拉尔斯·霍尔曼德尔Lars Hörmander

拉尔斯·霍尔曼德尔的父亲Armand Per Hilding Jönsson（1893-1981）是瑞典南海岸一个小渔村的学校教师，霍尔曼德尔就出生在那里。Armand Jönsson出生于克里斯蒂安斯塔德的Smedstorp，父母是Ola Jönsson和Hanna Nordqvist。霍尔曼德尔的母亲是Alma Elisabeth Kristiansson（1899-1983），出生于Mjällby。霍尔曼德尔是父母五个孩子中最小的；他的兄弟姐妹包括1921年出生的Oskar、1923年出生的Olof和Karin。霍尔曼德尔的母亲Alma是[4]：-

……一位聪明的农家女儿，她将短暂的求学岁月视为一生中的亮点。霍尔曼德尔的父母共同营造了一种以强烈求知欲和重视教育为特征的家庭氛围，五个孩子都很快从中学毕业，然后继续接受高等教育。

霍尔曼德尔在Mjällby上了第一所学校，然后每天乘火车到附近一个较大的城镇上更高级别的学校。下一阶段的学习，他不得不搬到隆德，从1946年起，他在文理中学（Gymnasium）就读，当时该校名为隆德私立小学，但自1986年起改名为Spyken。在那里，他得以投入大量时间学习大学水平的数学，这得益于一位教师Nils Erik Fremberg（1908-1952）的鼓励，此人也是隆德大学的讲师（Docent），曾是马塞尔·里斯的研究生。Fremberg在霍尔曼德尔就读隆德的中学期间，就教了他隆德大学第一学期的课程内容。

1948年，霍尔曼德尔从中学毕业，开始在隆德大学学习。他最初的打算是追随大哥Oskar的脚步学习工程学，但Fremberg说服了他，使他相信数学才是他应该学习的科目。在大学里，马塞尔·里斯教他，为他讲授经典函数论和调和分析。霍尔曼德尔于1949年秋获得学士学位，1950年春获得硕士学位。他开始在马塞尔·里斯的指导下进行研究，并于1951年10月以学位论文Applications of Helly's theorem to estimates of Chebyshev type获得副博士学位。两个月后，在圣诞聚会上，霍尔曼德尔遇到了Viveka Sonesson，两年后她成为了他的妻子。

1952年马塞尔·里斯退休并前往美国后，霍尔曼德尔开始研究偏微分方程理论。里斯出色的讲课为他研究这一课题做了充分准备，他还得到了两位新教授拉斯·戈丁和Åke Pleijel的支持，他们都在研究这一课题。1953年夏天，他在斯堪的纳维亚数学家大会上作了一场令人印象深刻的报告，这次大会在Lund举行。

在完成博士学位之前，霍尔曼德尔在1953-54年服兵役，但由于他主要在斯德哥尔摩从事国防研究，因此在此期间仍能继续研读数学。霍尔曼德尔在服兵役期间与Viveka结婚，他们的第一个孩子，女儿Gisela，于1954年10月出生。1954年秋回到隆德后，他在拉斯·戈丁的讨论班上授课，该讨论班每周举行一次。霍尔曼德尔要就尼古拉·布尔巴基风格的一般拓扑学做两次讲座，隆德的一位研究生Vidar Thomée听了他的第一次讲座。Thomée在[26]中解释说，霍尔曼德尔的第一次讲座之后他几乎没听懂什么，这事传到了拉斯·戈丁那里。问题由拉斯·戈丁解决了，他告诉Thomée，第二次讲座将由他来讲，而霍尔曼德尔会教他必要的数学。Thomée在[26]中写道：-

