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Terry Tao's generals (2012)
- jameyk 3y agohttps://terrytao.wordpress.com/ https://terrytao.wordpress.com/
- pfdietz 3y agoTao's blog is quite interesting.
- OldGuyInTheClub 3y agoI feel good about myself if I can understand more than the first sentence of any of his posts. Some days I'll even settle for only the first. Great mathematicians are said to be exceptional at visualization. I wonder if the modern greats think in TeX or LaTeX on top of that. It is amazing how fast they post fully-formed, meticulously referenced blog articles or upload preprints.
- I_AM_A_SMURF 3y agoAfter a while writing LaTeX becomes second nature. I used to type my master's classes notes straight in LaTeX.
- OldGuyInTheClub 3y agoImpressive. I can't think and type fast with accuracy nearly well enough to do that for regular text, let alone a markup/page definition language. Looking back I wish I had continued practicing typing after the one class I took in it. I should also have learned shorthand.
- ColinWright 3y agoTimes have changed. We didn't have laptops in classes ... laptops didn't exist. And I wrote my PhD in TeX. Not LaTeX ... that hadn't been invented.
- dctoedt 3y ago> Times have changed ... I wrote my PhD in TeX. Not LaTeX ... that hadn't been invented. Brings to mind when I first learned TOPS-20 Emacs in law school (long story), circa 1980-81: A few of the CS grad students tolerantly looked down on Emacs as a crutch for newbies, because any real programmer would use TECO instead .... (IIRC, that version of Emacs was written as a collection of TECO macros.) Obligatory XKCD: https://xkcd.com/378/ https://xkcd.com/378/
- OldGuyInTheClub 3y agoNo laptops in my era, either. I managed to do my thesis in LaTeX only because a student in another department made a stylesheet to the expected format. Figures and captions were a struggle back then. But, we were 'appy in them days, though we wuz poor.
- pfdietz 3y agoThis post on the cap set problem was fun. https://terrytao.wordpress.com/2016/05/18/a-symmetric-formulation-of-the-croot-lev-pach-ellenberg-gijswijt-capset-bound/ https://terrytao.wordpress.com/2016/05/18/a-symmetric-formul...
- ajtulloch 3y agoThere’s a big long list of these in https://web.math.princeton.edu/generals/ https://web.math.princeton.edu/generals/, not just Tao’s.
- krackers 3y agoBhargava's one is fun to read: https://web.math.princeton.edu/generals/bhargava_manjul https://web.math.princeton.edu/generals/bhargava_manjul
- zach 3y agoI noticed that Evan O’Dorney’s generals was chaired by Bhargava and this question was asked: What is Brauer's theorem? [I had no idea and they moved on] He clearly should have read Bhargava’s generals, where Andrew Wiles asks Bhargava the same question!
- xdavidliu 3y agoI don't like that question because it asks for recollection of a name, as opposed to taking the theorem "when X is true, then Y is true" and changing the question into the form "when X is true, ____???". Worst case I've seen of this was when I was in 9th grade and our geometry teacher required us to memorize the chapter and section names of theorems in the book when proving. For example, in our proofs about triangles, we had to write "theorem 12.5" or else we wouldn't get credit on the test, and here 12.5 was the chapter and section number in the particular textbook, which is an utterly useless piece of info. Of course, the name Brauer is not nearly as useless as a chapter name, but still being familiar with math history probably shouldn't be hard requirement for being a professional mathematician.
- zem 3y agothat was a very entertaining one!
- The_suffocated 3y agoI think his generals are easier than Tao's. I wonder how many "average" PhD candidates worldwide can answer the questions in Tao's generals satisfactorily without difficulties. Many of them just seem highly research-oriented.
- lr4444lr 3y agoI did state though that all second derivatives of u would have L^p norms bounded by that of the Laplacian. Kleinermann asked me for an elementary explanation of this, but I couldn't think of one. I haven't the foggiest idea of what an "elementary explanation" of anything during a math graduate degree committee assessment would even sound like.
- NotYourLawyer 3y agoElementary means not using complex numbers.
- superposeur 3y ago“Elementary” is a relative term!
- deleted 3y ago[deleted]
- teraflop 3y ago"Elementary" is a somewhat subjective term that roughly means "from first principles", i.e. not relying on a bunch of advanced theorems and concepts. It doesn't necessarily mean the explanation is simple or concise.
