A Mathematician's Knowledge

[mathwonk]

If this appeals to anyone, maybe we could induce some of the other mathematicians to post summaries of their expertise here as well. and physicists too. even if we have to change the title.

I have given you a survey of roughly 100 years of work on singularities of theta divisors on jacobians, and comparison with moduli of other abelian varieties with singularities on theta, from Bernhard Riemann to Aldo Andreotti and Alan Mayer, and Arnaud Beauville.

Thanks for your patience in letting me indulge myself in what I find interesting. Actually I sense from some of you it is not a crazy exercise, as you seem to sense the excitement of really blowing off the top and going for what researchers actually do.

This amy also help answer soem questions as to how to chose research projects, posed elsewhere. I have myself tried to egneralize the work of riemann, Andreotti mayer, Beauville, and others who have studied abelian varieties, jacobians, and their moduli.

 

[mathwonk]

And in reference to the original question, if you are a senior who has read the last few posts you may be better able to estimate the difference between the knowledge of an average mathematician and that of a senior in college. These posts concern topics of interest to me about 20 years ago. This is the first time in the three years or so i have posted here that I have discussed anything of actual concern to my research. Thank you for the opportunity to do so.

 

[andytoh]

This is the first time in the three years or so i have posted here that I have discussed anything of actual concern to my research. Thank you for the opportunity to do so.

You don't need an excuse to share with us your passion. For you, we are all ears, in any thread.

 

[mathwonk]

you are very kind. thank you.

 

[mathwonk]

andytoh, are you a college student? if you are fairly advanced and want to become a mathematician, you might consider applying for support to attend my birthday party. there you will see and hear speakers who are as far beyond me in knowledge as you may think i am beyond the typical senior. it should be inspiring. this is a great chance to get a view of what is out there. I hope matt grime may be there too.

 

[Gib Z]

I second what andytoh said. As to other mathematicians/physicists posting their expertise here, welcome, but i won't get any of it lol.

Edit: Hopefully you'll still be having these parties when I turn however old i need to be to get support. How old is that?

 

[Chris Hillman]

Stop fretting and just start exploring!

Hi again, Andy,

I'd just like to remind you that the essential point is this:

so just try to understand the math you are studying, and try to extend it a little. eventually you or someone advising you, will suggest a problem you can do that has not been done, anf you will get a thesis and be on your way.

To which I'd add that the best discoveries always begin with something really simple but really new. To mention one I witnessed at close hand: the Mandlebrot set.

 

[mathwonk]

Chris, could you tell us more about the genesis of the mandelbrot set?

 

[Chris Hillman]

Imaginative applications of techniques from algebraic curves?

Hi, mathwonk, and happy bday!

i feel however a certain insecurity at changing specialities, since i am a recognized specialist in my area, and if i change, i start over as a newbie.

The other coin finally dropped, so now I know we share some common interests. I really like that book by Herb Clemens on algebraic curves!

I'd like to see some gurus of algebraic geometry broaden their horizons in a less drastic manner than by actually switching fields. Namely, re-examine some of the foundations of the subject from a slighly different perspective. For example, I have been intrigued for some time by Cramer's paradox, filtered through a background in classical information theory. In another post, I just mentioned an interesting connection between Einstein's idea of the "wealth" of solutions in a physical theory (which he apparently came up while critiquing Nordstrom's theory of gravitation on the grounds that it has a paucity of solutions compared with Maxwell's theory of EM--- this was before he invented/discovered gtr, which as he expected has a much richer solution space than either Newton or Nordstrom gravity) and the Hilbert polynomial. In my view, these connections are grossly underdeveloped.

 

[Chris Hillman]

Genesis of the Mandlebrot set

Chris, could you tell us more about the genesis of the mandelbrot set?

I dropped out of college when biology proved too hard, but later decided to return to school. I declared myself a math major and announced that I was going to start right off with the most advanced courses on offer. I don't think anyone gave two cents for my chances, but to everyone's surprise I did very well. Probably because (with a few exceptions, as you would expect!) students with even modest mathematical talent were not thick upon the ground, so it was probably fairly easy to succeeed.

