奥斯卡·扎里斯基Oscar Zariski

奥斯卡·扎里斯基的父亲是Bezalel Zaritsky，母亲是Hannah Zaritsky。扎里斯基出生在这个犹太家庭，父母给他取名为Ascher Zaritsky。我们将在下面说明他的名字如何变成了如今熟悉的扎里斯基。

扎里斯基的母亲是确保年幼的儿子接受良好教育的人。从扎里斯基七岁起，家里就为他请了一位家庭教师，而扎里斯基在这位教师的指导下，在俄语和算术方面表现出非凡的天赋。当与第一次世界大战相关的战火蔓延到白俄罗斯时，扎里斯基一家逃往乌克兰的切尔尼戈夫。这是政治问题迫使他进行的数次搬迁中的第一次。

由于基辅大学数学系名额已满，他无法进入，于是改选了哲学。1918年至1920年，他在基辅学习哲学。不过他得以继续追求自己的数学兴趣，除哲学外还学习了代数和数论。然而扎里斯基是在基辅的一段动荡时期完成学业的。

1918年1月，乌克兰成为独立国家，以基辅为首都。次月，基辅工人发动的轻微起义被镇压，但红军部队进入基辅支持工人。随后基辅被德国人占领，但随着1918年11月战争结束，一个独立的乌克兰在基辅再次宣告成立。1919年11月，基辅一度被白军攻占，不久后又被红军取代。接着是俄波战争，1920年5月，波兰军队占领了基辅，但在一次反击中被逐出。对于扎里斯基来说，在这座被战争摧残的城市里生活实在太艰难了，于是他决定去意大利继续学业。

扎里斯基受到伟大的代数几何学家Castelnuovo、费代里戈·恩里克斯和弗朗切斯科·塞维里的影响。他于1924年在罗马获得博士学位，其学位论文的题目与伽罗瓦理论有关，是由Castelnuovo向他提出的。扎里斯基于1924年结婚；他在罗马结识了未来的妻子Yole Cagli，随后他们回到扎里斯基的家乡Kobin结婚。回到罗马后，他作为国际教育委员会会士留在那里直到1927年。

正是在扎里斯基逗留罗马期间，费代里戈·恩里克斯建议当时名叫Ascher Zaritsky的他改名为听起来像意大利语的扎里斯基。这是他首次发表论文时所用的名字，那篇论文是他与费代里戈·恩里克斯合写的。扎里斯基在2中写到他的数学兴趣如何不同于他的导师Castelnuovo和费代里戈·恩里克斯的兴趣：-

然而，即使在我逗留罗马期间，我的代数倾向也已显露，并被Castelnuovo清楚地察觉，他曾对我说：“你和我们在一起，但不是我们中的一员。”这话并非责备，而是善意而言，因为Castelnuovo本人一次又一次告诉我，意大利几何学派的方法已尽其所能，已走到尽头，不足以在代数几何领域取得进一步进展。

扎里斯基 为了逃避白俄罗斯和乌克兰的问题而去了意大利。然而，意大利的政治局势开始迅速恶化。1922年10月，墨索里尼组织了法西斯“向罗马进军”，并被要求组建政府。在18个月里，他以相当民主的方式治理国家，但在1925年至1927年间，他取消了言论自由权，并取缔了反对党和工会。法西斯对犹太人的仇恨使扎里斯基的生活因其犹太背景而变得特别艰难。

在所罗门·莱夫谢茨的帮助下，他于1927年逃离意大利的政治问题，前往美国。在那里，他在Johns 威廉·霍普金斯大学任教，担任Johnston学者直到1929年加入教师队伍。1937年，他成为Johns 霍普金斯的正教授。

Castelnuovo和弗朗切斯科·塞维里曾鼓励扎里斯基将所罗门·莱夫谢茨的topological方法视为代数几何的未来之路，因此在1927年至1937年间，扎里斯基经常访问普林斯顿的所罗门·莱夫谢茨。扎里斯基写道[2]：-