当然，这对我们两人来说都是艰苦的工作，但计划得以实施，我想我从这次经历中学到了重要的东西。

Viveka和霍尔曼德尔的第二个女儿是Sofia（生于1965年），她在[4]中写到她父亲撰写博士学位论文的情形：-

由于我母亲只能在最初几个月待在家里照顾Gisela，这篇学位论文的大部分实际上是在我姐姐待在他（拉出的）书桌抽屉里（抽屉的大小正好适合婴儿提篮）的情况下写成的。他显然并不觉得这让人分心，这正是我父亲的典型特点——打断似乎只让他损失了实际分心所用的那几分钟，仅此而已。

霍尔曼德尔于1955年完成博士学位，同年10月22日就该学位论文On the theory of general partial differential operators进行了答辩。他名义上的学位论文导师是拉斯·戈丁，但实际上霍尔曼德尔是独立工作的，自己寻找问题并加以解决。该学位论文于1955年发表在Acta Mathematica上。霍尔曼德尔写道[15]：-

它在很大程度上受到1954年公布的Bernard Malgrange的学位论文的启发，并结合了让·勒雷和拉斯·戈丁为双曲型微分算子发展的技巧。

答辩委员会的主席是拉斯·戈丁，三位学生对手是雅克-路易·利翁斯、Bent Fuglede和Nils Aslund。雅克-路易·利翁斯用法语提问，Bent Fuglede用丹麦语发言，而Nils Aslund用匈牙利语提出了他的第一个问题！霍尔曼德尔的学位论文和答辩均获得“优异”评价。11月4日，霍尔曼德尔作了一场讲座以证明自己具备讲师资格。他讲授了“约瑟夫·傅里叶变换”，并接受了化学家、副院长Erik Larsson和Åke Pleijel的考核。

获得博士学位后，他申请了斯德哥尔摩大学的教授职位。在教授职位做出决定之前，霍尔曼德尔前往美国，在芝加哥大学、堪萨斯大学和明尼苏达大学度过了一段时间。他还访问了数学科学研究所（现为理查·科朗特研究所），该研究所当时由理查·科朗特领导。回到瑞典后，他于1957年1月1日接受了斯德哥尔摩的教授任命，这一职位是在他在美国期间提供给她的。他接受该职位时年仅25岁，但正式就职时已26岁。斯德哥尔摩大学的学生报纸Gaudeamus发表了一篇题为“26岁数学教授庄严就职”的文章。霍尔曼德尔和Viveka 霍尔曼德尔的照片下方的文字写道：“瑞典最年轻的教授、数学天才霍尔曼德尔在蓝色大厅的庄严就职典礼上纵容地对他的甜美妻子微笑。”

他继续在美国度过时光，尤其是在斯坦福大学和高等爱德华·斯图迪研究所[18]：-

霍尔曼德尔在1960至1961年的夏天作为受邀教授在斯坦福大学度过，并利用这段时间完成了⟦E1⟧出版社的“施普林格基础教程系列”中关于偏微分方程的一本书的邀约。

在斯德哥尔摩，他不得不组建一个偏微分方程的研究小组。在[4]中，Jan Boman解释了当时的情况：-

由于当时斯德哥尔摩大学在偏微分方程方面没有活动，他不得不从头开始，讲授分布理论、约瑟夫·傅里叶分析和泛函分析。1961至1962年间，他讲授了后来成为他1963年著作《线性偏微分算子》的内容。他迅速聚集了一大群学生围绕在他身边。其中包括Christer Kiselman和Vidar Thomée。我们中的一人（Jan Boman）有幸成为该小组的一员。霍尔曼德尔的讲座非常精彩，他周围充满了兴奋和热情。他渊博的知识、敏捷的思维和惊人的工作能力激励了我们所有人，但有时也让学生们感到害怕。从他门后不断传来的打字机咔嗒声是出了名的。

