Terry Tao's generals (2012)(web.math.princeton.edu)
web.math.princeton.edu
Terry Tao's generals (2012)
https://web.math.princeton.edu/generals/tao_terence
144 comments
Here's some interesting backstory on how Terence almost failed this exam: https://www.ams.org/notices/202007/rnoti-p1007.pdf
> 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!
You and me both, buddy!
Thought 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
[30] from Wikipedia: https://content.time.com/time/subscriber/article/0,33009,853... --> p.4
"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
[30] from Wikipedia: https://content.time.com/time/subscriber/article/0,33009,853... --> p.4
So, one of my favourite stories is Witggenstein's Phd viva. He submitted the Tractatus Logico-Philosophicus as his PhD thesis.
Apparently on the way into his viva he said to Bertrand Russel "Don't worry, I know you'll never understand it.".[0] He passed with flying colours.
(Note: I tried to google and didn't find a primary source, but post saying the same thing I remembered, so possible citation needed here)
[0] https://news.ycombinator.com/item?id=33910633
Apparently on the way into his viva he said to Bertrand Russel "Don't worry, I know you'll never understand it.".[0] He passed with flying colours.
(Note: I tried to google and didn't find a primary source, but post saying the same thing I remembered, so possible citation needed here)
[0] https://news.ycombinator.com/item?id=33910633
I have tried to read the Tractatus a couple of times and whiffed majorly. I think Wittgestein foresaw Powerpoint culture and created a (the only?) major philosophical work[0] consisting of just bullet points.
[0] True, I didn't understand it but it clearly has had an impact!
[0] True, I didn't understand it but it clearly has had an impact!
My favorite? When Von Neumann(an Austrian aristocrat) walked into his oral exam, Hilbert asked another of the admistrators who his tailor was.
> Von Neumann(an Austrian aristocrat)
Usually Von Neumann is thought to be Hungarian. Born in Budapest, which at the time was part of the Austro-Hungarian Empire.
Usually Von Neumann is thought to be Hungarian. Born in Budapest, which at the time was part of the Austro-Hungarian Empire.
Also not really an aristocrat, more like an upper-middle class banker's family. The "von ..." was more of a faux title, although his family did have an honorary nobility title (a different one) received a few decades before.
In europe -every upstart family, at some point tried to get the "von und zu","van", "de" only exemption from this is the english aristocracy.. were you glue a "lord" or "lady" in front of the name, instead of a addition bewetween name and surname.
Born János Neumann
[deleted]
Other reference to Oppenheimer:
James Franck, who had emigrated to the US, advocated to not use the Atomic bomb against Japan:
https://en.wikipedia.org/wiki/Franck_Report
James Franck, who had emigrated to the US, advocated to not use the Atomic bomb against Japan:
https://en.wikipedia.org/wiki/Franck_Report
I 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?
That does make me wonder: why DOES the qualification/position/status name mention only philosophy?
What we today call science was once called natural philosophy. Philosophy was simply the systematic study of the world, including physics, logic, and math, as opposed to theology or learning an applied professional trade such as medicine, law, or architecture.
The terms we use to differentiate fields of study are still shifting. Nobody uses the term "naturalist" anymore, at least not to refer to a scholarly field. There's often a huge overlap between math and various other fields. I'm not sure the recent explosion in degree names is any better than using Ph.D. It's tempting to suggest that fields like Art History and English Literature should be differentiated from [modern] Physics, but ideally scholars in all those fields should be (and often are) using many of the same analytical techniques. And realistically, I doubt usage of a shared signifier, Ph.D, is causing any consequential confusion. Arguably, as specialization increases the value of using Ph.D increases--it general implies someone has attained the highest credential in their field of study, whatever the field.
