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

[Gib Z]

Anyone can suddenly change their area of expertise, but it will take time for them to work up to a certain point. A mathematician can change their area of focus to physics, but it will take some time before the mathematician is able to know as much as a fully fledged physicist?

As to you thinking you are wasting you time with some of the fields, I could not disagree more. It seems you care more about your PhD than the beauty of the mathematics. It does not matter if you ever need it again. Having the knowledge, and even better, an elegant proof is all i desire.

Study mathematics for the beauty of it. It is an art. It does not matter if it has no practicality, no physical usefulness, no human interpretation. If you find a field of mathematics that may be useful, learn it. But take the time to appreciate it.

 

[matt grime]

so what does a specialist mathematician do when after several years he gets bored in his specialized area? can an abelian varietist suddenly become a number theorist like you suggested?

Mathwonk didn't suggest changing speciality. He suggested learning about it and teaching it. Abelian varieties are algebraic geometry, a large field, which has links to algebraic number theory and arithmetic geometry. Fermat's last theorem is a proof that very much requires knowledge from such apparently unrelated-to-the-outsider areas.

If you want a reason to learn more than just one narrow area, then I suggest you read up on one of the Fields Medal winners, Terry Tao.

 

[Tom1992]

Study mathematics for the beauty of it. ...take the time to appreciate it.

perhaps you are right. i still need time to figure out which branch of math i really like and so should do some exploring with various courses to find out is right for me. and as mathwonk and mattgrime suggested, any math course we take may surprisingly turn out useful in whatever area we specialize in later on.

 

[mathwonk]

tom, i do get bored sometimes with my speciality. it does seem possible however to switch to number theory, from abelian varieties, since those subjects are closely related.

also algebraic geometry is so broad, that moving to many other fields, like diff geom, diff top, several complex variables, commutative algebra, number theory, maybe algebraic topology, or even mathematical physics such as string theory or quantum field theory, is quite feasible.

i have friends who have done such a transition. I myself have been an invited speaker at the institute for theoretical physics in trieste, while still a specialist in abelian varieties.

in fact as an algebraic geometer, i have learned and used almost all pure math fields.

it took me a while to choose a specialty as well, as i started in algebra, then algebraic topology, then several complex variables, then algebraic geometry.

i have also taught measure theory and functional analysis, and number theory, but never numerical analysis, lie groups, representation theory, or pde.

but i have used the heat equation in my research. so compared to some, certainly not all number theorists, my training is pretty broad.

 

[mathwonk]

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.

but hey, you have to go with what interests you before it is too late, right?

 

[mathwonk]

an algebraic function is also analytic, hence differentiable, hence continuous. thus algebraic geometry is a subspecialty of analytic geometry, analysis, differential topology, and topology. thus one can move backwards into any of those other fields, at least in principle.

 

[andytoh]

i certainly don't know all that stuff, especially not the pde...

i have used the heat equation in my research.

How did you incorportate the heat equation in your research when you don't feel up to par in pde's? Did you crash-read a pde's textbook out of interest and then the heat equation sparkled a light in your algebraic geometry lore? And to do research related to the heat equation, even if it isn't the main focus of your article, don't you have to have great in depth knowledge of the techniques of pdes in its solution methods? I don't get it.

 

[mathwonk]

i examined the heat equation and thought about it and its implications. you don't need a general education in an area to use specific topics from that area.

in particular i read carefully the paper of andreotti and mayer, on period relations for algebraic curves.

after many years, i have an ability to use things in my research that i barely know.

 

[mathwonk]

here is a little example: it is difficult to compute the circumference of a circle, but it is easy to prove that the area of a circle is (1/2) the product of the circumference times the radius.

i am like the guy who cannot compute the circumference, but who can prove that the area is the product of the radius by half the circumference.

