The publication [1] that improved on the MRI results was the famous paper that layed out the foundation of the what is called "Compressed sampling" / "Compressive sensing". This is the main field of research for Signal Processing folks since around 2006. (Before it was mostly "just" Wavelets for a few decades).
In MRI you sample in the frequency domain (FFT) and compressive sensing can be used since MRI signals (which are structured signals) are sparse in that domain.
[1] Stable signal recovery from incomplete and inaccurate measurements
I remember circa 2005 when I was graduating from college, I readed all I could find about wavelets. I was obsesed, but didn't find what I was looking for. 10 years later I find your comment and I'm ready to take the subject again.
Don't get the wrong impression from that though. Professors absolutely do NOT on average make enough money. Terence Tao is a rare case who have the potential to have their name lasted for the next thousand years in proof and theorem (emphasize potential). He's literally one of those "one in a millions" genius thing. If anything, paying for one of the best guy in the world 400k a year might be a wee bit too far on the low side ...
There is a general consensus that Terrence Tao is on a whole other level than any other mathematician alive. Compared to the average salary of baseball, basketball, football and soccer players, his salary is pathetically low.
Oh, it may be that those math professors considered the best will have their salaries approach those of sport stars while median pay for those who teach classes continues to drop.
This seems like part of the general trend to value the "rock stars" of every field while diminishing the field in general. Could mathematics get by with just the fifty or however many people there are now who are "truly on the next level" (or whatever is said). Perhaps when colleges have replaced with moocs (an appropriate name if you know rpgs).
> This seems like part of the general trend to value the "rock stars" of every field while diminishing the field in general.
This is precisely the topic of Tyler Cowen's Average is Over[0]:
> Cowen forecasts that modern economies are delaminating into two groups: a small minority of highly educated and capable of working collaboratively with automated systems will become a wealthy aristocracy; the vast majority will earn little or nothing, surviving on low-priced goods created by the first group, living in shantytowns working with highly automated production systems.
Cowen's blog, Marginal Revolution[1], is also worth a read.
Where I live in Canada, there are professors who aren't even close to the same caliber of proficiency or importance as Dr. Tao and earn more. Granted, they represent a small percentage of professors, but the point is that top end professor do end up making quite a bit of money.
Terence Tao is a great mathematician - accepted. But I wish journalists would use a fraction of the space they use for hyping Terence Tao to write about the mathematics he does.
Terrence Tao has a blog where he writes about his work. It definitely doesn't cover the depth of his work, but it shows the type of work he is doing. Many entries are very readable (of course, there is a lot of mathematics in it, but Terrence Tao does an excellent job explaining in a relatively simple way).
https://terrytao.wordpress.com/
In some sense I did (not Fields-medal level, but graduate math level) some years ago. The problem is not that it is impossible or that one needs stretched analogies etc., but that the mass is simply doesn't like learning about concepts that are foreign to them etc.
Or to explain it in different terms: When one asks me, what I'm doing as a mathematician (in research): When the other person works in a related area I can give a short, concise explanation that they will understand. On the other hand, if the counterpart hasn't that deep knowledge, I have to give longer explanations that are understandable for their level of knowledge. But this is not what they want to hear (and I have not yet found out, what they want to hear). For me it's quite illogical to ask what I'm doing research about, if they don't even want to hear the answer...
The question is vague. What is a genius? and what is "doing maths"? Are we talking about solving a simple linear differential equation, or getting a professor position in a reputable institution? or getting a field medal?
In any case, it seems to me that Tao is giving the politically correct answer, that is everybody can do anything provided they work hard enough. I wonder if he honestly believes what he's saying.
It may be pleasant to hear, but I'm convinced it's far from the truth. Some people are naturally much much more math inclined than others and no amount of extra work is going to make up for the difference in talent.
In the 5 minute interview video, he mentions that the most exciting uses for math are learned in college and that grade school math is very dry because the students don't see what it can be used for. However, the article says that his focus is on pure math, which doesn't necessarily lend itself immediately to real world application.
I'm wondering what his opinion is on how to encourage a love for studying abstract/pure math in spite of not seeing how it can be applied immediately.
The great thing about math research, or really research in any field, is that it's highly parallelizeable. There are a few geniuses like Terry Tao exploring specific directions of thought, but there are so many things to think about and problems to solve that even the best researchers can only focus on a very small fraction. For every genius there are hundreds of other researchers who are only, say, 10% as productive (this is maybe your typical professor at a top-20 school), but through virtue of numbers they can explore a much wider range of problems and have a large aggregate impact. Fetishizing of genius makes for fun pop-science articles but is not really reflective of how the bulk of research gets done.