The terms we use to differentiate fields of study are still shifting. Nobody uses the term "naturalist" anymore, at least not to refer to a scholarly field. There's often a huge overlap between math and various other fields. I'm not sure the recent explosion in degree names is any better than using Ph.D. It's tempting to suggest that fields like Art History and English Literature should be differentiated from [modern] Physics, but ideally scholars in all those fields should be (and often are) using many of the same analytical techniques. And realistically, I doubt usage of a shared signifier, Ph.D, is causing any consequential confusion. Arguably, as specialization increases the value of using Ph.D increases--it general implies someone has attained the highest credential in their field of study, whatever the field.
In the medieval university system, there were four fields of knowledge: medicine (MD), law (Juris Doctor, or JD), theology (ThD), or philosophy (PhD).
In the centuries since, new kinds of knowledge were discovered, but they generally got squeezed in as sub-categories of philosophy, so when you specialise in them, you get a PhD.
In the centuries since, new kinds of knowledge were discovered, but they generally got squeezed in as sub-categories of philosophy, so when you specialise in them, you get a PhD.
PhD is a license to teach (i.e. doctor) philosophy. Specifically, in "science" fields, natural philosophy.
Natural philosophy was the original (and i think better) name for science, especially physics. It was understood to be an important subbranch of philosophy.
I think its better name because it elucidates an important epistemological difference: science is a method, not knowledge. I learn science when I learn about experimental design. Im not learning science when I learn about evolution; rather Im learning a theory, very likely to be true, that is almost impossible to put under scrutiny using the scientific method.
The Pythagorean theorem has no science and is 100% true. In fluids, Bernoulli devised experiments to demonstrate to his calculus illiterate colleagues what he had already mathematically proven.
Science is a bad term.
Natural philosophy was the original (and i think better) name for science, especially physics. It was understood to be an important subbranch of philosophy.
I think its better name because it elucidates an important epistemological difference: science is a method, not knowledge. I learn science when I learn about experimental design. Im not learning science when I learn about evolution; rather Im learning a theory, very likely to be true, that is almost impossible to put under scrutiny using the scientific method.
The Pythagorean theorem has no science and is 100% true. In fluids, Bernoulli devised experiments to demonstrate to his calculus illiterate colleagues what he had already mathematically proven.
Science is a bad term.
> In fluids, Bernoulli devised experiments to demonstrate to his calculus illiterate colleagues what he had already mathematically proven.
Maybe I misunderstood you, but I have the feeling you're downplaying the importance of experiments in Physics. Once you mathematically prove something, you proved that a statement is true when given a certain set of axioms. This is enough in Mathematics, like your example of the Pythagorean theorem, but it isn't in Physics. The reason being that proving something Mathematically consistent isn't enough to prove that it reflects what happens in the real world. A famous example in pop science of this is string theory.
I also have some doubts about what you say regarding evolution theory, but I'm not familiar whit how biologists verified it. Maybe every time a new fossil is found we can consider it as an experiment that can add a data point in favor or against the theory?
Maybe I misunderstood you, but I have the feeling you're downplaying the importance of experiments in Physics. Once you mathematically prove something, you proved that a statement is true when given a certain set of axioms. This is enough in Mathematics, like your example of the Pythagorean theorem, but it isn't in Physics. The reason being that proving something Mathematically consistent isn't enough to prove that it reflects what happens in the real world. A famous example in pop science of this is string theory.
I also have some doubts about what you say regarding evolution theory, but I'm not familiar whit how biologists verified it. Maybe every time a new fossil is found we can consider it as an experiment that can add a data point in favor or against the theory?
Im about to jump in the shower, son cant go inti detail. Just that the best theories in physics, our best foundations, were not made by an incremental tic-toc theory-experiment cycle. There were bursts of genius that eventually got proven.
Im reading Truesdell's (America's greatest 20th century physicist?) book now where he goes through the history of fluid mechanics and the paucity of experiments. If I remember ill send you the reference
Im reading Truesdell's (America's greatest 20th century physicist?) book now where he goes through the history of fluid mechanics and the paucity of experiments. If I remember ill send you the reference
Truesdell is amazing, I like this quote:
"The computer: ruin of science and threat to mankind”
https://people.maths.ox.ac.uk/~ball/Miscelleaneous%20Article...