 

[andytoh]

i do get bored sometimes with my speciality. it does seem possible however to switch to number theory, from abelian varieties, since those subjects are closely related.

also algebraic geometry is so broad, that moving to many other fields, like diff geom, diff top, several complex variables, commutative algebra, number theory, maybe algebraic topology, or even mathematical physics such as string theory or quantum field theory, is quite feasible.

i have friends who have done such a transition.

How much time off does a specialist usually have to take to read up on the prerequisites in another (related) area to start doing research in this new area? (now here is where being a learning machine really helps)

 

[mathwonk]

I am noty sure, as it does seem daunting, but if I don't have the courage to do it now at 64, I may not get topo many more chances. academically, some universities give a year's support for study in a second disciplne.

the best way is to gradually train in the other area and then at some point, make the jump. Thats what David Mumford the famous fields medalist in algebraic geometry did. he became fascinated with comuters and began using them in his research on understanding and classifying the copmplexity of algebraic surfaces, a lifelong interest of his.

Then at some point he proved a beautiful theorem on moduli spaces, wiht joe harris, using comoputers again, then kleft the subject to go full time into artificail intelligence, and pattern recognition.

Mumford's is a hard act to follow, but maybe still a good example to aspire to.

 

[mathwonk]

here is a little crash course in the heat equation and its use in algebraic geometry.

it has long been known that a cubic curve X in the plane has the structure of a group. this is essentially because any two points determine a line, which meets the curve again in a third point, which determines the sum of the first two.

more topologically, the complex points on a smooth plane curve form a torus, or doughnut with one hole, as you can sort of see by looking at the simplest cubic, a triangle.

Now one can see that a torus can be made into a group as follows: take the complex plane and set equal to zero all points which are linear integral combinations of two vectors with different directions, say 1 and i. I.e. C is a group and {n + mi, for all n,m, in Z} is a subgroup and you take the quotient group C/{n+mi}, which as a group is a product of two circles.

topologically it is also the product of two circles, since it formed by gluing the opposite edges of the parallelogram formed by 0, 1, i, and 1+i, hence a torus. using the weierstrass P function and its derivative, one can embed this torus in the complex plane as a cubic. thus any lattice defines a plane cubic.now riemann or abel or someone back there, showed how to go backwards: i.e. given a complex plane cubic X, it inherits a complex and topological structure from the complex plane C^2, in which it lies, at least once it is compactifed at infinity, and hence it has two independent loops on it, one for each circle, i.e. a "homology basis" in fancy language, called say u and v.

There is also a single holomorphic differential dz, which is well defined on the torus, even though the coordinate z is not, because z is well defined up to translation by an element of the lattice {n + mi} and d of a constanT TRANSLATE IS ZERO.

so we get two complex numbers A and B by path integrating dz around u and around v, and Riemann showed these are independent complex numbers hence give a lattice {nA+mB} in C, which then determines a torus group C/{nA+mB}, which in fact is both analytically and group theoretically isomorphic to the original plane cubic X.

Now where does the heat equation come in? well first riemann showed one could normalize the complex generators A,B of the plane lattice so that one of them is always A = 1, and the other B = t, has positive imaginary part.

then one can write down a Fourier series using t which defines a "theta function". f(z,t). i.e. one first gives a quadratic non homogeneous polynomial with linear coefficient z and quadratic coefficient t, and then exponentiates it, and sums over all integer arguments. (see mumford's tata lectures on theta, where he credits me for this description, but i originally learned it from c.l.siegel.)

this gives one a function of the two variables z and t. we think of t as determining the complex structure of the curve (since from t, one can reconstruct the curve as C/{n+mt}), and z as a coordinate on the curve it self.

for fixed t, i.e. fixing the curve, the theta function is a function of z, hence on the curve, which is not well defined, since it is not doubly periodic, but its zero set is doubly periodic so it defines a well determiend zero locus on the curve which is only one point.

so we have a theta function f(t,z), a function of two complex variables (t,z, where t is thught of as determining a complex torus, and z as a point on the torus.

