's parents were Janet Vallance (born 1875), whose father William Vallance owned a confectionary business in Edinburgh, Scotland, and Archibald James Hodge (1869-1938) who worked in the property market as a partner in the firm Douglas and Company. Archibald Hodge and Janet Vallance were married in 1900 ([5] or [6]):-

At first the Vallance family objected strongly to the marriage. Although both families were by this time comfortably off, the Vallances, having carefully built up their savings, were distinctly wealthier and they suspected young Archibald of being after their daughter's money (which was quite untrue). In addition Archibald, being the only son of well-off parents, had led a gay life and acquired a taste for drink.

William was born at 1 Church Hill Place, Edinburgh, Scotland. He was the second of his parents' three children, having one older brother Archibald Vallance Hodge and one younger sister Janet Horsburgh Hodge. He began his schooling at the age of four and spent two years in the local kindergarten.

After leaving the kindergarten, Hodge's entire education was at George Watson's Boys College in Edinburgh, entering in 1909 and studying there until 1920. He performed well in his academic subjects at school and this was important to him for while his father and brother were excellent golfers, William displayed a distinct lack of ability on the golf course. His academic success was, therefore, a compensation to him and he determined to make the most of it. Hodge was eleven years old when World War I began and the next four war years were difficult ones. In the middle of these years, in 1916, his uncle William Hodge, died and this led to his elder brother Archibald Vallance Hodge leaving school to become an apprentice in the Vallance confectionary business. The family had decided that Archibald would, after training, take over the business. This had quite an impact on Hodge for now he had to make his own way at George Watson's and this led to him becoming friends with the academically inclined pupils. His performance improved markedly from this time but, having nobody to guide him, he made some poor choices of subjects. He opted to take optional courses on Latin and Greek instead of science. It was a decision that, in later life, he felt had been a major mistake.

In 1918 the war ended and the demand for housing rose sharply but, with many employees of the firm Douglas and Company still in the army, the family decided that Hodge should leave school and join the firm. At Easter 1918, he left George Watson's and took on a clerical position in Douglas and Company. Hodge did the work competently but felt that the rather trivial boring routine work did not challenge him in any way. After six months of this clerical job the family decided to allow him to return to George Watson's where he spent another two years. During these two years Hodge's family had many discussions about his future career, and eventually it was decided that he should aim at entering the Civil Service. This would require him to have a university degree so, as part of the plan, he prepared to sit the University of Edinburgh Bursary Competition. His best school subject was mathematics so his aim was to take a degree in mathematics and philosophy. His mathematical talents had been brought out by his teacher Peter Ramsay who ([5] or [6]):-

... had the gift of being able to stimulate any mathematical ability which could be found in his pupils in a manner which, as described by Hodge, was a remarkable combination of sarcasm and kindliness.

In June 1920, Hodge sat the Bursary Competition. He performed quite well and was ranked quite high on the list but he did not look outstanding. However, there was one paper which aimed to find the best mathematics student for the John Welsh Mathematical Bursary and in this special mathematics paper he did top the list. Winning this mathematical bursary made him financially secure and allowed him to progress to become a student at Edinburgh University. Another very positive aspect of his performance in the special paper was that, even before he began his studies, professor Edmund Whittaker was aware of his talented student. He matriculated at the University of Edinburgh in October 1920 and was taught there by Whittaker and also became a fellow student of Whittaker's son, the talented John Whittaker. Hodge, who had done well at school but only managed to be ranked somewhere in the top third of most classes, found himself easily ranked first in his mathematics classes at university. In addition to mathematics, he took classes in English, economics and physics but in these subjects his performance was less impressive. At this stage he was still intending to seek entry to the Civil Service after graduating but, since his achievements seemed to be solely dependent on his ability in mathematics, he began to feel that entering the Civil Service with only expertise in mathematics would limit his career. With strong encouragement from Whittaker he began to consider an academic career and Whittaker advised him to continue his studies at Cambridge. After graduating with First Class Honours in mathematics from Edinburgh in 1923, he entered St John's College, Cambridge, in October of that year.