"The computer: ruin of science and threat to mankind”
https://people.maths.ox.ac.uk/~ball/Miscelleaneous%20Article...
He nailed it for computers
"Experience, Theory and Experiment" from 1955 as found in "An Idiot's Fugitive Essay on Science" by Truesdell.
Bloody brilliant. 70 years ago he foresaw the problems of modern science.
Bloody brilliant. 70 years ago he foresaw the problems of modern science.
Holy cow, I had not until today realized that "doctor" comes from "docere" (to teach). Thanks for enlightening me!
Which makes you wonder why a medical doctor is called that?
Because doctor comes from the passive doctus ('having been taught', i.e. learned) rather than the active docens ('teaching, instructing').
Because physicians are licensed to teach the practice of medicine.
that's the root of the word. The original four-wise distinction seems to have been Doctor of Medicine/Law/Natural Phylosophy/Theology.
Just different kinds of knowledgeable people.
Interestingly enough I just realized in Italian there is an old word, "dotto" that literally would mean "someone who has been taught" but concretely is used in the sense of well educated, knowledgeable, wise, cultured.
Bologna, the home of the first university, has the nickname "la Dotta", i.e. the educated one.
Interestingly enough I just realized in Italian there is an old word, "dotto" that literally would mean "someone who has been taught" but concretely is used in the sense of well educated, knowledgeable, wise, cultured.
Bologna, the home of the first university, has the nickname "la Dotta", i.e. the educated one.
For the same reason social science is called science. When words start to become prestigious many will attach that word to what they do.
Social science is science. In what way is it not?
Lack of reproducibility results for one.
That's a problem in all the sciences right now: https://www.nature.com/articles/533452a
> Im not learning science when I learn about evolution; rather Im learning a theory, very likely to be true, that is almost impossible to put under scrutiny using the scientific method.
the very definition of a "theory" in the scientific sense of the word (e.g. "theory of evolution") includes the ability to test it using the scientific method.
the wikipedia page goes into great detail: https://en.wikipedia.org/wiki/Scientific_theory
and this is a good brief explanation: https://web.archive.org/web/20170709065046/http://science.ke...
the very definition of a "theory" in the scientific sense of the word (e.g. "theory of evolution") includes the ability to test it using the scientific method.
the wikipedia page goes into great detail: https://en.wikipedia.org/wiki/Scientific_theory
and this is a good brief explanation: https://web.archive.org/web/20170709065046/http://science.ke...
Indeed.
Jumping to specifics, Evolution Theory is relatively easy to demonstrate on short time scales of under a year with fruit flies, and can also be demonstrated on longer timescales of five, ten, thirty years given patience .. with ongoing field observations of some two centuries now in various locations.
https://www.the-scientist.com/news-opinion/fruit-flies-evolv...
Jumping to specifics, Evolution Theory is relatively easy to demonstrate on short time scales of under a year with fruit flies, and can also be demonstrated on longer timescales of five, ten, thirty years given patience .. with ongoing field observations of some two centuries now in various locations.
https://www.the-scientist.com/news-opinion/fruit-flies-evolv...
It wasn't a hit on evolution, I specifically stated that it is as true as anything lacking a mathematical proof can be.
And yes, I know about little bugs changing color and bacteria becoming resistant to antibiotics.
But, strictly speaking, that isnt what we're taught is "science" in 10th grade, is it? That's really an observation of what we cannot control fitting nicely into our theories.
The embarrassing thing about evolution is that biologists keep having to tweek it in ways that are bewildering if evolution is (I think better) understood as information theory. For example, why did biologists reject horizontal gene transfer?
Anyway the whole field is sitting atop of an ignored quantity-quality emergent behavior which is also the most interesting question.