But returning to the point, John Hubbard (Math, Cornell) was my first undergraduate advisor. This was at the very time when Douady was visiting and they had just proven their famous theorem (some background can be found at http://mathworld.wolfram.com/MandelbrotSet.html, and Hubbard was pretty much the first person to make those fabulous color computer pictures of Julia sets, the Mandlebrot set, etc., which everyone now takes for granted. (For example, http://linas.org/art-gallery/escape/ray.html and http://aleph0.clarku.edu/~djoyce/julia/julia.html) But in those days, no-one had ever seen anything like this, and Hubbard would show up with fabulous new computer pictures and wander the halls, eager to show them off to anyone who was interested, and many of us were very interested! I have a vivid memory of his explaining to me what we now know as the symbolic dynamics associated with Julia sets. (See the archived post quoted in full below for a bit about symbolic dynamics in the Mandlebrot set itself.) I could not have guessed then that I would wind up writing a dissertation on generalized Penrose tilings, a kind of symbolic dynamics (but, in some sense, at the "least chaotic" end of the spectrum, no pun intended).

An interesting fact about those early pictures: Cornell had a supercomputer, but high quality color printers did not yet exist. Hubbard would set up his camera on a tripod, aim at the monitor, take a photograph, and later make a slide in his home darkroom! Bye and bye he formed a small company which made postcards which were sold in the campus bookstore.

So, I knew Hubbard at a pretty happy time for both of us (I was succeeding as a math major, and he was discovering some really beautiful mathematics).

I remember another incident: Hubbard gave a talk which was in part a slide show. At one point he displayed slides showing the "explosions" which had just been discovered by Devaney (with assistance from undergraduates at Boston University!). There was a long silence, then someone said in an awed voice "there is a God!".

Here is the archived post by myself on the symbolic dynamics of the Mandlebrot set:

Newsgroups: sci.fractals
Subject: Re: Mandelbrot interior
Date: 16 Dec 1996 06:14:17 GMT
Organization: "University of Washington, Mathematics, Seattle"
Message-ID: <592pbp$dff@nntp1.u.washington.edu>
References: <32AD6BD1.2781E494@ic.ac.uk>
            <19961214003100.TAA00386@ladder01.news.aol.com>
Keywords: symbolic dynamics, shift map, zeta function, entropy

In article <19961214003100.TAA00386@ladder01.news.aol.com>,
swarsmatt@***.*** (SWars Matt) writes:

|> The dynamics of the set along the real axis are conjugate to
|> those of the logistic map studied by Feigenbaum, so by Sarkovskii's
|> theorem there are periodic points of all periods, and at the Feigenbaum
|> point chaotic dynamics begin.  In these points the orbit of the critical
|> point (zero of course) never is attracted to a periodic point.

For c in the interior of a lobe of the Mandelbrot set, x -> x^2 + c posseses
a superattracting cycle of a period which depends on the lobe.  The
open intervals between the points of the cycle 0, f(0), ... f^p(0) = 0 
can be taken as an "alphabet" and the itinerary of a "typical" x
is then associated with an infinite sequence according to which 
intervals it passed through.  The dynamical system defined by iterating
x -> x^2 + c on its Julia set then turns out to be topologically conjugate
to a "shift of finite type" (sft) which is defined by studying how these
intervals are mapped to one another by f_c(x) = x^2 + c.  This allows one
to compute the entropy, the number of points of each period, etc.,
by standard methods of symbolic dynamics.  Because the cycles are
attracting, and because the method employs only topological features
of how the intervals are mapped into one another, numerical inaccuracies
in determining the precise values of the roots do no harm--- only
the order in which x_1, x_2, ... x_p occur on the real line matters.

An example should suffice to indicate how these methods work.
First take f(x) = x^2 - 1.62541.  (This value of c belongs to
a "period five" lobe of the Mandlebrot set which sits over the
real axis, slightly behind the Feigenbaum point).  This has a
superattracting five cycle which is approximately

  (0, -1.62541, 1.01655, -0.592041, -1.2749)

Notice that the order in the cycle does not correspond to the order
along the real line.  Diagramatically, we have

           A       B        C         D
  -----|-------|-------|---------|--------|-----------
     f(0)   f^4(0)   f^3(0)      0      f^2(0)

This gives a four letter alphabet.  Since f is continuous, we can
see that f maps A onto D, B onto B union C, C onto A, D onto A union B.
This gives a graph

      
        A <-----  C
       ^ |        ^
       | |        |
       | V        |
        D ------> B 
                 / ^
                 |_|

which defines a shift of finite type which has NO three cycles
at all, but does have a five cycle ... CADBBC... and thus (by
Sarkovski's ordering) a point of every period other than three.
Here, the SFT is the space whose points are infinite sequences
of letters A, B, C, D, where each sequence is required to satisfy
the "transition constraints" given by the graph.  Thus, part of
one sequence might look like ... ADADBBCA ...  This space
can be given a metric in a standard way which makes the "shift map"
(which simply shifts each sequence one space to the right) into
a continuous map.  Moreover, the shift map acting on the space of
sequences defined by the graph (aka "the sft") is topologically
conjugate to f acting on its Julia set (here a Cantor set which
is a subset of the real line).  This means that the "interesting"
dynamical behavior of f is the same as that of the sft.  (The
"uninteresting" behavior of f involves the behavior of points
not in the Julia set, which is qualitatively easy to describe,
more or less by definition.)

