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Terence Tao's Raw Notes After His Princeton Comprehensive Exams (1999)
- dang 7y agoA thread from 2011: https://news.ycombinator.com/item?id=2771031 https://news.ycombinator.com/item?id=2771031
- tw1010 7y agoIn that thread someone says that Tao created his own lecture notes for a linear algebra course he was doing[1]. But that page is marked as coming from "2002", which would make him 27. But surely Tao didn't first take linear algebra at 27?? Can anyone make sense of this for me? [1] https://news.ycombinator.com/item?id=2772019 https://news.ycombinator.com/item?id=2772019
- j7ake 7y agoHe was probably teaching the linear algebra course.
- ginnungagap 7y agoThe post you linked says that Tao was teaching the class and made notes for it since he didn't like the book
- a1pulley 7y agoA friend from my Caltech undergrad did his PhD with Terence Tao. In harmonic analysis :-)
- iamcreasy 7y agoAny stories?
- a1pulley 7y agoNot a great one, but perhaps one useful to aspiring mathematicians! My friend, who is now an assistant prof at a major research university, said he could only do ~4 hrs of hard mathematical thinking per day. His colleagues and undergrad advisors all experienced the "four hour threshold." However, he said that Terence Tao was able to stay switched on for most of his waking hours, and that he'd structured his life - with nannies and a house close to UCLA - to take advantage of that. I guess the moral of the story is that normal mathematicians can be productive and quite happy with just a few hours of work per day, but that generation defining mathematicians have to put in longer hours :-)
- iamcreasy 7y agoI think it's a great one. Thank you for sharing! :)
- nisuni 7y agoLooks like the year is not correct. The text has clearly been written right after Terence Tao sat the exam, and given that he received his Ph.D. in 1996, most likely the notes are from 1993 or 1994.
- breck 7y agoThanks. I found it in a comment that said “20 years ago”, but wasn’t sure if that was an estimate or exact.
- hycaria 7y agoI guess it has been reposted because of his latest blog post. The source that sparked it is really interesting : https://www.ams.org/about-us/LivingProof.pdf#%5B%7B%22num%22%3A351%2C%22gen%22%3A0%7D%2C%7B%22name%22%3A%22FitH%22%7D%2C648%5D https://www.ams.org/about-us/LivingProof.pdf#%5B%7B%22num%22...
- kaitai 7y agoTo boost this, I do recommend checking out Living Proof, the book linked above, if you're interested in the mathematical life. Today's mathematicians discuss their challenges, successes, life, art, attitudes, etc. Especially if you're a student, check this out! You'll find some story that resonates with you.
- enriquto 7y agoFunny because it recalls an exam that he essentially aced as if it was a great failure.
- MRD85 7y agoHe was also only 16-17 years old at the time, completing an exam for university graduates. EDIT: This was his second year, so he was 17-18 years old. Also note this qoute: All in all, I probably only did about two weeks’ worth of preparation for the generals, while my fellow classmates had devoted months. Nevertheless, I felt quite confident going into the exam.
- hpcjoe 7y agoI remember doing something like this for my comps. I had left one school with a masters, looking for computational physics stuff I was interested in. Found another and an advisor who was interested in this work. I applied, got in, and started, while working full time. Graduate advisor called me up 2.5 weeks before starting and said "we want you to take the comprehensive exam in 2 weeks." After much swearing and cursing under my breath, I said "sure". I was told I had one of the highest scores in the written part. The oral part was just like this ... people asking me questions with vague definitions of various things. The example that sticks out to me was this one. "Is the atomic radius of an neutral atom a strong function of Z". Prof got annoyed when I asked them what "strong function" meant in this context; monotonically increasing/decreasing? Something else? I do remember being asked a few questions I had no idea how to answer, so I basically started from first principles and hashed out approximations/calculations very quickly. That was 29 years ago for me.
- nerdponx 7y agoThe most impressive thing about really smart people, to me, is how much they actually write down! I feel like writing is an underappreciated thinking tool. I should start doing it more.