And yes, I know about little bugs changing color and bacteria becoming resistant to antibiotics.
But, strictly speaking, that isnt what we're taught is "science" in 10th grade, is it? That's really an observation of what we cannot control fitting nicely into our theories.
The embarrassing thing about evolution is that biologists keep having to tweek it in ways that are bewildering if evolution is (I think better) understood as information theory. For example, why did biologists reject horizontal gene transfer?
Anyway the whole field is sitting atop of an ignored quantity-quality emergent behavior which is also the most interesting question.
Some of your points are very curious. Biologists did not reject horizontal gene transfer. Or at any rate it has been part of the canon for pretty much the last 60 years which is essentially forever in biology. (The mechanisms for heredity, i.e. DNA were only discovered in the early 1950s). Besides, scientists are always "tweaking" theories as you put it. More precisely, a fruitful and productive scientific program leads to more and more discovers and elaborations that have to be accomodated. Look at the changes in the Standard Model in particle physics. The current controversy is whether and why the Standard Model is no longer being "tweaked". In other words, some physicist wonder whether current research is no longer yielding discoveries that allow advances and hences changes or elaborations to the Standard Model.
Why specifically in science.
PhD is the degree you usually get at that level for English, Classics, Economics, Histrory etc..
Some Universities use DPhil (Oxford) for all these but I know of no other degree at that level except for those two (MD is effectively lower and DSc is usually some form of honourary degree and I am not certain what LLD is)
PhD is the degree you usually get at that level for English, Classics, Economics, Histrory etc..
Some Universities use DPhil (Oxford) for all these but I know of no other degree at that level except for those two (MD is effectively lower and DSc is usually some form of honourary degree and I am not certain what LLD is)
MIT seems to offer an ScD as an alternative to the PhD in some fields.
https://oge.mit.edu/graduate-admissions/programs/doctoral-de...
https://oge.mit.edu/graduate-admissions/programs/doctoral-de...
Because my post was wordy enough as it was and the OP was asking about a physics degree. I was going to be more specific, but it was late, I was tiered and.... meh
Historically a Masters degree was a license to teach, hence its name.
Apparently it is also a license for your brother in law to brag about having a PhD in economics, despite not being able to hold a job for more than 2 years and currently working as a manager at a pest control company after being laid off from his last job.
I can imagine that many great academics might have difficulties 'holding a job'. A PhD is certainly something to be proud of, just as running a marathon, raising a great child or having a great corporate career.
This is hilarious. Whereas I have to say, when holding a job means doing what other people want you to do, well, which person of high intelligence would be interested in that?
Physics was also philosophy for eons in that theories were on par with hard knowledge about the gods and mermaids.
Most early physic had surprisingly little explanatory or experimental connection to the actual physical world.
But it often made up for that with a solid grounding in popular mythology and the produce of some extraordinary imaginations.
You know, like philosophy!
Most early physic had surprisingly little explanatory or experimental connection to the actual physical world.
But it often made up for that with a solid grounding in popular mythology and the produce of some extraordinary imaginations.
You know, like philosophy!
Thats a gross mischaracterization of the ancients.
Anyway, those crazies came up with atomistic theory to explain silly myths.... and ended up explaining phase change as an emergent phenomenon.
Anyway, those crazies came up with atomistic theory to explain silly myths.... and ended up explaining phase change as an emergent phenomenon.
You are overlooking a lot. Then and today.
The ancients were not just the famous Greek, Roman, Babylonian, Indian and Chinese philosophers, who made progress in reasoning.
Human's across every culture have spun creative explanations for natural phenomena, going back as far as we have any records.
- Why does it rain? How can we make it rain?
- What are the planets? What do their cycles and alignments mean?
- What is an illness? How do we avoid it? Cure it?
Even those philosophers were not immune to this kind of thinking.
Non/Pre-scientific answers to unanswerable questions gave satisfied people’s need for order and gave them hope.