- eru 3y agoOften the advanced theorems and concepts make for a simpler and more concise explanation. I guess the closest equivalent in programming is "implement X without using any external libraries (or perhaps only the standard library)".
- adgjlsfhk1 3y agoElementary number theory means that you are doing number theory without complex analysis or lots of abstract algebra
- krsrhe 3y ago[dead]
- quickthrower2 3y agoI am dim today. Is this Tao getting rejected for some interview for something?
- barkingcat 3y agoit is an oral exam as part of a phd. this was years ago (1996?), so interesting as a retrospective.
- deleted 3y ago[deleted]
- acjohnson55 3y agoPrinceton PhD students take a general exam as one of the major milestones of their PhD candidacy. https://gradschool.princeton.edu/academics/degrees-requirements/general-exam https://gradschool.princeton.edu/academics/degrees-requireme...
- lmpdev 3y agoIt's his "general exam" to start his PhD at Princeton It's just faculty ensuring he has enough general knowledge of mathematics At this point in time he would've been a teenager
- mcphage 3y agoHe's one of the preeminent living mathematicians.
- pfdietz 3y agoFields Medal winner in 2006. (And, as listed on his wikipedia page, 27 other awards, medals, and prizes.)
- quickthrower2 3y agoI knew that, I was wondering if this was Tao rejecting someone or being rejected himself!
- analog31 3y ago
- OldGuyInTheClub 3y agoThought this Oppenheimer anecdote may amuse the HN contingent. "Oppenheimer obtained his Doctor of Philosophy degree in March 1927 at age 23, supervised by Max Born.[28][29] After the oral exam, James Franck, the professor administering, reportedly said, "I'm glad that's over. He was on the point of questioning me."[30] " Franck had won the Nobel two years earlier. Sources: [Refs] quoted directly from Wikipedia article https://en.wikipedia.org/wiki/J._Robert_Oppenheimer https://en.wikipedia.org/wiki/J._Robert_Oppenheimer [30] from Wikipedia: https://content.time.com/time/subscriber/article/0,33009,853367-4,00.html https://content.time.com/time/subscriber/article/0,33009,853... --> p.4
- NooneAtAll3 3y agoI keep making this mistake. I almost asked "why philosophy, didn't he study physics?" That does make me wonder: why DOES the qualification/position/status name mention only philosophy?
- liamwire 3y agoPartially, it’s tradition at this point, but it’s also worth pointing out that a PhD isn’t the only doctorate, see e.g. DBA, EdD, JD, MD, etc.
- jvvw 3y agoAnd some universities still award a DPhil rather than a PhD (in the UK I believe Oxford and Sussex do - I have an Oxford DPhil and I always have the very mild quandary of whether to call it a PhD or DPhil - everybody knows what the former is and it is equivalent, but I feel like I am being inaccurate if I call it that!)
- OldGuyInTheClub 3y agoI don't ask questions, I just use mine for what it's worth!: Not much and decreasing by the day. (Also not in math)
- lmm 3y ago
- fiforpg 3y agoMy first distracted thought upon reading this title on HN: cool, Terry Tao is finally getting his own army! Jokes aside, Tao also has a nice write-up in AMS Notices on this experience and how it influenced his career: https://www.ams.org/notices/202007/rnoti-p1007.pdf https://www.ams.org/notices/202007/rnoti-p1007.pdf
- akira2501 3y agoI misread it as Tery Tao's genitals. I had a very different experience.
- klyrs 3y agoI was hoping he was going to apply the combinatorial nullstellensatz to solve the Byzantine Generals Problem...
- svat 3y agoThat AMS article ("A Close Call: How a Near Failure Propelled Me to Succeed") is great and sobering. I am sure it has helped some people, by showing that even Terence Tao cannot succeed without working hard. > this was the first time I had performed poorly on an exam that I was genuinely interested in performing well in. But it served as an important wake-up call and a turning point in my career. I began to take my classes and studying more seriously. I listened more to my fellow students and other faculty, and I cut back on my gaming. I worked particularly hard on all of the problems that my advisor gave me […] In retrospect, nearly failing the generals was probably the best thing that could have happened to me at the time. See also https://terrytao.wordpress.com/career-advice/ https://terrytao.wordpress.com/career-advice/ which among other things links to "Work hard": https://terrytao.wordpress.com/career-advice/work-hard/ https://terrytao.wordpress.com/career-advice/work-hard/
- lordnacho 3y agoFascinating. I guess everyone has had that experience of "hey how does that guy never f*ck up his exams?". To an extent I was that guy in high school. In uni I wasn't, but I knew people who seemingly always got top marks. I guess everyone hits a wall at some point.
