王虹谈历练：年轻数学家的成长，不能只待在一个地方

上月，王虹成为首批获得菲尔兹奖的两位中国籍数学家之一。她表示，她在国外的时候，几乎所有的学校都希望学生或者年轻的数学工作者能在不同的学校都有历练。

Wang Hong, who last month became one of the first two Chinese nationals to receive the Fields Medal — often referred to as the "Nobel Prize" of mathematics — believes young mathematicians should embrace opportunities to study and work at different institutions.

“去不同的城市，去不同的学校历练是非常普遍的一个行为，大家也觉得这样对人的成长会比较好。”王虹在接受北京大学数学科学学院采访时说。

"Moving to different cities and schools is a very common practice, and people generally believe it is good for one's growth," Wang said in an interview with Peking University's School of Mathematical Sciences.

王虹如今是纽约大学柯朗数学科学研究所教授、法国高等科学研究所（IHES）数学科终身教授。她本人经历了从北大到法国、再到美国的多段学术旅程。

回顾在北大数院的岁月，王虹说：“我们级的学习氛围特别好，感觉想做数学、做科研的人特别多。大家有时候也会互相讨论，比如看什么课本比较好，课外再读一些什么书，有时候也会组织讨论班。”

她还记得，同学们一般大三就开始选研究生课，大四时已经修了不少，不仅基础知识扎实，对研究生级别的知识也了解较多。在北大数院接受的训练让她有了扎实的基本功，并且养成了一些好的学习习惯，这也让她在之后的课程学习中有了底气。

Looking back on her time at Peking University, where she transferred into the mathematics department, Wang recalled that students typically built a solid mathematical foundation while also getting an early start on graduate-level coursework.

在巴黎第十一大学读M2硕士时，王虹选修了帕斯卡•奥舍尔（Pascal Auscher）的调和分析课。她回忆：“每次上课都不带讲义，直接走上去就开始写，写得也非常流畅”，偶尔卡住时也能看到老师如何试图解决。

Wang later pursued a one-year master's program at Universite Paris-Sud, where she took a course on harmonic analysis taught by Pascal Auscher. She was struck by the lecturer's deep command of the subject — he never used notes and wrote fluently on the board.

但印象最深的是习题课。老师把近年调和分析的文章拆开，去掉能用课上知识证明的部分，让学生补全证明。“经常一个习题课做完，就相当于自己做完了一篇文章。”王虹说，“我觉得这个老师的设计非常好，而且我每次做作业都非常有成就感。”

她还提到，当时年级500人里只有五六个人想做纯数学，但教授们很愿意跟这些学生交流。

But what left the deepest impression were the tutorial sessions. The instructor would deconstruct recent papers in harmonic analysis, removing parts that could be proved using class knowledge and asking students to complete the proofs.

到麻省理工读博后，课程不再是重心，“最主要还是跟着导师做科研。”她的导师拉里·古斯对她影响非常大。

When Wang moved to the Massachusetts Institute of Technology for her PhD, coursework took a back seat. The most important part, she said, was conducting research with her adviser. Her adviser, Larry Guth, had a profound influence on her.

每次讨论前，她都要把导师可能问的问题都想过一遍，准备好回答，这样才能不挂黑板。这个习惯延续至今，让她和合作者的交流更有效率。

During their meetings, she had to anticipate every possible question he might ask and prepare answers in advance — a discipline that has shaped how she communicates with collaborators to this day.

古斯在每位博士生毕业时都会送一本书。王虹收到的是法文版《上学的烦恼》（Chagrin d'école），一本关于差生如何成长为教师的教育散文。“我读着觉得特别好。”

王虹坦言，长期卡壳是数学家的常态。“长时间没有做出让自己特别满意的工作，其实可能是一种常态，不只是我会经历，几乎所有人都会经历。”

Wang acknowledged that struggling with difficult problems for long periods is part of every mathematician's life.

她的应对方法是：先做一些“小”的东西，好好写下来。她把解决大问题比作拼图：“每一块拼图就是自己算例子和算子情形算出来的……当你把问题想足够久以后，你会知道最困难的地方在哪。”

She compared her approach to solving a large problem to piecing together a puzzle. Each piece comes from working out examples and special cases. Once you have thought about a problem long enough, you know where the real difficulties lie.

王虹表示，国内现在有很多非常厉害的老师，科研环境也很好。所以是否出国、什么时候出国要看自己所拥有的机会以及环境。

对于有志于数学的年轻人，她的寄语是：“希望大家按照自己的节奏去做自己喜欢的问题吧。”

For those pursuing mathematics, her parting advice was simple: "Follow your own pace and work on the problems you love."

记者：邹硕

来源：北京大学数学科学学院

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