他授课，指导研究生，撰写论文，并致力于1963年出版的Linear Partial Differential Operators一书[18]：-

那本书是偏微分方程研究的一个里程碑，广大数学界发现了L 霍尔曼德尔对该领域最新进展的阐述。首先，分布理论作为线性偏微分方程的完美工具的作用得到了强调。尽管偏微分方程弱解的概念在三十年代已为舍盖·索伯列夫和俄罗斯学派所知，但确实是洛朗·施瓦茨的分布定义创造了最佳视角，将泛函分析的抽象方面与傅里叶分析结合起来。霍尔曼德尔对洛朗·施瓦茨理论早已熟悉，但他注意到许多数学家，包括他的导师马塞尔·里斯，对此相当消极（至少可以说）。... 在学位论文工作中，霍尔曼德尔设法避免明确引用洛朗·施瓦茨理论，但在1963年，情况不同了，他选择将洛朗·施瓦茨分布理论作为他书中的基本泛函分析框架。

关于Linear Partial Differential Operators的评论摘录见THIS LINK。

当时他的一位博士生是 Christer Kiselman，他在17中写道：-

每周一到周五下午15:00，都会供应茶和面包。通常是由登记员 Anita Ahlbert（有人称她为 Ahlbert 的妻子，也有人称她为 Anita）来安排的。铃声响起，召集大家来聚会。当机构规模不太大时，这样的聚会对于营造工作共同体非常重要。霍尔曼德尔只要不出国就总会出席。讨论常常涉及数学。霍尔曼德尔总是可以随时咨询。学生们敲他的门；他能立刻回答一切问题。我和他讨论数学从未遇到过困难。他通常坐着写文章，或者写他那本1963年出版的书。那里有他打字机的咔嗒声。他还会写讲义和分发材料。……当我离开他的房间时，他在零秒之后又恢复了打字机的咔嗒声。（就我个人而言，我至少需要几秒钟才能找到可以继续的地方。）

1962年8月，国际数学家大会在斯德哥尔摩举行，霍尔曼德尔不仅深度参与了组织工作，还因在偏微分方程方面的工作而获得了约翰·查尔斯·菲尔兹奖章。拉斯·戈丁在描述霍尔曼德尔使其获得菲尔兹奖的贡献时，在9中写道，雅克·阿达马对线性微分算子的早期贡献导致了伊万·彼得罗夫斯基的第一个完整结果。他继续写道：-

……在1945年的一次演讲中，伊万·彼得罗夫斯基明确要求建立线性微分算子的一般理论，包括那些不出现在物理学数学模型中的算子。……在他关于广义函数的书洛朗·施瓦茨中……陈述了若干关于微分算子的问题。从那时起，一个相当全面的理论已经被发展出来。许多人做出了贡献，但最深刻和最重要的结果归功于霍尔曼德尔。

你可以在THIS LINK看到 Hormander 关于他因之获得奖章的工作的一些评论。

Nicolas Lerner 在18中写到了导致获得约翰·查尔斯·菲尔兹奖章的工作：-

他在偏微分方程方面的出色工作，特别是他对常系数次椭圆性的刻画以及他对汉斯·卢伊不可解现象的几何解释，无疑是授予他奖章的非常有力的理由。此外，他在偏微分方程上的新观点，将泛函分析与“先验”不等式结合起来，导致了对一大类方程的非常一般的结果，而这些在五十年代初期是遥不可及的。

1963年，霍尔曼德尔做出了一项安排，使他能够在斯德哥尔摩度过学年的教学时间，同时在春季和夏季在美国斯坦福大学度过。然而，他在[15]中写道：-

我刚到斯坦福，就收到邀请，请我去高等斯图迪研究所担任永久成员和教授。尽管我之前决心不离开瑞典，但在一个数学上非常活跃的环境中全职从事研究的机会实在难以抗拒。

霍尔曼德尔从1964年到1968年在普林斯顿度过，但感到全职研究职位的压力，因此于1968年回到瑞典，担任隆德大学的数学讲席。然而，他经常回美国访问，特别是高等斯图迪研究所和斯坦福大学。迈克尔·阿蒂亚在[4]中写道：-

