He has an IQ of 230, surpassing Hawking and Albert Einstein. At the age of 2, he could teach other older children to count by building blocks, but he said that he taught himself by watching "Sesame Street". He began to teach himself calculus at the age of 7. At the age of 12, he participated in the Mathematics Olympiad and won the gold medal, a record that has not been broken to this day.
He received his doctorate at the age of 21. At the age of 24, he was hired as a full professor at the University of California, Los Angeles. At the age of 31, he won the Nobel Prize "Fields Medal" in mathematics, and he was called "Mozart in mathematics". He was a likable mathematician, but he was also a mortal. When I was in school, I was bothered by games, and I cared more about him as a child than about his success stories as an adult. At the age of 15, when you and I were still playing with mud, how did he get into math?
本文选自 Living Proof ，记录了陶哲轩对自己当年入坑数学的回忆。
Gifted, he was different from a young age
Ever since I can remember, I've always been fascinated by numbers and formulas. At that time, I didn't really understand what math was. One of my earliest childhood memories was when my grandmother was arguing to clean the window and asking her to draw numbers on the window with detergent.
When I was a kid, whenever I was naughty, my parents would give me a math book for me to do exercises to distract me, which is something I was very willing to do. For me, math is an activity that I have fun with, and I can play with it for a long time. Maybe that's why I found that the math classes in school were quite simple, and even after I skipped a few levels, I still thought it was easy.
If I find a topic in class that interests me, I practice those handouts in class. Or find any of the methods that the teacher demonstrates on the whiteboard to prove in depth, or focus on some numbers and try to find solutions to some special example problems.
On the other hand, if it is a topic that I am not interested in, I will feel bored and bored just like other students. No matter what math problems I have, I have never taken particularly detailed notes, nor have I developed a particularly systematic study habit.
I prefer to improve my approach through my assignments and exams. For example, I don't memorize textbooks and sharpen my guns before the final exam. I might take out the parts of the lecture notes that I liked.
Until I graduated, it worked quite well. Most of the classes I like are A's, and the ones that I find boring almost barely pass or simply fail. (One of the classes was Fortran programming, which I didn't like because I had already learned to program in Basic.) The other was quantum mechanics, and before the end of the semester, I was told by my teacher to write a paper on the history of research on this topic, but I didn't remember it until the day of the exam, and this paper was going to account for half of the exam, and I couldn't cry so much that the teacher had to escort me out of the classroom. Not surprisingly, I failed. ）
Despite this, I eventually graduated from college with flying colors. In fact, in the same year, there were two other math honors students. When I was in graduate school at Princeton, I still had my study habits (or rather study habits less than perfect).
Genius has also missed his studies because of games
When I was studying at Princeton, there were no assignments or exams for graduate school, but there was only one important exam that everyone was afraid of. Exams usually last more than two hours, with one student facing three examiners, and are only available in the first and second year years.
To sum up, the questions generally come from these five areas: real analysis, complex analysis, linear algebra, and two topics of the student's choice. For most of my fellow graduate students, preparing for this general test was a top priority, reading textbooks from start to finish, organizing study groups, and simulating questions with each other.
It has become a tradition for every graduate student to jot down the content of the exam and then write down the answers to the questions for later students to practice with.
Sometimes there are even skits that imitate the invigilator. This "death committee", consisting of three teachers, was notorious for being particularly harsh on candidates, and I tried to ignore all of this.
I only go to classes that I like, and I skip classes if I don't like them. I also spent some time reading textbooks on and off, and in my early years in graduate school, I spent a lot of time online and playing computer games in my dorm room.
For my exam topics, harmonic analysis and analytic number theory were chosen, and I chose these two because I had already studied some of them before I went to Australia for my master's degree.
Feeling that analysis was my forte, I only spent a few days reviewing real, complex, and harmonic analysis. Most of his time is spent studying linear algebra and analytic number theory.
Overall, it only took me about two weeks to prepare for the general exam, while my classmates spent about a month preparing for it. Still, I felt quite confident during the exam.
The exam went smoothly because they asked me to demonstrate a pre-prepared harmonic analysis, which was mostly based on my master's thesis, specifically the theorem in harmonic analysis, known as the T(b) theorem.
However, when they don't ask about these topics, my lack of preparation is apparent. I can vaguely recall the basic results of the field, but I can't articulate it clearly, provide a correct proof, describe what it's used for, or what it's associated with.
I have a unique memory of the examiners, who asked a lot of very simple questions and tried to guide me to a point where I could give a satisfactory answer, for example, they took a few minutes to explain to me the source of the fundamental solution of the Laplace equation.
I love harmonic analysis, and I never pay attention to how it can be used in other fields, such as in some papers, or in complex analysis. For example, the Fourier multiplier was provided for the propagator of the wave equation, which I did not recognize at all and could not say anything interesting.
I was blessed by luck and began to study seriously
I was really lucky because their question turned to another topic of analytic number theory of mine, analytic number theory. And there was only one examiner with an extensive background in number theory, but he mistakenly thought that I had chosen algebraic number theory as my topic, so he felt that all the questions he had prepared were not appropriate.
Therefore, they only ask me very simple questions in analytic number theory (e.g., proving the prime number theorem, Dirichlet's theorem, etc.). These are the topics I actually prepared so I was able to answer them easily.
The next exams went very quickly, as none of the examiners prepared really challenging algebra questions. After many painful closed-door "interrogations", the examiner decided to let me pass. But my mentor gently stated that he was disappointed with my performance and hoped that I would do well in the future.
I'm still in a state of shock, it's the first time I've had a bad performance on an exam, and I really want to do well. But it was also a major turning point in my career and a wake-up call. I started taking my lessons seriously and studying harder.
I listened to my classmates and other teachers and reduced the amount of time I spent playing games. I took the task given to me by my mentor very seriously, hoping to let him see my efforts in this way.
Of course, I don't always succeed in this area – for example, the first project my supervisor gave me didn't really solve until five years after I graduated from my PhD. But in the last two years of my graduate studies, I put a lot of effort into writing my dissertations, as well as a number of publications, and started my career as a professional mathematician.
In retrospect, the near-failed final exam was probably the best thing that ever happened to me. I recorded the experience of this failed exam and put it online, you can still find it now. I was told that this was well known to all the graduate students at Princeton University.
Postscript: Tao Zhexuan said in the preface to "Tao Zhexuan Teaches You to Learn Mathematics": "I was pleasantly surprised to find that even those very complex and esoteric results can often be deduced using some fairly simple, even common-sense principles. When you comprehend one of these principles and suddenly see how it illustrates a vast mathematical system, you can't help but shout 'aha' in surprise. It was truly an unusual experience. ”
I hope you and I can experience such "aha" moments!
Recommended Books for this issue
Tao Zhexuan Teaches You to Learn Mathematics
作者:[澳]陶哲轩(Terence Ta)
Translator: Li Xin
Reprinted from: Turing Editorial Department
Editor-in-charge: Dong Xiaoxian