- alkibiades 3y agoso has he accomplished anything big/groundbreaking other than being a child prodigy and having a high iq? serious question. i only know his name for being smart. not anything specific
- bglazer 3y agoHe won the fields medal, which is the effectively the Nobel prize for math, and has been cited nearly 100k times. Academically, these are extremely high level achievements. He’s contributed across very diverse fields of math, so it’s difficult to pin down his contributions to one thing in particular, but his most cited papers are fundamental to compressed sensing, which is both theoretically interesting and practically useful
- vietvu 3y agoBeing a great mathematicians not all about solving Millennium Prize Problems or being a household name like Einstein.
- Nevermark 3y agoI don’t know, I keep hearing about the Tao of this, or the Tao of that. I am sure no one has ever ever called him “The Tao of Math” before, but they have now! He is in it. Everywhere! Legacy Achievement: locked in :)
- wging 3y agoOff the top of my head, two big accomplishments are compressed sensing and the existence of arbitrarily long arithmetic progressions in the primes (the Green-Tao Theorem, https://en.wikipedia.org/wiki/Green–Tao_theorem https://en.wikipedia.org/wiki/Green–Tao_theorem). But there is more than that. It’s a very googleable question.
- eru 3y agoHave you tried asking your favourite search engine or even chat bot? See eg https://en.wikipedia.org/wiki/Terence_Tao#Research_contributions https://en.wikipedia.org/wiki/Terence_Tao#Research_contribut...
- jebarker 3y agoThis is the first time I've read a detailed account of "generals" in a US math PhD. I'm surprised how broad the topics covered seem. I did my PhD in the UK and the transition to fully fledged PhD candidate after the first couple of years was significantly less traumatic than this seems!
- deleted 3y ago[deleted]
- bmitc 3y agoGranted, this is at Princeton and for a child prodigy. Certainly only a few schools in the U.S. will have oral exams like this. Also, there's a potentiality of an unreliable narrator and certainly a modicum of figurative dick measuring on both sides of the examination table.
- jebarker 3y agoFor sure, definitely not equating my PhD experience to Tao's. I felt like even in my viva the examiners didn't push me too hard and stuck close to the material in the thesis. Probably due to my decidedly non-prodigy status!
- bmitc 3y agoI didn't think you were equating at all. I went to graduate school for math in the U.S., and we didn't even cover a lot of the topics in the first two years I see discussed in these general exams, much less be examined on them.
- stanford_labrat 3y agoIn life sciences the qualifying exam is an oral exam where they can in theory ask you anything. In comparison with for example the physics qualifying exam which (as I’m told) is a traditional paper exam, just ridiculously difficult.
- ishan0102 3y agoHere's some interesting backstory on how Terence almost failed this exam: https://www.ams.org/notices/202007/rnoti-p1007.pdf https://www.ams.org/notices/202007/rnoti-p1007.pdf
- rcpt 3y ago> Talk about primes in arithmetic progressions Turns on super saiyan mode for the first time
- CraigRo 3y agoFirst of all, nobody ever passed these exams with flying colors unless the examiners wanted that to happen or were having a bad day. Assume that there are three full professors testing you, and they have an average of 20 years each of post-graduate experience. That's 60 years of reading -- no way that a 2nd or 3rd year graduate student can keep up. Part of the process is humiliating the students a bit -- they did this for everyone. They also had a language requirement, which was often fulfilled by memorizing the 500 most common words in a foreign language as well as 300-500 common math terms in that language; there was a library of prepared crib sheets for these. It was kind of a running joke in that it tested your ability to translate an article from German/Russian/French into English sufficiently well to explain it to a bored examiner, but most of the students kind of looked it as a kind of giant 'Wheel of Fortune' exercise in intelligently guessing between the revealed clues. Second, some of the other graduate students basically went insane studying for this -- one locked himself in a room with a 30 day supply of microwavable meals, and another locked himself in his dorm and (having read MicroSerfs) only ate 2 dimensional food slipped under the door. A lot of this was immature students trying to out-do each other in commitment and probably didn't markedly improve pass rates. Tao matured, but mostly because he got older -- there were finally reasonably smart people his age studying with him. He also used to hang out with John Conway a lot, and Conway was going through his own problems; he spent more time with the 'more conventionally normal' faculty and that probably rubbed off.
