How Much More Knowledge Does a Mathematician Have Than a Math Graduate?

[mathwonk]

Matt, Is Terry Tao the Tao in the reference in this UGA seminar talk today? Sounds interesting.

3:30pm, Room 304
Speaker: Kevin Purbhoo, University of British Columbia
Title of talk: Horn's conjecture
Abstract: I will talk about two problems, which at first glance appear to be unrelated.
The first is a linear algebra problem that dates back to the 19th century, known as the Hermitian sum problem. It asks: If the eigenvalues of two Hermitian matrices are known, what are the possible eigenvalues of their sum? The second is a fundamenatal question concerning the geometry and combinatorics of Grassmannians and related spaces: When does a collection of Schubert varieties (in general position) have a non-empty intersection?

Both these questions are interesting in their own right, and have a long and rich history. However, in the 1990s it was shown that these two problems are connected in deep and remarkable ways. This revelation gave rise the first complete solution to the Hermitian sum problem [Klyachko 1994], and proved that this solution satisfied a mysterious recursion which had been conjectured by Horn in the 1960s [Knutson-Tao 1999]. To the uninitiated, Horn's conjecture may seem a little strange. However, I will explain why it is at the heart of this story, and how our understanding of it sheds light on the whole picture. Finally I will discuss a few of the directions in which these results have been refined and generalized.

 

[matt grime]

Yep, that will be the Tao I mentioned. Actually, by pure coincidence, I happened to be in a seminar today that referenced the Knutson-Tao result.

http://www.ams.org/notices/200102/fea-knutson.pdf

is the paper, and my 'interest' in it is in the Littlewood-Richardson rules.

 

[Chris Hillman]

Baez can explain anything to anyone

so chris, you understood EVERYTHING that mathwonk was talking about throughout all this posts?

No, not everything, in fact I started a new thread to pick his brains. I know enough to be confident I can understand it with some help from mathwonk, though. See this:

And here is Part IV of mathwonk's minicourse, followed by some suggested (broadly relevant) background reading: ...

mathwonk mentioned a talk whose abstract included the sentence:

The second is a fundamental question concerning the geometry and combinatorics of Grassmannians and related spaces: When does a collection of Schubert varieties (in general position) have a non-empty intersection?

Note that one series of posts by Baez cited in the background reading I threw together explains what Schubert cells and Grassmannians are. (I could give more formal citations, but I think this TWF is better!) BTW, I've been trying to find out for years if anyone knows whether Hermann Schubert, the mathematician, was related to Franz Schubert, the composer. It is also worth noting that Baez sometimes hangs out with Terrence Tao, and some of TWF concerns Tao's work.

having only 3rd year math knowledge at age 14, most of it just flew right by me.

3rd year undergraduate at age 14? Don't worry, no one will dismayed that you can't follow much right now; fear not, in a year or two (assuming you plan to enter graduate school at 16) you will find it much easier to begin to follow stuff like this. If you can't wait (heh! --- hooray for impatience!), try the postings by John Baez which I cited in the other thread.

 

[Chris Hillman]

More shameless name dropping

Yep, that will be the Tao I mentioned. Actually, by pure coincidence, I happened to be in a seminar today that referenced the Knutson-Tao result.

Purely by coincidence, someone mentioned the axiom of choice. Allen Knutson is a graduate of the same high school as Paul Cohen. Also Peter Lax, Bertram Kostant, Elias Stein, Melvin Hochster, Robert Zimmer, David Harbater, Eric Lander, and Noam Elkies. Also (to add a few physicists to this list), Rolf Landauer, Richard H. Price, Brian Greene, and Lisa Randall.

 

[matt grime]

One heck of a high-school. If we're going for trivia, then if

1) I have kids
2) I don't move

then there is zero chance of said kids being the best mathematicians to attend the local primary school (elementary for the Americans), no matter what I do, since they have to cope with Ben Green and Paul Dirac as alumni.

 

[andytoh]

So Chris or Matt, why don't you also regale us with an anecdote from your research past? Perhaps this long thread can go down in PF history as a sort of time capsule that contains a pedagogical anecdote from every professor that stepped foot in PF.

 

[Chris Hillman]

One heck of a high-school. If we're going for trivia, then if

1) I have kids
2) I don't move

then there is zero chance of said kids being the best mathematicians to attend the local primary school (elementary for the Americans), no matter what I do, since they have to cope with Ben Green and Paul Dirac as alumni.

LOL, only a mathematician could love a well-ordering which doesn't put their own children first! (Even hypothetical children.)

 

[Chris Hillman]

Professor?

So Chris or Matt, why don't you also regale us with an anecdote from your research past? Perhaps this long thread can go down in PF history as a sort of time capsule that contains a pedagogical anecdote from every professor that stepped foot in PF.

I'm not a professor. As for anecdotes from my past, I already described my memories of John Hubbard at the birth (or rather, the modern rebirth) of complex analytic dynamics.

This might be a place to mention an anecdote from my present: see Eq. (1) of http://www.arxiv.org/abs/gr-qc/0701081 (version of 15 Jan 2007). I went back and forth for several minutes, trying to find the definition of \lambda_{rm S}. OK, r for some kinda radius, m for mass, S for Schzarzschild or maybe surface, somehow analogous to \lambda_{\rm S} = -\log(1-2 M_{\rm S}/r ), but what could it be? I was considering the possibility that this might somehow refer to the initials of Reinhard Meinel (see the acknowledgments), when I finally remembered that I know latex, whereupon I immediately deduced the nonpresence of a missing hidden backslash! The moral is: just another reason why andytoh's list should have included latex as necessary background for all serious math students! You don't just need to know latex to write papers, you bloodly well need to know latex to read them!