If in the product CxC with coiordinates (t,z) we mod out by the family of alttices {n+mt}, we get a family of tori, one over each point t, and aglobal theta function whose zeroes determine one point no each torus.

the t line is a moduli space for 1 dimensional tori, and over each number t, we have a copy if the corresponding torus and a distinguished point.

as you may know, this theta function is a characteristic solution of the heat equation, so that pde must contain some useful informaton about curves.

 

[mathwonk]

theta functions and the heat equation, part 2

This really comes into its own in higher dimensions and genera. I.e. Riemann generalized this construction to assign a group to each curve of any genus > 0, as follows: he proved a curve of genus g, i.e. a doughnut with g holes, has g independent holomorphic differentials w1,...,wg, and a homology basis of 2g loops u1,...ug, v1,...vg, and thus determines a g by 2g matrix of path integrals [A, B]. he showed one can again normalize the bases wi and ui,vj, so that the matirx contains a gbyg identity matrix, and another gbyg complex matrix t, with pos. def. imaginary part, i.e. [I, t].

then he wrote down "riemanns theta function" f(t,z) of g complex variables z, and apparently g^2 complex variables t, and if one mods out C^g by the lattice of semi periods i.e. by n + mt, where now n,m are integer g-vectors, one gets a complex g dimensional torus C^g/{nI+mt].

HE ALSO SHOWED THAT THE period matrix t is symmetric so there are really only (g)(g+1)/2 variables t. thus the riemann theta function is a holomorphic function on the product space of points (z,t) in C^g x C^(gxg). Again we can mod out this product to form a family of complex tori, and the theta function determines a family of hyperurfaces, one in each torus. these hypersurfaces are called theta divisors.

Now the inverse problem above is of interest. I.e. given a g diemnsional complex torus, when does it arise as above from a genus g complex curve? This is called the Schottky problem. presumably if so, it should be visible from looking at the theta divisor of the corresponding torus.

Now curves depend on 3g-3 parameters, so In genera 1,2, and 3, essentially all "indecomposable" tori do arise from curves, but in genus 4, curves only have 9 parameters and 4 dimensional complex tori have (4)(5)/2 = 10.

so there is one condition that should specify whether or not a complex 4- torus comes from a genus 4 curve. Riemann shoiwed that tori coming from curves in fact have "singular" theta divisors, i.e. if the torus comes from a curve, there is a kink or node on the theta divisor. This raises the opposite question, do all 4-tori with singular theta divisors come form curves? (those which do are called jacobians, so we are trying to recognize jacobians among all complex tori.)

In his thesis at Columbia, Allan Mayer showed about 1960 that at least locally near a 4 diml jacobian, there is a nbhd where this is true. he did it by observing that jacobians J form a hypersurface of codimension one in the space of all 4 dimensional complex tori, and J is contained in the set N of tori with singular theta divisors, so all he had to do was show that N is also a hyperurface of copdimension one.

But the cauchy data for the heat equation implies that if all theta functions satisfying the ehat equation had singular zero loci, then the theta function would be the identically zero solution of the ebnat equation, and it isnt.

this story goes on. Mayer and Andreotti showed in 1967 that in all genera, jacobians are acomponent of N. then in 1977, Beauville showed that in genus 4, N has exactly one other component, thus completely describing 4 dimensional jacobians geometrically.

More recently Robert Varley and I gave a shorter proof of this corollary of Beauville's more extensive work.

Varley and then i used the ehat equation to show that also in genus 5, N has exactly 2 components, and computed the multiplicity of jacobians J on the correspoing component of N, but did not uniquely specify J there.

if you look at the heat equation you see it equates a second derivative of theta wrt z to a first derivative wrt t. As Andreoti and mayer showed, this gives a geometric relation between tangent directions in the moduli space of tori, with quadratic tangent cones to th theta divisor in the torus itself.