- kodz4 7y ago"Reading maketh a full man, conference a ready man, and writing an exact man", said Mr. Knowledge is Power, "And, therefore, if a man writes little, he had needed have a great memory; if he confers little, he had need of a ready wit; and if he read little, he had need of much cunning to seem to know that he knoweth not”
- Bakary 7y agoThe greatest advantage of writing is that it exposes your own thinking and ideas as sloppy and confused.
- klyrs 7y agoI feel personally attacked by this. I should probably write more
- audiometry 7y agoWhen you have a genius-prodigy guy like this being 'examined' -- the professors ask him questions, some (many?) he doesn't know the answer to, or perhaps an incomplete answer to, while they (presumably) do know. Is this because they're equally genius as him? or are 'merely' much-better read than him, although maybe not as gifted in whatever other magical ways he is?
- petters 7y agoMathematics is not only brain teasers. I remember my first lecture in topology. Our professor said that while we may be smarter and beat our teachers in things like group theory and algebra, in this subject no one was going to be better than him. Experience also matters a lot in some subjects.
- enriquto 7y agoOf course the professors were better read and had much more experience and knowledge. But you con infer from that report (even if the author tries to hide it) that the professors were extremely impressed by the young candidate.
- aetimmes 7y agoTerrence's own recounting of the event [1] seems to contradict you: "After many nerve-wracking minutes of closed-door deliberation, the examiners did decide to (barely) pass me; however, my advisor gently explained his disappointment at my performance, and how I needed to do better in the future." [1] https://www.ams.org/about-us/LivingProof.pdf#%5B%7B%22num%22%3A351%2C%22gen%22%3A0%7D%2C%7B%22name%22%3A%22FitH%22%7D%2C648%5D https://www.ams.org/about-us/LivingProof.pdf#%5B%7B%22num%22... (found elsewhere in the thread)
- enriquto 7y agoThe sentence before the one you quote is this: "The rest of the exam then went fairly quickly as none of the examiners had prepared any truly challenging algebra questions." Thus, he emptied the pool of questions prepared by the professors. Notice that your quote does not really contradict a good impression upon the examiners. The nerve wracking is self reported, as is the word ``barely''. The later comments by the examiners may be also a generic sentence they say to everybody. Regardless of your level, you can always benefit for being told to do better. Sure, if you ask these professors today they will probably say that they were very impressed by the young genius... but this does not really say much.
- lonelappde 7y agoIs this a type-up of handwritten notes, or did he type them originally?
- kaitai 7y agoThis is part of the Princeton notes on the general exams, just an effort by grad students to support each other by sharing knowledge. It's not just helpful for them -- when I did my exams, I looked up every question they posed in algebraic geometry & my minor topic area and used it as a question bank, despite not going to Princeton :) https://web.math.princeton.edu/generals/ https://web.math.princeton.edu/generals/
- madcaptenor 7y agoIn particular this would probably have been written up a few days after the exam - these are oral exams. I took a similar exam at a similar program. We also had a repository of exam writeups like this. But I never got around to writing mine up - there were so many things I had put off until "after orals" that the last thing I wanted to do was relive them. (Like Tao, I passed; like Tao, I muddled through large portions of my exam.)
- lonelappde 7y agoThis teenager's document is a great counter point to Tao's naive claim that he is not a native genius and that anyone can do what he does if they merely study as much as he did growing up.
- deleted 7y ago[deleted]
- umanwizard 7y agoWhere does Tao claim that "anyone can do what he does if they merely study as much as he did growing up" ? I find that hard to believe. Unless you have an actual citation, it's much more plausible to me that he said hard work is important in addition to raw talent.