Today, many people still take such answers seriously where there is no science (afterlife, cosmic justice, ...), and even where there is (astrology, crystals, etc.)
The ancients were not just the famous Greek, Roman, Babylonian, Indian and Chinese philosophers, who made progress in reasoning.
Human's across every culture have spun creative explanations for natural phenomena, going back as far as we have any records.
- Why does it rain? How can we make it rain?
- What are the planets? What do their cycles and alignments mean?
- What is an illness? How do we avoid it? Cure it?
Even those philosophers were not immune to this kind of thinking.
Non/Pre-scientific answers to unanswerable questions gave satisfied people’s need for order and gave them hope.
Today, many people still take such answers seriously where there is no science (afterlife, cosmic justice, ...), and even where there is (astrology, crystals, etc.)
The term philosophy is used in the original Greek sense, "love of knowledge". As opposed to, Wikipedia tells me: theology, medicine and law. I guess it also has something to do with philosophy in the early sense of the word, before natural philosophy branched off and became what we today call the sciences.
The "philosophy" in PhD refers to philosophy/wisdom in general (philos=love, sophia=wisdom) not the specific academic discipline.
Partially, 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.
And 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!)
Philosophy is the study of everything that is not well understood or is difficult to understand, even the study of study itself. A good way to think of it is, "philosophy exists only in the gaps", the gaps in our understanding. What is the nature of matter, can you take clay and keep dividing it in half and in half? When we didn't know the answer, only speculated about it, it was philosophy. Now that we know the answer, it's physics. But unknown answers in physics are still considered philosophy, such as whether or what would it mean if time can run backward.
This implies philosophy and philosophers will disappear once we know everything, which could by some be considered one of the rewards of this quest for truth.
BTW, many institutions today offer the ScD degree, doctor of science; it is generally equivalent to a PhD, though in some places it can be considered "higher"
This implies philosophy and philosophers will disappear once we know everything, which could by some be considered one of the rewards of this quest for truth.
BTW, many institutions today offer the ScD degree, doctor of science; it is generally equivalent to a PhD, though in some places it can be considered "higher"
Early in the tradition of thinking, practitioners of thinking were called philosophers. In those times there was no distinction between the physics of the world and of the mind, physics and metaphysics were one in the same. As philosophy evolved, new branches became distinct, mathematics, physics, and later chemistry and others.
The basis of all of these branches of the sciences owe their place to the work done by the early thinkers who established the ways of knowing. It is in the method of knowing that the highest degree in academic sciences is a Doctor of Philosophy. By obtaining the degree you have proven you are capable of establishing "knowing". Which is to say, you can reason.
The basis of all of these branches of the sciences owe their place to the work done by the early thinkers who established the ways of knowing. It is in the method of knowing that the highest degree in academic sciences is a Doctor of Philosophy. By obtaining the degree you have proven you are capable of establishing "knowing". Which is to say, you can reason.
> Which is to say, you can reason.
He took the doctorate degree in Europe. He gained a PhD in Physics.
"Doctor of Philosophy" (PhD): a doctorate in just about any subject except law or medicine.
"Doctor of Letters" (D.Lit.): a doctorate in Law.
"Doctor of Medicine" (D.Med.): a doctorate in medicine.
"Doctor of Philosophy" (PhD): a doctorate in just about any subject except law or medicine.
"Doctor of Letters" (D.Lit.): a doctorate in Law.
"Doctor of Medicine" (D.Med.): a doctorate in medicine.
Philosophy literally means the love to knowledge, and was (in some way still is) the father/mother of all subjects we today call science. Before "sub-genres" like Math, Physics, Chemistry, Biology were large enough to warrant their own subjects, they were all just part of philosophy. Today, we speak of philosophy if we mean topics that are not part of these subjects, altough I feel a fair argument could be made that they could be still considered part of philosophy. The name just is something left of that time.