Of course, even if it doesn't require genius, making serious research contributions in mathematics does typically require many thousands of hours of training and hard work (usually in the form of a PhD) which most people are not going to dedicate their lives to. Given this, I really like Bill Thurston's perspective, from the famous essay "On proof and progress in mathematics" (http://arxiv.org/abs/math/9404236), which is that proving new theorems is only a small part of the project of mathematics. The real goal is to increase humanity's mathematical understanding, which does not necessarily grow monotonically: there are plenty of results proved 100 years ago that are not known or understood by any currently living mathematician, thus have been "lost" to the mathematical community. And of course there are new people born every day who know no math as all, so understanding of mathematics will tend to decrease unless people actively work to sustain it. In this view, understanding and communicating mathematical insights is just as important and valid a part of the intellectual process as proving new theorems, and requires a different set of talents. Plenty of great researchers are not great communicators, and vice versa. (Terry Tao actually is an excellent writer, but again he's only one person and there's a ton of math out there to be shared). You may not know as much math as Terry Tao, but anyone on HN probably knows more than the vast majority of adults (the bar here is really low), and the world at this moment is certainly not suffering from a surplus of mathematical thinking. So whatever your background and abilities, there are almost certainly lots of useful ways to contribute.
We all have our intellectual strengths and weaknesses. Do not worry too much about how much smarter a man like Terence Tao is compared to you (or me).
Work hard and learn what you need to learn to do what you want to do. The rest should fall into place.
Some vanishingly small percentage of people are prolific geniuses. If you aren't, no real sense worrying about it! Just hang in there and keep learning and doing.
On a similar note, how should one feel when reading/ learning about people like this?
I mean, on one hand, it's rather inspiring to read about the amazing thing that another human is achieving. On the other hand, I can't help but feel ... helplessness, like whatever I do, it doesn't really matter (add on with the guilt of squandering my time, and wondering if I could ever have been that good).
When I was younger and have less (or no) ambitions, it was easy to brush off everytime a story like this comes up: well, they're genius, and ... so what? But interestingly, since I've learned that I might be able to do more than I thought I could, everytime I read a story like this, I just feel extremely mediocre, even pathetic of myself.
I've been dealing with this kind of existential anxiety myself lately. Nothing anyone does matters on a universal scale, and I realized that was only painful because I had expectations for myself that were incongruent with reality. It's okay to not be a genius or contribute something significant, you only need to look for happiness and stability, which could be fading into obscurity and enjoying your time with friends and family. Personally, the times when I feel most relaxed are the times when I am most creative and do some of my best work anyway.
Another perspective is that this type of genius is a narrow measure of intelligence and an even narrower predictor of "success".
In leadership for example, do you think CEOs have the highest IQs in the company? How about Churchill, King, Lincoln? No doubt very smart people but relied just as much on other abilities.
What about the arts? How many of your favorite musicians or comedians are great at math? Arguably these abilities are orthogonal to what Tao can do, yet are clearly a form of genius.
The innateness/validity of EI is not yet settled but emotional intelligence enables people in ways that just are not possible with only pure problem solving prowess. How long would most math PhDs last as politicians, if they were somehow motivated to do so?
Speaking of motivation, how many IQ points past our own does someone of average ambition have to reach before we can't just out work them? Quite a few in my experience.
You need to ask the philosophical question of how you determine your self-worth. Is it discovering something new, learning...or is it more important that people, the large majority of whom you will never meet, consider you to be brilliant? Is a really a search for knowledge that you are undertaking, or a search for adulation from others. If it's the former, there is more than enough to learn for a number of lifetimes before advancing any branch of mathematics. If it is the latter, I can only pity you, it's a recipe for a sad life.
The brain is surprisingly plastic...all the way into 40s and beyond. If you have the passion, will, and the discipline to learn what you want to learn then amazing things can happen.
Five years is generally how long it takes to be noted as talented in something...so I guess patience is another virtue.
> The brain is surprisingly plastic... all the way into 40s and beyond
I'll give you the benefit of the doubt and it is probably factual. But it reads like 40s is "advanced". What should we consider the other 50% of our lives?
I'm in my early 30s, so maybe I am sensitive. But then early 20s is not far from early 30s is not far from ...