Chris May have understood it quite well because his field of expertise, which I believe is relativity, requires knowledge in the particular field mathwonk was referring to. I am not sure if you know much physics tom, but Chris is really good at what he does :p

Thanks! But I am a gtr amateur, in fact I have no formal coursework in physics. Come to that, I really have no formal coursework in dynamical systems either, in fact just about everything I mention in the other thread I learned from books, not from classwork. However, I had the benefit of a fine undergraduate education, and then a fairly standard first year of graduate school, which gave me the foundation needed to learn other all this other stuff. My primary reason for growing to graduate school was to learn enough to learn, in this sense. (Although I confess I also had the ambition of writing as often and as well as Baez... but he's a very tough act to follow in this regard.)

In another PF post I described in some detail how I happened to pick up gtr without really trying. But a more important formative influence for me was probably encountering (via the astronomer Martin Harwit) the work of Shannon on "communication theory". The only reason I yak endlessly about gtr is that so many people seem inordinately interested in this (admittedly interesting) topic, and I tend to take pity on those I notice are confused about something I happen to understand.

Something I thought about mentioning to andytoh earlier in the thread, which I just alluded to above: I think that most academics would probably agree that the most influential moments in their classroom experience (as students) tends to be random remarks or "sidelines". For example, in my first year complex analysis course, Scott Osborne happened to mention non-Hausdorff sheaves, which then led me, via a book I stumbled upon by accident, to my fascination with category theory!

Speaking of library anecdotes: I was once describing how I rediscovered E^{2,2} in the context of two by two real matrices to Noel Brady (Mathematics, Univ. of Oklahoma) in the UC Berkeley math library (which is huge). I was saying that I had been unable to find this amusing observation anywhere when he reached up and pulled down a book which contained exactly the construction I had in mind! This led to a letter to the author of the book in question, which I mentioned in another ancedote in an earlier PF thread, because a polite reply eventually arrived from a surprising and currently notorious location.

Trivia item for those who know Martin Harwit as author of Astrophysical concepts: which U.S. Senator denounced him (as a Scotsman!) on the floor of the U.S. Senate and why?

Hint: this is actually connected to the previous anecdote, via the general topic discussed in an apparently little noticed news story, which IMO should be of grave concern to everyone everywhere (not that there is really very much we can do to prevent the inevitable third use of nuclear weapons against a civilian population): http://news.bbc.co.uk/2/hi/south_asia/6264173.stm

 

[andytoh]

Chris, you are the epitome of every interested learner! You have proven to everyone else who thought you were a scientist that if one has the desire to learn, then nothing should stand in his/her way. You have just given me more incentive to pick up those books and I will follow my desire to learn whether or not I make it as a mathematician. It is the love of learning, not just the prospect of the profession, that will henceforth give me inspiration everyday.

 

[Chris Hillman]

Wow, andy, you made my day! Thanks!

 

[J77]

Then there is zero chance of said kids being the best mathematicians to attend the local primary school (elementary for the Americans), no matter what I do, since they have to cope with Ben Green and Paul Dirac as alumni.

Nah - with the new government places-from-lottery incentives they've an equal chance to end up in Hartcliffe, then they're likely to be the best if they get a GCSE

 

[Gib Z]

Tell me what e^{i\pi +2ki\pi} is equal to then :)

Edit: Holy whack there's been a lot of posts today, I was replying to people the page before, about ln -1...

 

[nightdove]

Ok, this may sound like a stupid question, but I really, really need to know.

How much more mathematical knowledge does a typical mathematician have than a math specialist student who has just finished fourth year university math courses?

Before you criticize my question, let me explain why I ask. I want to become a mathematician, so I need to get a feel of how much knowledge I need to acquire before I can become one. Now, I know that knowledge is not everything. Indeed, it is problem solving skills and generation of ideas that makes a true mathematician. I agree! Nevertheless, one must have immense prerequisite knowledge before they can come up with original ideas and solve open problems.

If the answer is, say, 3 times as much. Then I can focus on my problem solving skills, read thoroughly the proofs of theorems, etc..., and build my knowledge at the pace of a regular student. If, however, the answer is, say, 100 times as much, then I will know that I have to step up on my reading. So this question, I think, is important in order for me to get a sense of how much and in what manner I should self-study.

My guess is that a typical mathematician has 50 times as much knowledge as a math student who has just graduated from university. Any other ideas? A mathematician's honest answer would be greatly appreciated (and I won't think you are being arrogant).

Psychologists estimate 10,000 hrs before you can match the "masters" in a field and begin producing novel work. I think university math needs to be distinguished from pre-university math as the two are entirely different animals.

How many hrs did we spend in university studying Math independently?

3 years (UK system) x 30 weeks x 6 days x 4 hrs/day = 2160 hrs.

That's 21.6% of the way to becoming a first class professional mathematician.

Of course, if you're one of those hardworking kids who did 8 hrs a day, day in, day out, then it'll have been 4320 hrs, and you'll be considerably closer to your goal. (43.2%).

Add in a masters and postgraduate degree and you'll probably have clocked in 10,000 hrs, which is what you need to do to become a first-rate mathematician.

 

[nesp]

<My guess is that a typical mathematician has 50 times as much knowledge as a math student who has just graduated from university. Any other ideas?>

It's impossible to quantify but if you want numbers, my guess is that a working mathematician has many times more knowledge in their specialty than a new graduate, but probably less in adjacent areas due to forgetfulness. It's like that old saying, an expert has forgotten more than a non-expert ever knew.

But I don't think that's the right measure. A better measure is the hard to define quality of mathematical maturity. A working mathematician typically has much higher level of maturity than a new student, allowing him or her to gain insight and assimilate new material more rapdidly, and to quickly extract the essence of a theorem or proof. This is a mental activity that only comes with experience.