His studies at Cambridge were financed by a van Dunlop bursary which he had won from the University of Edinburgh and a £60 scholarship from St John's College. The University of Cambridge came as a bit of a shock to the young Hodge who had been taught a rather old fashioned mathematics course at Edinburgh. The Mathematical Tripos at Cambridge had just undergone a major revision and Hodge was taught by J E Littlewood and F P White (who lectured on projective geometry). He was also taught by Henry Baker and Hodge, poorly prepared for this material, had a hard struggle to keep up with the demanding courses. However, it was White whose course most inspired Hodge ([5] or [6]):-

Fairly soon after arriving in Cambridge Hodge had to decide what subject he intended to specialize in for schedule B. In the event this decision turned out to be extremely easy. From the moment he joined F P White's course on projective geometry he never gave a thought to any other topic. This may seem surprising since White, though an enthusiastic geometer and a charming man, made no claims to being a brilliant lecturer. The explanation lies in the contrast between the way projective geometry was treated in Edinburgh and Cambridge. Whereas in Edinburgh it was an appendage to Euclidean geometry, White's course gave a comprehensive account of the synthetic approach, based on the axioms of incidence, and its relation to coordinate geometry. This opened up a whole new world to Hodge and, in his own words, he was promptly "hooked".

In 1921 Alan Broadbent had matriculated at St John's College and, once Hodge came to the same College, the two became friends. They were almost exactly the same age, Hodge being about two weeks younger than Broadbent. After gaining distinction in the Mathematical Tripos of 1925, Hodge went on to win a Smith's Prize and spent a further year undertaking research at Cambridge financed by a Ferguson scholarship. His friend Broadbent was also undertaking research at this time advised by J E Littlewood but Hodge seems to have had no intention of spending more than a year undertaking research so he only had advice from H F Baker on an informal basis. In some ways the year was not a great success for Hodge did not find any problem he could make progress with for most of the year. However, in other ways the year was useful since after the stressful Tripos years when he was having to put in vast efforts to achieve top quality results, now the pressure was off. He lived in rooms in St John's College in 1925-26 with Broadbent and the two would remain close friends for the rest of their lives. In fact Broadbent married Hodge's sister, Janet (Nita) Hodge, in 1930.

Hodge was appointed to an assistant lectureship at the University of Bristol in 1926 and spent five years there. It was a very profitable period for him and, realising his research potential, the head of department Henry Ronald Hassé gave him a light teaching load so that he could concentrate on research. He began publishing papers during his time at Bristol, namely Linear systems of algebraic curves on a plane and on a cone (1928), The isolated singularities of an algebraic surface (1929), On multiple integrals attached to an algebraic variety (1930), and Further properties of abelian integrals attached to algebraic varieties (1931). It was the 1930 paper which brought him international attention. In it he applied topological ideas from a 1929 paper by Solomon Lefschetz who had studied integrals on a curve. Hodge was able to solve a problem about integrals on a surface which had been posed by Francesco Severi using Lefschetz's topological methods. That same year he gained election to a fellowship at St John's College, Cambridge.

In the previous year, on 27 July 1929, he had married Kathleen Anne Cameron, the daughter of Robert Stevenson Cameron, the manager of the Edinburgh branch of Oxford University Press. They had one son and one daughter. In 1931, after winning an 1851 Exhibition Studentship, he went to Princeton so that he could work with Lefschetz whom he greatly admired. His initial contact with Lefschetz had been difficult since Lefschetz refused to believe that Hodge's results in On multiple integrals attached to an algebraic variety were correct [5]:-

At first Lefschetz insisted publicly that Hodge was wrong and he wrote to him demanding that the paper should be withdrawn. Eventually Lefschetz and Hodge had a meeting in May 1931 in Max Newman's rooms in Cambridge. There was a lengthy discussion leading to a state of armed neutrality and an invitation to Hodge to spend the next academic year at Princeton.

It took Hodge the whole of his first month at Princeton before he was able to convince Lefschetz that he was right. Once convinced, Lefschetz publicly admitted his error and was very generous in his praise of Hodge's work. While in the United States Hodge spent two months at Johns Hopkins University studying with Oscar Zariski. This had been at the suggestion of Lefschetz who said that Hodge would gain much useful insight into algebraic geometry from Zariski.

After his visit to the United States, Hodge returned to Cambridge, England, in 1932. He was appointed as a university lecturer in the following year and, in 1935, was elected to a fellowship at Pembroke College, Cambridge. He had a very high teaching load but, nevertheless, during this period he developed the relationship between geometry, analysis and topology, and produced some of his best remembered work on the theory of harmonic integrals. For these contributions Hodge won the Adams Prize in 1937 and Hermann Weyl described this contribution as:-

... one of the great landmarks in the history of science in the present century.

Hodge published a polished account of his important theory in 1941 in the book The theory and applications of harmonic integrals. This work marked an important change in direction for the Cambridge school of geometry which, under Baker's leadership, had become somewhat isolated from other areas of mathematics.