I don't ask questions, I just use mine for what it's worth!: Not much and decreasing by the day. (Also not in math)
Philosophy used to include all of science, up until "Natural Philosophy" got big enough to separate out on its own.
Because academics love to cling to outdated and today factually incorrect terminology, while arrogantly berating the general public when they do the same.
"Actually, it's not called the Sapir–Whorf hypothesis anymore. The preferred term is 'linguistic relativity' now."
... while you keep using titles that have been nonsensical for 200 years. Got it.
Academia is 99% power games. The emperor wears no clothes.
"Actually, it's not called the Sapir–Whorf hypothesis anymore. The preferred term is 'linguistic relativity' now."
... while you keep using titles that have been nonsensical for 200 years. Got it.
Academia is 99% power games. The emperor wears no clothes.
The preferred term for the Sapir–Whorf hypothesis is "uninformed nonsense mostly due to Whorf which really shouldn't have Sapir's name attached to it".
First 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.
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.
>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
who cares?
10 juniors with year of experience don't make it even close to decade of experience
Yes, what you're pointing out is that it doesn't scale linearly. But the point still stands.
that's just guessing. the point could also sit. what you mean to say is that it is still there. how is not relevant to your argument.
Typically exams are 2 full professors and maybe one asst prof. That asst prof will be 3-4 years in, plus 4-5 years of grad school + often a postdoc. And that asst prof will, on average, be better than the graduate student.
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There’s a big long list of these in https://web.math.princeton.edu/generals/, not just Tao’s.
Bhargava's one is fun to read: https://web.math.princeton.edu/generals/bhargava_manjul
I 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]
I 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.
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.
I 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.
that was a very entertaining one!
Quite a few future field medalists in that list. The collective math IQ of the group is off the charts!
This 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.
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.
If you want to end up with no candidates, go on.
Sounds like a great way to hire academics rather than people with industry experience.
Yeah, 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.
Reading 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!
Not something fun to go through, that's for sure!
This 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!
Granted, 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.
For 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!
I 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.
In 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.
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Sorry for asking, but I'm not familiar with this term. "Generals" means "final exam"? For PhD?
This other article by Tao explains the situation: https://www.ams.org/notices/202007/rnoti-p1007.pdf
I think these are normally referred to as oral exams. Most PhDs go from qualifying exams to oral exams to thesis defense.
I'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/
[0] https://web.math.princeton.edu/generals/
The exams would be at the beginning to middle of the program.
I 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.
I 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
I went to MIT for a physics doctorate, and in 2009 their generals were done in three parts. Part 1 was given in August of 2009, basically immediately after orientation, so I had spent the entire summer studying for it.
> graduate students
As far as I understand, in America graduate student can mean both a M.Sc. student and a Ph.D. student.
As far as I understand, in America graduate student can mean both a M.Sc. student and a Ph.D. student.
My 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
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
That 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/ which among other things links to "Work hard": https://terrytao.wordpress.com/career-advice/work-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/ which among other things links to "Work hard": https://terrytao.wordpress.com/career-advice/work-hard/
Fascinating. 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.
I was hoping he was going to apply the combinatorial nullstellensatz to solve the Byzantine Generals Problem...
I misread it as Tery Tao's genitals. I had a very different experience.
> 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
TT was 21 when he got his PhD, according to wikipedia.
https://en.wikipedia.org/wiki/Monster_group
TT was 21 when he got his PhD, according to wikipedia.
I am dim today. Is this Tao getting rejected for some interview for something?
These go by different names at different schools, but a grad student has to pass an oral exam, roughly at the end of their coursework and beginning of their dissertation phase. Sometimes called "orals," "comprehensive exams," etc.
By tradition, they can ask you anything. But in practice, they're not out to kill you. I went to a second tier university, and it was generally believed that the questions at a top school tended to be harder. I have no recollection of mine.
At this time, you also typically present a proposal for your thesis project. Passage of these hurdles makes you a "candidate," i.e., a candidate for a PhD upon defense of your thesis.