In particular, the topological entropy (a measure of the
"unpredictability") and the number of points of each period
(represented by a power series called the zeta function, which
is in fact related to the famous Riemann zeta hypothesis) are the
same for f acting on its Julia set and for the sft. In turns out
that both the entropy and zeta function of the sft are easily
computed from the adjacency matrix of the graph:

           A  B  C  D
        A  0  0  0  1
   M =  B  0  1  1  1
        C  1  0  0  0
        D  1  1  0  0

The characteristic polynomial is det(t I - M) = t^4 - t^3 - t^2 + t - 1.
On the other hand, the zeta function is

                   1                  1
  zeta(t) = -------------- = ------------------------
             det(I - t M)    1 - t - t^2 + t^3 - t^4

Expanding log zeta(t) in a McLaurin series gives the number of points
of each period

  log zeta(t) = log det(I - t M)
              = t + 3 (t^2/2) + 1 (t^3/2) + 7 (t^4/4) + 6 (t^5/5) +
                    15 (t^6/6) + 15 (t^7/7) + 31 (t^8/8) + ...

That is, there is one fixed point, the sequence ...BBBBB...,
two points of period two, ..ADAD.. and ..DADA.. (plus the fixed point,
which also has period two), no point of period three (except the fixed point),
four points of period four (plus the two points of the two-cycle and
the fixed point), and so forth.

In other words, in addition to the superattracting five cycle we
started with, f(x) = x^2 - 1.62541 also has an infinite number of
repelling cycles, specifically one fixed point, one two cycle,
no three cycles, one four cycle, two six cycles, two seven cycles, etc.
You can verify that this is correct by examining the graph and
looking for cycles.  Converting between the number of p-cycles
and the number of points of period p can be systematized using
the Moebius inversion formula.

The topological entropy turns out to be log lambda, where lambda is
the largest eigenvalue of the characteristic equation, in this case
about 1.51288.   Because M^6 has all positive entries, we also know
that this sft is "topologically mixing", a strong property which
implies "topological transitivity".  Since every sft has sensitive
dependence on initial conditions and dense periodic points, a
topologically transitive shift is chaotic according to the definition
proposed by Devaney.  It turns out that the sft defined as above
by a graph is transitive iff the graph is transitive in the sense
that you can get from any vertex to any other vertex.

For a given p, it is easy (in principle) to determine the c values
corresponding to superattracting cycles of period p, one of whose
elements is the origin. For example, for p=3, set

         f_c[f_c[f_c[0]]] = 0

This can be solved numerically to find the real root -1.75488.
This degree of accuracy is enough to determine the three cycle

         (0, -1.755, 1.325)

to sufficient degree of accuracy to obtain the diagram

            A          B
     ---|--------|---------|-----
       f(0)      0       f^2(0)

This leads to the graph

                   <------
                  A -----> B
                 / ^
                 |_|

which defines an sft called "the golden mean shift",
because its entropy is log 1.61803.  The entropy and zeta function
for this sft can be computed from the adjacency matrix as above.
This sft is conjugate to f_c(x) = x^2 + c acting on its Julia set,
for c in the "period three" lobe of the Mandlebrot set.  (This lobe
appears as a "bug" sitting about halfway along the "tail".)

In this way (a program such as Mathematica is helpful here), you
can explore how the sft's change as you move down the real line,
starting from the trivial one point sft defined by the graph

                  A 
                 / ^
                 |_|

obtained from the two cycle (0,-1) associated with x^2 - 1,
which belongs to the "period two" lobe attached to main cardiod.
As you move along you come next to a period four lobe.  We now
have the four cycle

     (0, -1.3107, 0.4072, -1.145)

associated with x^2 - 1.3107.  This gives the diagram            A        B     C
     ---|--------|-----|------|-----
       f(0)    f^3(0)  0     f^2(0)

which gives the graph

                  B ---> A <---- C 
                 / ^       ---->
                 |_|

This defines an intransitive sft with entropy zero.  Continuing
along the period doubling cascade, we find that the graphs become
more complicated but the entropy remains zero until the Feigenbaum
point at about c = -1.46692, where the entropy goes positive with
the onset of chaos.  For instance, x^2 - 1.47601 has a superattracting
six-cycle and is conjugate to an sft with entropy about log 1.27.
Continuing, we find that the entropy increases monotonically,
remaining constant on the "periodic windows".  A typical such
periodic window begins with the golden mean shift at about
x^2 - 1.75488.  This is the first in a cascade of lobes with
associated intransitive sft's whose entropies agree with that
of x^2 - 1.75488, namely log 1.61803.  Eventually, at the
tip of the tail of the Mandlebrot set (c = -2), we obtain
the "full two shift" which has entropy log 2.