- deleted 3y ago[deleted]
- tester756 3y ago>Assume that there are three full professors testing you, and they have an average of 20 years each of post-graduate experience. That's 60 years of reading who cares? 10 juniors with year of experience don't make it even close to decade of experience
- GTP 3y agoYes, what you're pointing out is that it doesn't scale linearly. But the point still stands.
- hhcoder 3y agoI don't know any of those theories and I feel quite ignorant and humbled. Extremely smart.
- j7ake 3y agoI found these questions excellent prompts for chatgpt. I imagine using chatgpt plus these questions would greatly help a grad student study for their comps.
- hintymad 3y agoThis seems a great way for system design interviews. Interviewer: what do you know about transactions? Candidate: Transactions can be serializable. Interviewer: great, tell me the difference between final state serializability and view state serializability and prove that they are the same if neither of them has dead step. Candidate: yada yada yada. Interviewer: Can you tell me how to use such concepts in implementing a production database system? Please come with a few short illustrative examples too.
- siva7 3y agoIf you want to end up with no candidates, go on.
- tourist2d 3y agoSounds like a great way to hire academics rather than people with industry experience.
- hintymad 3y agoYeah, my comment was a half-hearted joke. That said, the idea should work though: ask the candidate of fundamentals in their line of work, and see if the candidate can have coherent understanding of the landscape and can dive into critical details if needed.
- nvarsj 3y agoReading TFA I actually was reminded of doing system design interviews. It has a pretty similar vibe. Although the interviewers here seemed a lot more lenient than ones I've experienced in tech (Tao saying he "barely passed" kinda confirms it). Not something fun to go through, that's for sure!
- parentheses 3y agoTFA?
- 3y ago
- fnord77 3y ago> primes of the form 4n - 3. I think they mean 4n + 3 unless this is a joke I'm missing
- NotYourLawyer 3y agoThere are infinitely many of them either way.
- yu3zhou4 3y agoSorry for asking, but I'm not familiar with this term. "Generals" means "final exam"? For PhD?
- bmitc 3y agoI think these are normally referred to as oral exams. Most PhDs go from qualifying exams to oral exams to thesis defense.
- GTP 3y agoI'm not familiar either, but here they are talking about graduate students [0]. I guess it's a sort of final cumulative exam at the end of the programme. [0] https://web.math.princeton.edu/generals/ https://web.math.princeton.edu/generals/
- bmitc 3y agoThe exams would be at the beginning to middle of the program.
- GTP 3y agoI mean a final exam at the end of the program covering not just a single subject but potentially anything you studied during the program. But in practice it has to be more restricted, that's why they ask him what he prepared: it could be potentially anything, but to practical reasons they have to restrict it to a few subjects. Still more fields than what is covered by a single course. Or at least this is my interpretation.
- bmitc 3y agoI knew what you meant, but as I said, these type of exams are not at the end of a PhD program. At Princeton, it appears that the general exam occurs after year one or two: https://www.math.princeton.edu/graduate/requirements https://www.math.princeton.edu/graduate/requirements
- xdavidliu 3y ago
- tjrgergw 3y ago> What other simple groups do you know. (The monster? But I backed down quickly, saying I knew not how to define it.) https://en.wikipedia.org/wiki/Monster_group https://en.wikipedia.org/wiki/Monster_group TT was 21 when he got his PhD, according to wikipedia.
- supernova87a 3y ago> Stein then asked me what other boundedness theorems I knew. I couldn't think of any concrete ones off hand, but I mumbled about a convolution operator being bounded if its Fourier transform was bounded and had some smoothness condition, but I couldn't recall any details. You and me both, buddy!
- aquafox 3y ago> they decided to pass me, though they said that my harmonic analysis was far from satisfactory. :( Funny, because he received the Fields medal 1o years later for "his contributions to partial differential equations, combinatorics, harmonic analysis and additive number theory". So either his knowledge wasn't that bad or he is a very quick learner. ;)