我非常感谢[所罗门·莱夫谢茨]的启发式指导和鼓励。

在此期间，扎里斯基撰写了Algebraic Surfaces，该书于1935年出版。他在[2]中解释了撰写这部专著如何改变了他工作的方向：-

当时（1935年），现代代数已经诞生（通过埃米·诺特的工作和巴特尔·伦德特·范德瓦尔登的重要论著），但当巴特尔·伦德特·范德瓦尔登将其应用于代数几何基础的某些方面时……就代数探索而言，双有理代数几何的更深层次方面……在很大程度上，甚至完全是处女地。在[代数曲面]中，我尽力呈现了意大利几何学家处理曲面理论这些更深层次方面时所使用的巧妙几何方法和证明的基本思想……我开始对原始证明的严谨性感到明显不满（丝毫没有减少我对贯穿这些证明的富有想象力的几何精神的钦佩）；我确信整个结构必须用纯代数方法重新做一遍。

1939年至1940年间，扎里斯基在约翰霍普金斯大学实施了他将现代代数应用于代数几何基础的计划。他研究了正规簇理论、局部单值化以及代数簇奇点的消解。

对扎里斯基来说，1945年是重要的一年，他在圣保罗度过。在那里，他每周讲三天课，听课的只有安德烈·韦伊，没有别人。扎里斯基和安德烈·韦伊在讨论——常常是争论——扎里斯基所讲内容的过程中都学到了很多。在伊利诺伊大学度过1946-47学年后，扎里斯基被任命为哈佛大学的讲席教授，他在那里一直待到1969年退休。从1970年代末起，他患上了阿尔茨海默病，最后几年随着健康状况恶化而十分艰难。

1981年，扎里斯基因其全部数学研究的累积影响，被美国数学会授予凯瑟琳·斯蒂尔奖。该奖项的颁奖词总结了扎里斯基一生对数学的贡献[2]：-

1924年在意大利开始工作时，扎里斯基的风格非常接近“意大利代数几何”，但他意识到整个学科需要适当的基础。因此，在1927年至1937年期间，他首先转向拓扑问题，然后在1937年开始为其学科奠定交换代数基础。他的拓扑工作主要集中在基本群上；他开创的许多思想既是拓扑学也是代数几何的创新，此后在这两个领域中独立发展。

1937年，扎里斯基彻底调整了研究方向，开始将抽象代数的思想引入代数几何。事实上，他与B L 巴特尔·伦德特·范德瓦尔登和安德烈·韦伊一起，在不使用拓扑或分析方法的情况下，彻底重写了这门学科的基础。他在代数几何中对整相关、赋值环和正则局部环等概念的使用被证明特别富有成果，并使他达到了诸如1944年特征0的三维簇奇点消解、1947年对单点概念的澄清，以及任意基域上代数簇的全纯函数理论等高峰。扎里斯基于1965年开创的等奇异性与饱和理论也具有很大的影响力和重要性。

扎里斯基的所有工作都为代数几何当前的繁荣奠定了基础，当前的学派在该学科的现代发展中运用了他的工作和思想。

扎里斯基最著名的书是Commutative Algebra，这是一部与皮埃尔·萨米埃尔合著的两卷本著作。第一卷于1958年出版，第二卷于1960年出版。

American Mathematical Society在扎里斯基的一生中发挥了重要作用，他多年来也为该学会做出了巨大贡献。他于1960年至1961年担任该学会副主席，并于1969年至1970年担任该学会主席。

扎里斯基在被任命为正教授后，在数学出版方面发挥了重要作用。他于1937年至1941年担任American Mathematical Journal的编辑，于1941年至1947年担任Transactions of the American Mathematical Society编辑委员会成员，还曾在Annals of Mathematics和American Journal of Mathematics的编辑委员会任职。

1927年前往美国后，扎里斯基在美国和其他国家的其他大学进行了相当长时间的讲学。我们已经提到，他于1945年在圣保罗担任访问教授，并于1946-47年在伊利诺伊大学担任访问教授。在此之前，他于1936年在莫斯科大学讲学。后来，他先后在京都（1956年）、高等科学研究所（1961年和1967年）以及剑桥大学（1972年）讲学。

除了上述的斯蒂尔奖之外，他还因其工作获得了许多荣誉。他于1944年因四篇关于代数簇的论文获得美国数学会颁发的法兰克·尼尔森·寇尔代数奖，其中两篇于1939年和1940年发表在American Journal of Mathematics上，另外两篇也分别于1939年和1940年发表在Annals of Mathematics上。他于1965年获得国家科学奖章。

许多科学院和学会通过选举他为成员来向他表示敬意，包括美国国家科学院（1944年）、American Academy of Arts and Sciences（1948年）、美国哲学学会（1951年）、Brazilian Academy of Sciences（1958年）和Accademia dei Lincei（1958年）。

Oscar Zariski's father was Bezalel Zaritsky and his mother Hannah Zaritsky. Oscar, born into this Jewish family, was named Ascher Zaritsky by his parents. We will comment below on how his name came to be changed to the now familiar version of Oscar Zariski.