Time is not linear, what and how you learn in your life will evolve. I was a lot 'smarter' (read faster/reckless) in my early 20s, I felt creative, in flow. In the late 20s I realized how much I didn't see, history, paradigms, abstractions.
Alan Kay had a quote, something like "perspective matters much than intelligence".
In most fields, raw brainpower isn't what's needed to make an impact. Grit, determination, empathy, creativity, exploration, there are numerous attributes. Hell, even in D&D they separated intelligence and wisdom into two attributes!
My experience was similar to his in one respect, specifically the quote that suggests he first had a fairly normal life when he went to grad school at age 17. I left home for the first time to go to college at age 16, specifically graduate school in mathematics at Harvard. In retrospect, it seems to have been a fairly normal experience in leaving home and going to college.
It helped me that I was friends with Ran Donagi (started in my department at age 17, had the office next to mine, and accepted my invitation to be my roommate), and that Ofer Gabber (started grad school at 16) was around in the next dorm room to the right, being even more awkward than I was. (At least I didn't have a language barrier ...)
It's common in math to get theorems that prove existence without giving any explicit examples (nonconstructive proofs). A big research area is coming up with efficient ways to construct such examples.
Definitely true, but these things are of a different "kind" (in my opinion). Often such a proof implies, for instance, a really inefficient algorithm, such as "try all the random possibilities and with positive probability such an example exists.
They also tend to have a flavor of a somewhat complicated object whose existence you are proving, and they don't tend to be phrased as algorithms (especially not efficient ones).
Here the primality test algorithm is an efficient algorithm, and doesn't seem natural to view as an existence/nonexistence proof of an object, in the same way. Instead it's exploiting the number-theoretic properties of primality in a weird way. Hope that makes sense, but anyway just trying to explain my intuition for why this is very different from the phenomenon you mentioned.
This top-rated story is kinda low even for "Hacker News". Written in the ritual heroic allegory style. A puff-piece with supposed facts about some dude's childhood and what people think about him, patronizing even his own opinions as modesty-only-rivalled-by-his-genius.
I'd at least understand such articles about Sophie Germain or Alexander Grothendieck, who achieved "genius" despite not being coddled quite so much by every superficial social system arbitrarily impressed by early child-development timelines. For example, Germain hacked the system as a child: "When night came, [her parents] would deny her warm clothes and a fire for her bedroom to try to keep her from studying, but after they left she would take out candles, wrap herself in quilts and do mathematics."
Then, "She used the name of a former student Monsieur Antoine-August Le Blanc, 'fearing,' as she later explained to Gauss, 'the ridicule attached to a female scientist.'"
Social engineering? One thing hackers do.
As for Tao as a child wanting something in shape of arabic numerals, or as an adult imagining himself as some mathematical transform? His wife giving up "paid employment to manage the household, shield Tao from mundane matters and buy the polo shirts he wears"? Seems less hackerish. If Tao has done anything beyond what society shaped him to do, straying from the path he was given in an intensely hierarchical society, I don't recall seeing it in this article.
I thought it was an interesting/fine article. If you want to submit articles about other people to HN, go for it (your anecdotes sound interesting). But there's no reason that every interesting person has to fit some "hacker" archetype you seem to have in mind.
I was fine with it until the video of the 'behind-the-scenes' further down in the article that shows the camera crew telling him to write math on a piece of glass for pictures.
I don't expect to read a good article about math from any general-purpose publication, but goddamn am I sick of the glass-with-formulas cliche. It's as tired as the "hooded man in dark room behind a green phosphor terminal" image that is thrown about any time there is a db leak or black hat scandal of some sort.
Poster from Adelaide here. A friend of mine went to university and did some courses with Terry; he commented to me back in those days about 'some smartarse guy who interrupted the lecturer and argued with them'. Little did he know at the time what a bright spark Terry would turn out to be!
This is a bit nitpicky, but I was annoyed with the shifting between using "math" and "maths" in the article - it's annoying enough to see "maths" used, having formerly been in the academic math world, but to waffle between the two is just confusing.
The publication [1] that improved on the MRI results was the famous paper that layed out the foundation of the what is called "Compressed sampling" / "Compressive sensing". This is the main field of research for Signal Processing folks since around 2006. (Before it was mostly "just" Wavelets for a few decades).
In MRI you sample in the frequency domain (FFT) and compressive sensing can be used since MRI signals (which are structured signals) are sparse in that domain.
[1] Stable signal recovery from incomplete and inaccurate measurements