By tradition, they can ask you anything. But in practice, they're not out to kill you. I went to a second tier university, and it was generally believed that the questions at a top school tended to be harder. I have no recollection of mine.
At this time, you also typically present a proposal for your thesis project. Passage of these hurdles makes you a "candidate," i.e., a candidate for a PhD upon defense of your thesis.
> Sometimes called "orals," "comprehensive exams," etc.
— other names I have heard people use for these: prelims, quals, ...
— other names I have heard people use for these: prelims, quals, ...
"Prelim" also sometimes refers to an exam taken shortly after entering a PhD program to ensure you belong there.
One of the grad schools I attended had both a written "prelim" in year 1 and an oral "qual" in year 2/3. The other just had an oral "general" in year 2.
One of the grad schools I attended had both a written "prelim" in year 1 and an oral "qual" in year 2/3. The other just had an oral "general" in year 2.
"viva" in India
Princeton PhD students take a general exam as one of the major milestones of their PhD candidacy.
https://gradschool.princeton.edu/academics/degrees-requireme...
https://gradschool.princeton.edu/academics/degrees-requireme...
It'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
It's just faculty ensuring he has enough general knowledge of mathematics
At this point in time he would've been a teenager
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I think when it is written "They passed me" it meant that he passed the exam.
it is an oral exam as part of a phd. this was years ago (1996?), so interesting as a retrospective.
He's one of the preeminent living mathematicians.
Fields Medal winner in 2006.
(And, as listed on his wikipedia page, 27 other awards, medals, and prizes.)
(And, as listed on his wikipedia page, 27 other awards, medals, and prizes.)
I knew that, I was wondering if this was Tao rejecting someone or being rejected himself!
> 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. ;)
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. ;)
> Talk about primes in arithmetic progressions
Turns on super saiyan mode for the first time
Turns on super saiyan mode for the first time
I found these questions excellent prompts for chatgpt.
I imagine using chatgpt plus these questions would greatly help a grad student study for their comps.
I imagine using chatgpt plus these questions would greatly help a grad student study for their comps.
I don't know any of those theories and I feel quite ignorant and humbled. Extremely smart.
Tao's blog is quite interesting.
I 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.
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.
This post on the cap set problem was fun.
https://terrytao.wordpress.com/2016/05/18/a-symmetric-formul...
https://terrytao.wordpress.com/2016/05/18/a-symmetric-formul...
After a while writing LaTeX becomes second nature. I used to type my master's classes notes straight in LaTeX.
Impressive. 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.
Looking back I wish I had continued practicing typing after the one class I took in it. I should also have learned shorthand.
Times 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.
And I wrote my PhD in TeX. Not LaTeX ... that hadn't been invented.
No 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.
But, we were 'appy in them days, though we wuz poor.
> 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/
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/
> primes of the form 4n - 3.
I think they mean 4n + 3 unless this is a joke I'm missing
I think they mean 4n + 3 unless this is a joke I'm missing
There are infinitely many of them either way.
I 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.
I haven't the foggiest idea of what an "elementary explanation" of anything during a math graduate degree committee assessment would even sound like.
"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.
Often 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)".
I guess the closest equivalent in programming is "implement X without using any external libraries (or perhaps only the standard library)".
Elementary number theory means that you are doing number theory without complex analysis or lots of abstract algebra
Elementary means not using complex numbers.
“Elementary” is a relative term!
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so 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
serious question. i only know his name for being smart. not anything specific
He 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
Off 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). But there is more than that. It’s a very googleable question.
Have you tried asking your favourite search engine or even chat bot?
See eg https://en.wikipedia.org/wiki/Terence_Tao#Research_contribut...
See eg https://en.wikipedia.org/wiki/Terence_Tao#Research_contribut...
Being a great mathematicians not all about solving Millennium Prize Problems or being a household name like Einstein.
I 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 :)
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 :)