- amch 7y agohttps://terrytao.wordpress.com/career-advice/does-one-have-to-be-a-genius-to-do-maths/ https://terrytao.wordpress.com/career-advice/does-one-have-t... Not exactly "anyone can do this," but the first paragraph in the above link: "The answer is an emphatic NO. In order to make good and useful contributions to mathematics, one does need to work hard, learn one’s field well, learn other fields and tools, ask questions, talk to other mathematicians, and think about the “big picture”. And yes, a reasonable amount of intelligence, patience, and maturity is also required. But one does not need some sort of magic “genius gene” that spontaneously generates ex nihilo deep insights, unexpected solutions to problems, or other supernatural abilities."
- umanwizard 7y agoThis is about whether any intelligent person can "do mathematics", not about whether anyone can be like Terence Tao, one of the most brilliant mathematicians in the world.
- BeetleB 7y agoNowhere in there does he claim he is not a native genius. Nor does he say that anyone can do what he does if they merely study as much as he did.
- mb_72 7y agoMy Mum attended one maths course at university (as a mature age student) with Tao, and I distinctly remember her coming home and complaining about some 'smart-arse kid who wouldn't stop challenging the lecturer'. I guess it can be tough being very smart and capable and also young at the same time.
- matz1 7y agoWhats the relation of someone complaining and tough in this context ?
- hojjat12000 7y agoIt's tough to be smart, young, and capable. Because people will call you smart arse and not like you.
- matz1 7y agoIf you are smart then you are smart enough to ignore that.
- munificent 7y agoIf this were true, many many smart people would have much less anxiety and self-doubt.
- waterhouse 7y agoA lesson which many highly intelligent persons never learn as long as they live is that human beings in general are incorrigibly very different from themselves in thought, action, and desire. Many a reformer has died at the hands of a mob which he was trying to improve. The highly intelligent child must learn to suffer fools gladly--not sneeringly, not angrily, not despairingly, not weepingly--but gladly, if personal development is to proceed successfully in the world as it is. Failure to learn how to tolerate in a reasonable fashion the foolishness of others less gifted leads to bitterness, disillusionment, and misanthropy, which are the ruin of potential leaders. As a form of failure to suffer fools gladly, negativism may develop. The foolish teacher who hates to be corrected by a child is unsuited to these children. Too many children of IQ 170 are being taught by teachers of IQ 120. Into this important matter of the selection of the teacher we cannot enter, except to illustrate the difficulty from recent conversation with a ten-year-old boy of IQ 165. This boy was referred to us as a school problem: "Not interested in the school work. Very impudent. A liar." The following is a fragment of conversation with this boy: What seems to be your *main* problem in school? Several of them. Name *one*. Well, I will name the teachers. Oh, boy! It is bad enough when the *pupils* make mistakes, but when the *teachers* make mistakes, oh, boy! Mention a few mistakes the teachers made. For instance I was sitting in 5A and the teacher was teaching 5B. She was telling those children that the Germans discovered printing, that Gutenberg was the first discoverer of it, mind you. After a few minutes I couldn't stand it. I am not supposed to recite in that class, you see, but I got up. I said, "No; the Chinese *invented*, not discovered, printing, before the time of Gutenberg--while the Germans were still barbarians." Then the teacher said, "Sit down. You are entirely too fresh." Later on she gave me a raking-over before the whole class. Oh, boy! What teaching! It seemed to me that one should begin at once in this case the lesson about suffering fools gladly. So I said, "Ned, that teacher is foolish, but one of the very first things to learn in the world is to suffer fools gladly. The child was so filled with resentment that he heard only the word "suffer." "Yes, that's it. That's what I say! Make 'em suffer. Roll a rock on 'em." I quote this to suggest how negativistic rebels may seize on the wrong idea. Before we finished the conversation Ned was straightened out on the subject of who was to do the suffering. He agreed to do it himself. I will cite another conversation, this time with a nine-year-old, of IQ 183. What seems to be the *main* trouble with you at school? The teacher can't pronounce. Can't pronounce *what*? Oh, lots of things. The teacher said "Magdalen College"--at Oxford, you know. I said, "In England they call it Môdlin College." The teacher wrote a note home to say I am rude and disorderly. She does not like me. -- Leta Hollingworth, "Children Above 180 IQ Stanford-Binet"
- mistermann 7y agoAnyone know of a decent documentary on Terence Tao? Not finding much on Youtube.