我认为从普林斯顿到隆德的过渡对霍尔曼德尔来说并不容易。他在瑞典乡村和文化中感到自在，但在数学上他有些孤立，尽管他享有盛名并有国际联系。

在16中，霍尔曼德尔描述了他获得约翰·查尔斯·菲尔兹奖章后的研究方向。他特别描述了其四卷本著作The analysis of linear partial differential operators所涵盖的主要领域是如何发展的，该著作的各卷于1983年至1985年间出版。这部作品更新了他最初的著作Linear Partial Differential Operators（1963年），其中包含导致他获得约翰·查尔斯·菲尔兹奖章的研究成果。American Mathematical Society于2006年授予霍尔曼德尔勒罗伊·P·凯瑟琳·斯蒂尔奖：-

……因他的著作⟦E1⟧。

引文如下：-

在数学史中，很难找到任何可与之相比的“阐述性”著作，涵盖如此多的材料，并且对如此广阔的数学领域有如此深度和理解。

1978年，悲剧降临这个家庭，霍尔曼德尔的长女吉塞拉自杀身亡。她年仅二十三岁。

霍尔曼德尔在15中描述了他职业生涯的后期阶段：-

在花费五年时间撰写一部关于线性偏微分算子的四卷本专著之后，我在1984-86学年担任斯德哥尔摩约斯塔·米塔格-莱弗勒研究所所长。我只接受了两年的任命，并从隆德请假，因为我怀疑那里的许多行政职责不会很适合我。这个预感是对的，自1986年以来我一直在隆德，并于1996年1月成为荣休教授。

霍尔曼德尔的著作An Introduction to Complex Analysis in Several Variables已成为一部经典，论述多复变函数论。它源于他1964年在斯坦福大学讲授的一门课程的讲义，两年后以书的形式出版。1973年和1990年出版的扩充版中增加了额外材料。1994年，他出版了Notions of Convexity一书。你可以在THIS LINK上读到这些书评的摘录。

在[3]中，Jan Boman谈到了霍尔曼德尔的数学方法：-

关于霍尔曼德尔在数学中追求正确性的方法，我学到了很多。首先，他有一种非凡的能力，能够避免犯大大小小的错误。其次，他对不完全准确的数学极为挑剔。

如上所述，1996年1月1日，霍尔曼德尔退休，并被隆德大学授予荣休教授称号。

我们应该稍微谈谈霍尔曼德尔的个性。他的女儿Sofia在[4]中写道，生动地描绘了霍尔曼德尔家的家庭生活：-

小时候我很崇拜父亲，我们一起度过了无数个小时，用乐高积木搭建、拼拼图、玩西洋双陆棋和拼字游戏（后者母亲也会加入），或者做木工。我父亲喜欢动手，尤其是木工，他小时候曾考虑过将来当木匠。除其他东西外，他还为我建了一个芭比娃娃屋，有四个房间，每个房间都装了电，家具齐全——我姐姐也帮忙做了一些较小的细节，比如纺织品或陶器。我喜欢圣诞节前的那几周，他会在地下室里花很长时间为我制作礼物——我不能加入，这让人沮丧，但同时又令人兴奋。有一年，我被地下室里传来的刺耳声音深深吸引。原来他为娃娃屋建了厨房，配有冰箱、炉灶、水槽、橱柜、抽屉，甚至还有一个扫帚柜，而那奇怪的声音来自打磨四枚硬币，把它们做成炉盘。我非常幸运，父亲总是有很多空闲时间，多到我母亲常常抱怨她跟不上他，夏天更是如此，那时他会长时间不工作。在瑞典西海岸外的Askerön岛上，我父母买了一栋海边的避暑别墅，我们在那里一起度过了很多户外时光。要么在森林里——一个亮点是一个鱼鹰巢，我们多年来每天都去看——要么在海上——我父亲喜欢航海。我很小的时候，我们有一艘SeaCat，令我非常懊恼的是，后来它被换成了一艘小得多的船，只用于一日游。我们还和一个朋友及其孩子们一起用网捕鲽鱼。当不积极参与家庭或房子的事务时，我父亲会把空闲时间用来阅读。他总是读很多书，小说和纪实都有，主要是历史或科学。他有一颗非常好问的心，直到生命的最后几天都喜欢学习新事物。在这一切中，他出色的记忆力帮了大忙；他似乎无法忘记任何学过的东西，而且不幸的是，他似乎无法意识到别人没有他那样的记忆力……他对我可能相当不耐烦。我问他问题时必须小心：总有风险会听到一场冗长而热情的讲座，包含的信息比我想要的要多得多。……在他去世前，父亲常对我说，他对自己的生活感到满意，那是一段美好的生活。我确实相信他去世时是一个快乐的人。