For some extracts from reviews of Hodge's 1941 monograph, see THIS LINK.

In March 1936 Hodge had been appointed as Lowndean Professor of Astronomy and Geometry, succeeding Henry Baker, and he held this chair at Cambridge until 1970. He spoke of his appointment nearly 20 years later [12]:-

I well remember that when I became Professor at Cambridge in 1936 [Hardy] took great pains to help me to encourage developments in geometry along the lines that seemed best to me, and gave me much advice which proved of the greatest value.

When he was appointed to the chair, he also became a professorial fellow of Pembroke College and so continued his association with that College. He continued to work at Cambridge during World War II, but took on extra duties to compensate for the shortage of staff who were away in the forces; in particular he acted as bursar of Pembroke. The rules did not allow a professor to be a bursar but this was got round by calling the post Steward of Pembroke [6]:-

As Steward, Hodge was responsible for the domestic side of College affairs. It was his first taste of administration and he soon found, to his surprise and amusement, that he was coping very efficiently with his new responsibilities. This proved a turning point in Hodge's life. Once it was discovered that he both enjoyed and was efficient at administration he was constantly called on to serve in various capacities both within and outside the University.

During the war, in 1941, Hodge began a collaboration with Daniel Pedoe who had undertaken research at Cambridge for his Ph.D. under H F Baker. Hodge had been one of the examiners of Pedoe's thesis The Exceptional Curves on an Algebraic Surface in 1937. The collaboration between Hodge and Pedoe led to the three-volume work Methods of Algebraic Geometry.

For some extracts of reviews of these three volumes, see THIS LINK.

Hodge visited Harvard in 1950. He was an invited plenary speaker at the International Congress of Mathematicians in Cambridge, Massachusetts in September of that year where he gave the address The Topological Invariants of Algebraic Varieties. However, he spent the year from January 1950 at Harvard where Zariski had arranged for him to be a visiting lecturer. Hodge went with his family to the United States and every Sunday they dined at the Zariski home. On these occasions Hodge wished to be sociable and did not discuss mathematics with his host.

In 1958 he was appointed as Master of Pembroke, holding the post until he retired from university life in 1970.

Hodge was one of the originators of the British Mathematical Colloquium, an annual conference which visits different British universities. He also played a major role in setting up the International Mathematical Union in 1952, being elected as vice-president from 1954 to 1958. He was chairman of the International Congress of Mathematicians in Edinburgh in August 1958. In 1959 the London Mathematical Society awarded him their De Morgan Medal. He was elected to the Royal Society of London in 1938, received the Society's Royal Medal in 1957, and was vice-president from 1959 to 1965. The Royal Medal was awarded in:-

... recognition of his distinguished work on algebraic geometry.

He was also awarded the Copley Medal by the Royal Society in 1974:-

... in recognition of his pioneering work in algebraic geometry, notably in his theory of harmonic integrals.

He received many other honours. He was elected to the Royal Society of Edinburgh, and the American National Academy of Sciences. He was awarded honorary degrees by many universities including Bristol (1957), Edinburgh (1958), Leicester (1959), Sheffield (1960), Exeter (1961), Wales (1961), and Liverpool (1961). He was knighted in 1959.

He is described in [4] as follows:-

Hodge was very unlike the conventional picture of a mathematician. Jovial, informal and down-to-earth, he could easily have passed for a businessman.

Michael Atiyah, who was one of Hodge's student, writes that Hodge was:-

... modest and unassuming. Genial in manner and temperament, endowed with sturdy Scots common sense, he got on well with his colleagues and students.

Let us end by quoting Hodge's own assessment of how the position of geometry changed from the time he was an undergraduate at Cambridge [12]:-

The last thirty years [1925 to 1955] have seen an enormous improvement in the position of geometry as a branch of mathematics, or, rather, have seen the re-integration of geometry into the main fabric of mathematics. Indeed, one can go further and say that with the restoration of geometry to its rightful place in the mathematical scheme the process of fragmentation which had been doing so much harm to mathematics has been reversed, and we may look forward to the day in which there are no longer analysts, algebraists, geometers and so on, but simply mathematicians. Mathematical research has two aspects, motivation and technique, and when the latter gains control the result is apt to be excessive specialisation. The revolution of geometrical thought, and the reinstatement of geometry as one of the major mathematical disciplines, have helped to bring about a unification of mathematics which we may justly regard as one of the major contributions of the last quarter century to the subject