- Balgair 7y agoHe's a middle aged man. Doing a docu on him today isn't likely a good idea, he's still got a lot of life left to live. Numberphile on YouTube has some good stuff on him though. https://www.youtube.com/results?search_query=numberphile+tao https://www.youtube.com/results?search_query=numberphile+tao
- jason_slack 7y agoTao is the reason I recently started taking math classes again. I wanted to be able to at least understand some of what he talks about. :-).
- dpflan 7y agoThis caught my attention: """ Then they asked how Dirichlet got an explicit formula for this when \chi was a real character. I was going to write a messy (but finite) expression involving sines and logs, but then I realized that they were talking about the class number formula. (I said carelessly though that "this was a disgusting way to do it", since I was still thinking about the sine-log formulas. Then they made a comment that "This would put thousands of people out of work", or something like that.) """ Can someone explain the comment made by the examiners? > "This would put thousands of people out of work". Thanks!
- robinhouston 7y agoI read this as a joking reference to the fact that class field theory is an important area of contemporary mathematics, which presumably employs many people (though perhaps not literally thousands). I’d be curious to know if anyone reads it differently.
- 7402 7y agoI don't know much pure math. If I pretend I had never heard of Tao, then I would have completely believed these notes if they had ended with "And that's how I failed my comprehensive exam," instead of, "After this, they decided to pass me." It's funny to me that just going by the tone (without having any domain knowledge), it's hard to tell how well he was doing.
- kkylin 7y agoThese exams often work by having the examiners ask questions until one reaches the boundary of the student's boundary, partly to see where that boundary is and partly to see how they handle things they do not know. Ideally, anyway. The oral examination format can be very intimidating, and committees often don't care to (or don't know how) to put students at ease. So how well performance correlates with actual ability really varies.
- peteretep 7y agoIt feels like a common idea that the worse you felt you did in Oxbridge entrance interviews, the better you probably did because the interviewers were trying to stretch you.
- ska 7y agoFor what it's worth, I'm nowhere near as bright as Terence (which puts me in copious company, of course) but notes of my comprehensive exams would have looked kind of similar in tone if not content. Having given oral exams as well - often you are trying to find the boundary where certainty breaks down to "on the fly" thinking for the candidate. You can get a pretty accurate view of how well someone knows the material quite quickly this way, but you certainly have to account for "nerves" also. I remember being asked a question and just having no idea how to answer it - another examiner jumped in with a `different' question which I answered, then the 1st came back with "can you show how that is equivalent to what I asked" and it took me two seconds to realize they were basically the same question. That stuff can really throw you off.
- svat 7y agoThis is part of the Princeton math department Graduate Students' Guide to Generals: https://web.math.princeton.edu/generals/ https://web.math.princeton.edu/generals/ There you can see similar notes made by other students; the standard is really high. E.g. here are notes by Manjul Bhargava: https://web.math.princeton.edu/generals/bhargava_manjul https://web.math.princeton.edu/generals/bhargava_manjul
- urmish 7y agoI can only comment on the first two subjects in the second link, but they look like standard graduate level coursework questions. The standard here and the standard of special topics questions Terrance Tao was asked seem quite different.
- dpflan 7y agoVery interesting. Does such a detailed list of accounts like this exist for other fields/departments/at other institutions? I would like to see for graduate CS departments.
- deleted 7y ago[deleted]
- jdoliner 7y ago> Talk about primes in arithmetic progressions. Tao would later go in to prove that there are arbitrarily long arithmetic progressions of primes in his most famous work to date, The Green-Tao Theorem. https://en.wikipedia.org/wiki/Green%E2%80%93Tao_theorem https://en.wikipedia.org/wiki/Green%E2%80%93Tao_theorem