我们在上文提到了霍尔曼德尔获得的若干奖项。还有一个重要奖项必须提及，即1988年授予霍尔曼德尔的沃尔夫奖。他于1976年当选为国家科学院会士，并于1987年至1990年担任国际数学联盟副主席。

Lars Hörmander's father Armand Per Hilding Jönsson (1893-1981) was a school teacher in the small fishing village on the south coast of Sweden where Lars was born. Armand Jönsson had been born in Smedstorp, Kristianstad to parents Ola Jönsson and Hanna Nordqvist. Lars' mother was Alma Elisabeth Kristiansson (1899-1983) who was born in Mjällby. Lars was the youngest of his parents five children; his siblings included Oskar, born 1921, Olof, born 1923, and Karin. Lars mother Alma was [4]:-

... a bright farmer's daughter who recalled her few years in school as a highlight of her life. Together [Lars's parents] created an atmosphere characterized by a strong thirst for knowledge and education, and all five children graduated quickly from secondary school and then continued with higher education.

Lars attended his first school in Mjällby, then made a daily train journey to attend a higher level school in a larger nearby town. For the next stage in his schooling he had to move to Lund where, from 1946, he attended the gymnasium which, at that time, was called the Lund Private Elementary School but since 1986 has been called Spyken. There he was able to devote much time to studying university level mathematics, encouraged by a teacher, Nils Erik Fremberg (1908-1952), who was also a docent at Lund University and had been a research student of Marcel Riesz. Fremberg taught Hörmander the material covered in the first semester of Lund University while he was at the gymnasium in Lund.

In 1948 Hörmander graduated from the gymnasium and began his studies at the University of Lund. His original intention had been to study engineering, following in the footsteps of his oldest brother Oskar, but Fremberg had persuaded him that mathematics was the topic he should study. At university he was taught by Marcel Riesz who lectured to him on classical function theory and harmonic analysis. Hörmander was awarded a bachelor's degree in the autumn of 1949 and a master's degree in the spring of 1950. He began to undertake research supervised by Marcel Riesz and was awarded his licentiate degree in October 1951 with a thesis Applications of Helly's theorem to estimates of Chebyshev type. Two months later, at the Christmas party, Hörmander met Viveka Sonesson who, two years later would become his wife.

After Marcel Riesz retired in 1952 and went to the United States, Hörmander began working on the theory of partial differential equations. He had been well prepared to study this topic from Riesz's excellent lectures and he was also supported by the two new professors, Lars Gårding and Åke Pleijel who were both working on that topic. He delivered an impressive lecture in the summer of 1953 at the Scandinavian Congress of Mathematicians, which on this occasion was held in Lund.

Before completing his doctorate, Hörmander spent the year 1953-54 doing military service but, since he was mainly undertaking defence research in Stockholm, he was able to continue reading mathematics during this time. Hörmander had married Viveka while he was on military service and their first child, a daughter Gisela, was born in October 1954. When he returned to Lund in the autumn of 1954 he lectured in Lars Gårding's seminar which was held once a week. Hörmander was to give two lectures on Bourbaki style general topology and one of the postgraduate students at Lund, Vidar Thomée, attended his first lecture. Thomée explains in [26] that after Hörmander's first lecture he had understood very little of it and this got back to Gårding. The problem was solved by Gårding who told Thomée that he would deliver the second lecture and Hörmander would teach him the necessary mathematics. Thomée writes in [26]:-

Of course it was hard work for both of us, but the programme was implemented, and I think I learned something significant from the experience.