Later Welters gave a completely algebraic proof of this version of the heat equation, so that it makes sense in characteristic p geometry, and Varley and I used that version to generalize a famous result of Mark Green on theta divisors of complex Jacobians, to characteristic p > 2.

thus the heat equation has a completely geometric interpretation that can be used to reason about it, independently of knowing analysis or pde.

 

[andytoh]

Totally amazing! A person who did not read earlier in this thread would think that you are a worldwide guru in both algebraic geometry and pde's.

 

[mathwonk]

a mathematician gets very familiar with his own specialty and the tools that are used in it. Algebraic geometry is unusual in that it concerns the study of a specific class of objects, algebraic varieties, rather than the use of a specific tool.

So algebraic geometers use synthetic gometry, commutative algebra, complex and real analysis, algebraic topology, group theory, differential geometry, and differential equations, to study algebraic varieties. Hence they tend to have a broader acquaintance with other fields than some specialists.

What I outlined above is pretty well known stuff, and many people have a far wider knowledge of these areas than I. But you learn one topic at a time. After a long while it adds up. And the better are the people you read and listen and talk to, the more you benefit.

Many of these famous people are very generous with their time. When I took leave in the early 1980's to go to Harvard to study, David Mumford kindly gave me his prepublicatioon notes for part of his three volume book on theta functions and I lectured on them for an audience including him. That was very enlightening.

He later shared preprints by other mathematicians specializing in theta functions, including Igusa, which contained ideas that came in handy later in some of the work mentioned above.

Over the years my colleague Robert Varley and others have patiently shared their knowledge. When you are in a math department, you have the luxury of learning by asking questions, which is faster than reading, but there is no substitute for lifelong consistent reading of work by experts.

 

[mathwonk]

theta functions and the differential equations, part 3

The Schottky problem of characterizing Jacobians among all complex algebraic tori, also called abelian varieties, was originally an analytic or algebraic question, that of giving actual equations in some appropriate coordinates, such as the matrices t, for the moduli space of abelian varieties that vanish exactly on jacobians.

The problem was given its impetus 100 years ago by Schottky who wrote down some relations which he proved were indeed satisfied by jacobians, but it was hard to show even that these relation were not identically zero, much less that they vanished only on jacobians.

In the 1970's Igusa annunced he could prove in genus 4, that (the closure of) jacobians was the only component of the zero locus of the one genus 4 Schottky relation, and in about 1981 he wrote down the proof. He used a differential equation satisfied by hyperellipic jacobians, to deduce that every possible component of the Schottky locus must pass through the "boundary" locus of degenerate 4 dimensional abelian varieties, i.e. products of 4 elliptic curves (genus one curves).

Then he only had to count the number of components through that locus, which he did by explicitly computing the tangent cone at that locus and showing it was defined by an irreducible polynomial. Since every component of the Schottky locus must contribute at least one component to the normal cone, the irreducibility of the normal cone implied irreducibility of the Schottky locus.

In dimension 5 Varley and I were trying to show the Andreoti Mayer hypersurface N parametrizing 5 dimensional abelian varieties with singular theta divisor, had just 2 components, as Beauville had shown in dimension 4. So we used a modification of Igusa's idea, namely we showed all possible components of N had to pass through the locus of Jacobians having an "even vanishing theta null", and then we were reduced to finding the number of components of N that did pass through that locus.

Unlike Igusa's case we knew there were at least 2 components so we needed a way to count them. Unlike his case also, the normal tangent cone to this locus had an "multiple" component, i.e. one whose algebraic equation had multiplicity greater than one, which we needed to understand, since that can increase the number of normal cone components over the number of actual discriminant locus components.

The classical study by Lefschetz of moduli of singular hypersurfaces with only isolated singular points had been completed by Teissier and Le. Their theory showed that one could compute the multiplicity of the tangent cone at a point of the moduli variety of singular hypersurfaces, i.e. of the "dscriminant locus", using "Milnor numbers", which are a count of the homology cycles in the hyperurface that vanish into the singularity as the hypersurface acquires a pinch or singularity.