To sum up--- life inside the Mandlebrot set is indeed interesting.

Hope this has been intriguing!  If so, here are some references.

For an introduction to sft's defined by graphs, entropy, zeta functions,
full shifts, and more, see the recent undergraduate textbook

Author:       Lind, Douglas A.
Title:        An introduction to symbolic dynamics and coding / Douglas Lind,
              Brian Marcus.
Pub. Info.:   Cambridge ; New York : Cambridge University Press, 1995.
LC Subject:   Differentiable-dynamical-systems.
              Coding-theory.

For more on Julia sets, superattracting and repelling cycles, the Mandelbrot
set, interval maps and graphs, etc., see

Title:        Chaos and fractals : the mathematics behind the computer
              graphics / Robert L. Devaney and Linda Keen, editors ; [authors]
              Kathleen T. Alligood ... [et al.].
Pub. Info.:   Providence, RI : American Mathematical Society, 1989, c1987.
LC Subject:   Computer-graphics -- Mathematics.
              Fractals.

For a more sophisticated analysis of interval maps, the basic
reference is a famous paper by Milnor and Thurston,
"On Iterated Maps of the Interval I", which was written in 1977 but
unpublished for many years until it finally appeared in the book

Title:        Dynamical systems : proceedings of the special year held at the
              University of Maryland, College Park, 1986-87 / J.C Alexander,
              ed.
Pub. Info.:   Berlin ; New York : Springer-Verlag, 1988.
LC Subject:   Topological-dynamics -- Congresses.
              Ergodic-theory -- Congresses.

For Moebius inversion see almost any combinatorics text, for instance

Author:       Cameron, Peter J. (Peter Jephson), 1947-.
Title:        Combinatorics : topics, techniques, algorithms / Peter J
              Cameron.
Pub. Info.:   Cambridge ; New York : Cambridge University Press, 1994.
LC Subject:   Combinatorial-analysis.

The graphical method employed here was introduced in the very readable
article

P. D. Straffin, Jr., "Periodic points of continuous functions",
Mathematics Magazine 51 (1978), 99-105.

More recent developments may be found in the book

Author:       Alseda, L.  Libre, M.  Misiureqicz, M.
Title:        Combinatorial Dynamics and Entrophy in Dimensions One 
Publisher:    World Scientific Publishing Company, Incorporated
Year:         1993
Series:       Advanced Series in Nonlinear Dynamics
Pages:        344p.
ISBN/Price:   981-02-1344-1 Cloth Text $74.00
Subj (BIP):   ENTROPY.  COMBINATORIAL-ANALYSIS

Chris Hillman

This illustrates some stuff I said in another post recently about the relationship between topological entropy and measure-theoretic entropy. See "All Entropies Agree for an SFT" at http://www.math.uni-hamburg.de/home/gunesch/Entropy/dynsys.html
(Unfortunately, due to harrassment from cranks I took down most of the expository papers I wrote when I was a graduate student, but this website still exists courtesy of Roland Gunesch.)

 

[Chris Hillman]

Superadditivity of knowledge

Ok, here's my definition of a senior unit of knowledge. The mathematical knowledge consisting of:

[snip UG curriculum (from MIT?)]

Have I missed anything?

Yes! Picking up on mathwonk's remark about Mumford, you didn't mention computers. Familiarity with latex has been essenital for a long time, and with standard CAS like Mathematica, Maple, and some speciality systems like GAP or macaulay2 is increasingly important, in fact I would say, "essentially, essential".

OTH, you should regard the curriculum you quoted as a list of what is on offer to diligent undergraduates at your school. If you add up the credits you'll probably soon see that comparatively few math majors actually take all those courses. In another thread, I argued that because mathematical knowledge (in particular) builds upon itself, social engineers should recognize that, if we accept the premise that what the world needs most is more math (self-evident, no doubt, to everyone here!), then the standard time alloted for UG education should be increased to six years, to give hard working and well prepared students time to actually take all those courses. I do feel that it is best if you learn as much as you possibly can as early as you possibly can, but you will many contary opinions, and I agree there is a trade-off involved.