Viveka and Lars Hörmander's second daughter was Sofia (who was born in 1965) and she writes in [4] about her father working on his doctoral thesis:-

Since my mother had only been able to stay at home to take care of Gisela for the first few months, much of the dissertation was actually written with my sister in his (pulled-out) desk drawer (which was just the right size for the carriage insert). He apparently didn't find this distracting, which was typical of my father - the only effect an interruption seemed to have on his work was the few minutes lost to the actual distraction, nothing more.

Hörmander's doctorate was completed in 1955 when he was examined on this thesis On the theory of general partial differential operators on 22 October of that year. His formal thesis advisor had been Gårding but in reality Hörmander had worked independently, finding his own problems and solving them. The thesis was published in Acta Mathematica in 1955. Hörmander writes [15]:-

It was to a large extent inspired by the thesis of Bernard Malgrange which was announced in 1954, combined with techniques developed for hyperbolic differential operators by Jean Leray and Lars Gårding.

The chair of the examining panel was Gårding and the three student opponents were Jacques-Louis Lions, Bent Fuglede and Nils Aslund. Jacques-Louis Lions asked questions in French, Bent Fuglede spoke in Danish and Nils Aslund asked his first question in Hungarian! Hörmander was awarded a "distinction" for both the thesis and for his defence. On 4 November, Hörmander gave a lecture to prove his abilities for a lectureship. He lectured on "The Fourier transform" and was examined by Erik Larsson, a chemist who was assistant dean, and by Åke Pleijel.

After the award of his doctorate, he applied for a professorship at the University of Stockholm. Before a decision was made on the professorship Hörmander left for a visit to the United States where he spent time at the universities of Chicago, Kansas and Minnesota. He also visited the Institute for Mathematical Sciences (now the Courant Institute) which was directed at that time by Richard Courant. Back in Sweden he took up the appointment of professor at Stockholm on 1 January 1957 which had been offered to him while he was in the United States. He was only 25 years of age when he took up the position but was 26 by the time he was officially installed. The University of Stockholm's student newspaper, Gaudeamus, published an article headed "26-year-old mathematics professor solemnly installed". The text beneath a photograph of Lars and Viveka Hörmander reads: "Sweden's youngest professor, mathematics genius Lars Hörmander, smiles indulgently to his sweet wife during the solemn installation ceremony in the Blue Hall."

He continued to spend time in the United States, particularly at Stanford University and the Institute for Advanced Study [18]:-

Hörmander spent the summers 1960-61 at Stanford University as an invited professor, and took advantage of this time to honour the offer of the 'Springer Grundlehren series' of publishing a book about partial differential equations.

In Stockholm he had to set up a research group in partial differential equations. In [4] Jan Boman explains the situation:-

Since there was no activity in partial differential equations at Stockholm University at this time, he had to start from the beginning and lecture on distribution theory, Fourier analysis, and functional analysis. In 1961-62 he lectured on the material that was to become his 1963 book, 'Linear Partial Differential Operators'. He quickly gathered a large group of students around him. Among them were Christer Kiselman and Vidar Thomée. One of us (Jan Boman) had the privilege to be a member of that group. Lars's lectures were wonderful, and there was excitement and enthusiasm around him. His vast knowledge, fast thinking, and overwhelming capacity for work inspired all of us, but also sometimes scared his students. The clattering from his typewriter, which was constantly heard through his door, is famous.