Oscar's mother was the one who ensured that her young son had a good education. A tutor was provided for Oscar from the time he was seven years old and Oscar, under the guidance of the tutor, showed remarkable aptitude for the Russian language and for arithmetic. When the fighting associated with World War I reached Belarus, Oscar's family fled to Chernigov in Ukraine. It was the first of several moves forced on him by political problems.

Unable to enter the faculty of mathematics at the University of Kiev as all the places were full, he chose philosophy instead. He was a student of philosophy at Kiev from 1918 to 1920. However he was able to pursue his mathematical interests and studied algebra and number theory in addition to philosophy. However Zariski had carried out his studies through a period of turmoil in Kiev.

In January 1918 Ukraine had become an independent state with Kiev as its capital. In the following month minor uprisings by workers in Kiev were suppressed but Red Army troops entered Kiev to give support to the workers. Kiev was then occupied by the Germans, but with the end of the war in November 1918, an independent Ukraine was declared again in Kiev. In November 1919 Kiev was briefly taken by the White armies, soon after to be replaced by the Red Army. There then followed the Russian-Polish War and, in May 1920, the Polish army captured Kiev but were forced out in a counterattack. Life for Zariski was just too difficult in this city so devastated by war, so he decided to go to Italy to continue his studies.

In Rome Zariski came under the influence of the great algebraic geometers Castelnuovo, Enriques and Severi. He obtained a doctorate from Rome in 1924 for a doctoral thesis on a topic related to Galois theory which was proposed to him by Castelnuovo. Zariski married in 1924; having met his future wife Yole Cagli in Rome they returned to Zariski's home town of Kobin to marry. Returning to Rome he remained there as a fellow of the International Education Board until 1927.

It was while Zariski was in Rome that Enriques suggested that Ascher Zaritsky, as he was then called, change his name to the Italian sounding Oscar Zariski. This was the name which he used on his first publication which was a joint paper with Enriques. Zariski wrote in [2] of how his mathematical interests differed from those of his supervisors Castelnuovo and Enriques:-

However, even during my Rome period, my algebraic tendencies were showing and were clearly perceived by Castelnuovo who once told me: "You are here with us but are not one of us." This was said not in reproach but good naturedly, for Castelnuovo himself told me time and time again that the methods of the Italian geometric school had done all they could do, had reached a dead end, and were inadequate for further progress in the field of algebraic geometry.

Zariski had gone to Italy to escape the problems in Belarus and Ukraine. However, the political situation in Italy began to deteriorate rapidly. In October 1922 Mussolini organized the Fascist "March on Rome" and he was asked to form a government. For 18 months he ran the country in reasonably democratic way but, during the years 1925 to 1927, he removed the right of free speech, and removed opposition parties and trade unions. The Fascist hatred of Jews made life for Zariski, because of his Jewish background, particularly difficult.

Helped by Lefschetz, he escaped from the political problems of Italy in 1927 and went to the United States. There he taught at Johns Hopkins University, being a Johnston Scholar until 1929 when he joined the Faculty. He became a full professor at Johns Hopkins in 1937.

Castelnuovo and Severi had encouraged Zariski to view Lefschetz's topological methods as being the road ahead for algebraic geometry, so between 1927 and 1937 Zariski frequently visited Lefschetz at Princeton. Zariski wrote [2]:-

I owe a great deal to [Lefschetz] for his inspiring guidance and encouragement.