(If I appear to be somewhat contradicting my advice to stop fretting and just plunge right into thinking about problems, well, whenever you take any math course you should probably just plunge right in. Good things come to those who are reckless. So do horrific wrecks. There's no predicting. I think there's no "safe" way to pursue a career in such a challenging discipline as mathematics, so if you accept this at the outset, you should just forge ahead and hope for the best.)

Multiply this list by 100, and you have the knowledge of a 50 year-old mathematician. Right?

No, if you accept that whatever knowledge is, by axiom it should be superadditive. I tried to explain earlier my view that linear effort invested leads to exponential growth in ability to learn/use increasingly abstract or novel mathematics. If you ask around, I think you'll find that this expectation is generally supported by experience.

 

[Chris Hillman]

Good story...

Andy, hope you noticed the following:

In my thesis i was studying the degree of a mapping of moduli spaces, and the usual technique for that is to use "regular values", but i did not have any at my disposal. It turned out the inverse and implicit function theorems could be applied to the normal bundle of a fiber to substitute for them.

This method was very effective, and had not been used before. It only came to mind because years before i had thought long and hard about those theorems from advanced calculus. I was using them in algebraic geometry but the ideas were the same, once understood deeply.

The point is that your struggles to really understand something in a relatively "junior" math class today can pay off years later in utterly unanticipated ways!

the idea behind groups is just that of symmetry, which is why it is useful in many places especially physics.

Hear, hear! Which suggests another bit of good advice for all ambitious and systematic math students: keep a notebook in which you jot down "big ideas" like symmetry, good problems, random thoughts, etc. (don't forget to date your notes-to-self!).

 

[Gib Z]

I have a book where I take down good Theorems, handy integrals and things like that, along with their proofs. However I myself never have these ideas, pity...The only thing that was mine was that I worked out the formula for the sum of a geometric series, at which time I was 10 and didnt know the name, let alone the formula.

 

[J77]

Chris - do you work in dynamical systems? Which field?

mathwonk - these later posts should be merged to your "who wants to..." thread.

 

[Chris Hillman]

But enough about me...

Chris - do you work in dynamical systems? Which field?

I aim to be the "Poor Man's John Baez": I try to think about whatever interests me at the moment. Things which interest me typically involve at least two of entropy, symmetry, invariants, enumerative combinatorics, graph theory, categorification, the interaction of algebra-analysis-geometry-logic-probability, differential equations, computational group-theory/algebraic-geometry, that kind of thing. I like theories which start with a really good definition or two, and build machinery which enable one to compute quantities which manifestly go to the heart of the matter. I like surveys which discuss at least two simple nontrivial examples illustrating the power of the theory. (Shannon 1948 is my shining example of The Perfect Paper.) I like learning about unexpected connections between seemingly very different phenomena. I like things I can compute and interpret, axioms I can mull, diagrams I can draw, pictures I can plot, grand visions I can popularize, good textbooks I can study, papers written to be read, devices that work, youth, intelligence, library research, community, things like that.

mathwonk - these later posts should be merged to your "who wants to..." thread.

I hope everyone noticed that I collected mathwonk's "short course" and added some suggested reading, and plan to follow up with some glosses and questions.

 

[Tom1992]

Hi, mathwonk, and happy bday!



The other coin finally dropped, so now I know we share some common interests. I really like that book by Herb Clemens on algebraic curves!

so chris, you understood EVERYTHING that mathwonk was talking about throughout all this posts? having only 3rd year math knowledge at age 14, most of it just flew right by me.

 

[Gib Z]

I second that as well :p Flew right fast me, though I have no idea what tom means by 3rd year math knowledge...Im assuming this is some american standard?

Edit: O if it meant 3rd year in College or University level, I wouldn't say I am up to that yet :) Not to mention, I seem to enjoy different topics to tom.

 

[Gib Z]

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

Edit: On another topic, please don't mind me putting it here, if anyone wants as a little excercise in some Number THeory/ Analysis, I have been sent a proof of the original Riemann Hypothesis, you could try to disprove it :P No offence to the author, just that the odds are against him, seeing as he thought the natural log of -1 was i*pi...:'(

 

[yenchin]

No offence to the author, just that the odds are against him, seeing as he thought the natural log of -1 was i*pi...:'(

So what should it be?

 

[J77]

So what should it be?

Yeah.

$$e^{i\pi}=-1$$ last time I checked

 

[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 :tongue:

 

[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.