He lectured, supervised research students, wrote papers and worked on the book Linear Partial Differential Operators which was published in 1963 [18]:-

That book was a milestone in the study of PDE, and a large mathematical public discovered L Hörmander's exposition of recent progress in the area. In the first place, the role of Distribution Theory was emphasized as the perfect tool for linear PDE. Although the notion of weak solution for a PDE was already known to Sergei Sobolev and to the Russian school in the thirties, it is indeed Laurent Schwartz's definition of distributions which created the best perspective, combining abstract aspects of functional analysis with Fourier analysis. Lars Hörmander had been familiar for quite a long time with Schwartz theory, but he had noticed that many mathematicians, including his mentor Marcel Riesz, were rather negative (to say the least) about it. ... During his thesis work, Lars Hörmander managed to avoid explicit reference to Schwartz theory, but in 1963, it was a different story and he chose to present Schwartz Distribution Theory as the basic functional analytic framework of his book.

For extracts of reviews of Linear Partial Differential Operators see THIS LINK.

One of his Ph.D. students at this time was Christer Kiselman who writes in [17]:-

Every Monday to Friday at 15:00 tea and buns were served. It was usually the Registry clerk Anita Ahlbert, called by some the wife of Ahlbert, and by others Anita, who arranged it. A bell tolled and called everyone to the meeting. Such meetings were very important to create a working community when the institution was not too large. Lars was always present if he was not abroad. The discussions often touched on mathematics. Lars was always available for consultations. Students knocked on his door; he could answer everything at once. I never had trouble discussing mathematics with him. He was usually sitting and writing articles, or writing his book, the one that was published in 1963. There was the clatter of his typewriter. He would also write lecture notes and handouts. ... When I left his room, he resumed the clatter of his typewriter after zero seconds. (Personally, I would have needed at least a few seconds to find the place where I could continue.)

In August 1962 the International Congress of Mathematicians was held in Stockholm and Hörmander, as well as being heavily involved in the organisation, received a Fields Medal for his work on partial differential equations. Lars Gårding, describing Hörmander's contributions that earned him the Fields Medal, writes in [9] of early contributions by Jacques Hadamard to linear differential operators which led to the first complete results by Ivan Georgievich Petrovsky. He continues:-

... in a lecture in 1945 [Petrovsky] explicitly asked for a general theory of linear differential operators including those which do not appear in the mathematical models of physics. ... In his book on distributions Laurent Schwartz ... stated a number of problems about differential operators. Since then a rather comprehensive theory has been worked out. Many people have contributed but the deepest and most significant results are due to Hörmander.

You can see some comments by Hormander on the work for which he was awarded the medal at THIS LINK.

Nicolas Lerner writes in [18] about the work which led to the award of the Fields Medal:-

His impressive work on Partial Differential Equations, in particular his characterization of hypoellipticity for constant coefficients and his geometrical explanation of the Lewy non-solvability phenomenon were certainly very strong arguments for awarding him the Medal. Also his new point of view on Partial Differential Equations, which combined functional analysis with 'a priori' inequalities, had led to very general results on large classes of equations, which had been out of reach in the early fifties.

In 1963 Hörmander made an arrangement which allowed him to spend the academic teaching year in Stockholm while he spent spring and summer at the University of Stanford in the United States. However, he writes in [15]:-

I had barely arrived at Stanford when I received an offer to come to the Institute for Advanced Study as a permanent member and professor. Although I had previously been determined not to leave Sweden, the opportunity to do research full time in a mathematically very active environment was hard to resist.

Hörmander spent from 1964 to 1968 at Princeton but felt the pressure of a full time research position so returned to Sweden to take up the chair of mathematics at the University of Lund in 1968. However he returned for frequent visits to the United States, in particular to the Institute for Advanced Study and to Stanford. Michael Atiyah writes in [4]:-

I think the transition from Princeton to Lund was not easy for Lars. He was at home in the Swedish countryside and culture, but mathematically he was somewhat isolated, despite his fame and his international connections.

In [16] Hörmander describes the direction of his research after the award of the Fields Medal. In particular he describes how the main areas developed which are covered by his four volume text The analysis of linear partial differential operators the volumes of which appeared between 1983 and 1985. This work updates his original book Linear Partial Differential Operators (1963) which contained the results which led to the award of his Fields Medal. The American Mathematical Society awarded Hörmander its Leroy P Steele Prize in 2006:-

... for his book "The Analysis of Linear Partial Differential Operators".