The multiplicity of the discriminant locus at a point corresponding to a hypersurface with finitely many singularities equaled the sum of the Milnor numbers at all singularities. We had to generalize this to the case of infinitely many singularities, i.e. a positive dimensional famnily of singular points.

We showed that in the isolated case the sum of the milnor numbers equaled the change in the global euler characteristic of the hypersurface as it acquired a singularity. This version made sense in the infinite singularities case. I.e. we defined the global milnor number to be the change in the euler characteristic, and then showed that we could meaure the multiplicity of the components of the normal cone by this new global milnor number.

Strangely we got multiplicity 3, at the point where we expected only two components to pass. But an interesting phenomenon for theta divisors that is not true for general hypersurfaces, is that on the component containing jacobians, there are in general two ordinary double pointsof the theta divisors. We could show this even by looking at a Jacobian, where there are infinitely many, because we could look in a normal direction and see that only two singularities persisted in a given normal direction under deformation.

To carry out this calculation, we used the geometric interpretation of the heat equation, to study the geometry of the family formed by the union of all the singular loci of all theta divisors, the so called "critical locus".

Still this only handled components that met the one we knew to contain jacobians, so we had to show in fact all divisors on the moduli space of abelian varities must meet. For this we worked out statement by mumford that the Picard group, was isomorphic to Z, and this could be comoputed from the second cohomology group, which in turn was linked to a group cohomology calculation for the "symplectic group" Sp(2g), one of the famous classical matrix groups defined by the standard symplectic form. It also required some homotopy calculations using postnikov towers that one learns about in algebraic topology.

Finally it followed that in fact there were only two global components to the discriminant locus of singular theta divisors in dimension 5, but one of them had "Milnor multiplicity" 2 and the other had multiplicity one. The latter result answered a question attributed to Igusa, by proving that a general abelian variety (of dimension 5) having a vanishing even theta null, only has one of them.

This theory of positive dimensional Milnor numbers was later generalized by Parusinski. you can learn the classical theory, isolated singularity case, from milnor's book on singularities of complex hypersurfaces. Using a different but related technique, involving degeneration to lower dimensions, a sort of geometric induction method, DeBarre later proved the discriminant locus of abelian varieties with singualr theta diviusors has 2 components in all dimensions. I believe he used a beautiful computation of the monodromy group of the Gauss map of a smooth theta divisor.

 

[mathwonk]

geometric heat equation part 4

here is the geometry of the heat equation following andreotti and mayer.


recall the product space C^g x C^(g^2) can be viewed, by projecting on the second coordinate, as a family of complex g spaces, one over each gbyg matrix t.

If we mod out the space C^g over t, by the lattice {nI+mt} where n,m, run over all integer vectors, we get a family of complex tori, one over each matrix point t.

If we restrict this family over only the set H in C^(g^2), of matrices which are symmetric and have positive definite imaginary part, we can write down a convergent Fourier series defining a theta function f(z,t), hence defining a hypersurface f(z,t) = 0, in the 2 vbls (z,t),which can be viewed as a family of hypersurfaces in z, one over each t in H.

now we have a family of g-1 diml hypersurfaces, one over each point t in H. Most of these hypersurfaces are non singular, i.e. smooth, but a codimension one family of t's have a singular hypersurface over them. this codimension one set of t's is called D, the discriminant hypersurface in H, for the family.

the set C of singular points on all theta hypersurfaces, is the common zero locus of the functions f =0, and of the partials of f wrt z. since there are g+1 such functions, the common zero locus does not meet every g diml hypersurface, but does meet a closed subvariety of them.

the closed set D in H, the discriminant locus, consists of those t such that the corresponding theta divisor has at least one singular point, usually only one or two. There is a projection down p:C-->D in H, and we can look at its derivative.