During this period Zariski wrote Algebraic Surfaces which was published in 1935. He explains in [2] how writing this monograph changed the direction of his work:-

At that time (1935) modern algebra had already come to life (through the work of Emmy Noether and the important treatise of B L van der Waerden), but while it was being applied to some aspects of the foundations of algebraic geometry by van der Waerden ... the deeper aspects of birational algebraic geometry ... were largely, or even entirely, virgin territory as far as algebraic exploration was concerned. In [Algebraic Surfaces] I tried my best to present the underlying ideas of the ingenious geometric methods and proofs with which the Italian geometers were handling these deeper aspects of the whole theory of surfaces ... I began to feel distinctly unhappy about the rigour of the original proofs (without losing in the least my admiration for the imaginative geometric spirit that permeated these proofs); I became convinced that the whole structure must be done over again by purely algebraic methods.

At Johns Hopkins University between 1939 and 1940 Zariski carried out his project of applying modern algebra to the foundations of algebraic geometry. He worked on the theory of normal varieties, local uniformisation and the reduction of singularities of algebraic varieties.

An important year for Zariski was 1945 which he spent in São Paolo. There he gave a lecture course three days each week which was attended by André Weil and nobody else. Both Zariski and Weil learnt much in discussions, often arguments, about the material that Zariski was presenting. After spending the year 1946-47 at the University of Illinois, Zariski was appointed to a chair at Harvard where he was to remain until he retired in 1969. From the late 1970s he suffered from Alzheimer's disease and his last few years were difficult ones as his health failed.

In 1981 Zariski was awarded the Steele Prize by the American Mathematical Society for the cumulative influence of his total mathematical research. The citation for the prize summarised Zariski's contributions to mathematics throughout his life [2]:-

After beginning his work in Italy in 1924 very much in the style of "Italian algebraic geometry," Zariski realised that the whole subject needed proper foundations. Thus in the period 1927 to 1937 he turned first to topological questions and then in 1937 he began to lay the commutative algebraic foundations of his subject. His topological work concentrated mainly on the fundamental group; many of the ideas he pioneered were innovations in topology as well as algebraic geometry and have developed independently in the two fields since then.

In 1937 Zariski completely reoriented his research and began to introduce ideas from abstract algebra into algebraic geometry. Indeed, together with B L van der Waerden and André Weil, he completely reworked the foundations of the subject without the use of topological or analytic methods. His use of the notions of integral independence, valuation rings, and regular local rings, in algebraic geometry proved particularly fruitful and led him to such high points as the resolution of singularities for threefolds in characteristic 0 in 1944, the clarification of the notion of simple point in 1947, and the theory of holomorphic functions on algebraic varieties over arbitrary ground fields. The theory of equisingularity and saturation begun by Zariski in 1965 has also been of great influence and importance.

All of Zariski's work has served as a basis for the present flowering of algebraic geometry and the current school uses his work and ideas in the modern development of the subject.

Zariski's most famous book is Commutative Algebra, a two volume work written jointly with P Samuel. The first volume appeared in 1958, the second in 1960.

The American Mathematical Society played a large role in Zariski's life and he contributed greatly to the Society over many years. He was vice president of the Society between 1960 and 1961 and president of the Society from 1969 to 1970.

Zariski played an important role in mathematical publishing after his appointment as a full professor. He was an editor of the American Mathematical Journal from 1937 to 1941, served as a member of the editorial committee of the Transactions of the American Mathematical Society from 1941 to 1947, and also served on the editorial boards of the Annals of Mathematics and the American Journal of Mathematics.

After going to the United States in 1927 Zariski spent considerable periods lecturing at other universities both in the United States and in other countries. We have already mentioned that he was a visiting professor at São Paulo in 1945 and a visiting professor at the University of Illinois in 1946-47. Before that, in 1936, he had lectured at the University of Moscow. Later, he lectured at Kyoto (1956), the Institut des Hautes Études Scientifiques (1961 and again 1967), and the University of Cambridge (1972).

He was awarded many honours for his work in addition to the Steele Prize described above. He was awarded the Cole Prize in Algebra from the American Mathematical Society in 1944 for four papers on algebraic varieties, two published in the American Journal of Mathematics in 1939 and 1940, and the other two in the Annals of Mathematics also one in 1939 and the second in 1940. He was awarded the National Medal of Science in 1965.

Many academies and societies have honoured him by electing him to membership, including the U.S. National Academy of Sciences (1944), the American Academy of Arts and Sciences (1948), the American Philosophical Society (1951), the Brazilian Academy of Sciences (1958), and the Accademia dei Lincei (1958).