The citation reads:-

In the history of mathematics, one is hard-pressed to find any comparable 'expository' work that covers so much material, and with such depth and understanding of such a broad area of mathematics.

Tragedy struck the family in 1978 when Hörmander's eldest daughter Gisela took her own life. She was twenty-three years of age.

Hörmander describes in [15] the later stages of his career:-

After five years devoted to writing a four volume monograph on linear partial differential operators, I spent the academic years 1984-86 as director of the Mittag-Leffler Institute in Stockholm. I had only accepted a two year appointment with leave of absence from Lund since I suspected that the many administrative duties there would not agree very well with me. The hunch was right, and since 1986 I have been in Lund where I became professor emeritus in January 1996.

Hörmander's text, An Introduction to Complex Analysis in Several Variables, has become a classic dealing with the theory of functions of several complex variables. It developed from lecture notes of a course which he gave in Stanford in 1964 and published in book form two years later. Extra material was added to later expanded editions of the work which appeared in 1973 and in 1990. In 1994 he published the book Notions of Convexity. You can read extracts from reviews of these books at THIS LINK.

In [3] Jan Boman spoke about Hörmander's approach to mathematics:-

I learned a lot about Lars' approach to correctness in mathematics. First, that he had an unusual ability to avoid making mistakes, both large and small. Second, he was extremely critical of mathematics that was not entirely accurate.

As noted above, on 1 January 1996 Hörmander retired and was made professor emeritus by the University of Lund.

We should say a little about Hörmander's personality. His daughter Sofia, writing in [4], paints an excellent picture of family life in the Hörmander household:-

I adored my father as a child, and we spent countless hours together building with Legos, doing jigsaw puzzles, playing backgammon and Scrabble (for the latter my mother would also join), or doing carpentry. My father loved to work with his hands, especially with wood, having contemplated a future as a carpenter as a child. Among other things he built a Barbie dollhouse for me with four rooms, each fitted with electricity and completely furnished - my sister also helping with some of the smaller details, like textiles or earthenware. And I loved the weeks before Christmas when he would spend long hours in the basement crafting presents for me - it was frustrating that I couldn't join, but at the same time exciting. One year I was deeply intrigued by screeching sounds coming from the basement. It turned out that he had built the kitchen for the doll house, complete with refrigerator, stove, sink, cupboards, drawers, even a broom closet, and that the strange sounds came from sanding down four coins to make them into stove plates. I was very fortunate that my father always had a lot of spare time, so much so that my mother used to complain that she couldn't keep up with him, and this was even more so in the summers when he would take long periods away from work. On the island of Askerön off the west coast of Sweden my parents had bought a summer house by the sea, and there we spent much time outdoors together. Either in the forest - a highlight was an osprey nest that we would visit every day for years - or on the sea - my father loved sailing. When I was very young we had a SeaCat, which, to my great chagrin, was later traded in for a much smaller boat for day trips only. We also fished for plaice, with nets, together with a friend and his children. When not actively engaged with the family or the house, my father would spend his spare time reading. He always read a lot, fiction as well as fact, mostly history or science. He had a very inquisitive mind and loved to learn new things up until the last days of his life. In all this he was much aided by his excellent memory; he seemed unable to forget anything he had learned and, unfortunately, unable to realize that other people did not have quite his powers of memory ... he could be quite impatient with me. And I had to be careful when asking him a question: there was always the risk of a long and enthusiastic lecture, containing a lot more information than I wanted. ... Before his death my father often said to me that he was content with his life, that it had been a good life. I do believe he died a happy man.

We have mentioned above a number of awards that Hörmander received. There is one other major award that we must mention, namely the Wolf Prize which was awarded to Hörmander in 1988. He was elected to the National Academy of Sciences in 1976 and he served as a vice president of the International Mathematics Union from 1987 to 1990.