I.e. the total family C of all singular points on all theta divisors, is itself usually smooth, and we can look at the derivative of p as a linear map from the tangent space to C at a singular point (z,t), down to the tangent space to D inside the tangent space to H at t. C and D have the same dimension, one less than H.

now remember we have three nested families upstairs lying over H. we have a family of smooth tori, and in that a family of not always smooth theta divisors, and in that, a family C of singular points on non smooth theta divisors.

look at the projection from these various families down to H. and at the derivative of the projections. since the tori are all smooth, the derivative of the big projection is surjective. since however the theta divisors are not all smooth, the derivative of the projection restricted to the family of theta divisors, fails to be surjective exactly at a singular point (z,t) of a singular theta divisor.

so we could also define the critical locus C, as the points upstairs where the derivative of the restricted projection from the family of theta divisors down to H, has non surjective derivative. now in general the discriminant locus downstairs is a smooth hypersurface D in H, and the image at t in D of the derivative of the restricted projection, is just the tangent space to D at t.

moreover the equation of that tangent hyperplane is the vector of first partials of the theta function wrt t. This is what comes up on one side of the heat equation.

Now at a general singular point upstairs on the singular theta divisor over t in D, the singular point is a double point, at which the first non zero terms of the taylor series are quadratic, and the matrix of this quadric is the symmetric matrix of second partials of the theta function wrt z. that is what is on the other side of the heat equation.

now notice that the tangent space to H consists of symmetric matrices, and on the other hand the tangent space to the vertical space C^g over t in H consists of g diml vectors z1,..zg. Now a quadric tangent cone to a double point of the theta divisor over t, thus is a symmetric matrix of second partials of f wrt z. i.e. a quadratic homogeneous polynomial in the vbls zi.

this quadric cone in C^g, on the other hand can be looked at as a determining a symmetric matrix and hence a tangent vector to H, i.e. as the coordinates of a vector in t space.

the heat equation says these are the same.

i.e. the geometric heat equation says the symmetric gbyg matrix determined by the quadric tangent cone to a double point of the singular theta divisor lying over the point t of D, is the same as the vector in g(g+1)/2 space determining the tangent plane at t to the discriminant locus D in H.


If the theta divisor over t has several singular points, then you get several quadrics and several tangent planes at t in H. The intersection of those tangent planes gives the tangent directions in H of the locus of t's whose theta divisors have as many singular points as does the one at t.

hence if you can compute the dimension of the intersection of those tangent planes at t, you can see how large is the locus of t's having the same number, or same dimension, of singular points as the one at t.

AM showed that near a jacobian period matrix t, the locus of matrices with g-4 dimensional singularities on theta, was 3g-3, exactly the dimension of the set of jacobian matrices. thus near a jacobian, one can recognize another jacobian because the theta divisor has the same size singular locus at as t.

they made the computation by using the heat equation to equate it with the computation of how many quadrics contained a certain canonical model of the curve X defining the jacobian matrix. thus classical geometry in projective space enabled a tangent computation in the moduli space of abelian varieties.


Is not this amazing?

 

[Gib Z]

If I understood more of it, I bet my jaw would be on the ground. From what I *think* you have shown, its pretty amazing.

I'm not sure about which field I want to go into, I was thinking Number Theory but that sounds extremely difficult...You need to know a vast range of mathematics, and I don't have that special gift of seeing a theorem when I see one. For Example, Fermat Would just notice, or intuitively think, about a theorem, and then prove it. Its easier to prove them if you know them. But deriving Number Theorems are extremely difficult, even elementary ones. I don't know something worthwhile when I see it.

 

[mathwonk]

well none of us has to come anywhere near fermat to make a contribution to ma thematics. but we can try to emulate him, and see where it leads us. you can do your own mathematics.this beautiful work of AM just described here, is very unusual in its originality and scope.

i would guess, and it is an informed guess, however that they came up with it by reading riemann and other great 19th century workers.

 

[mathwonk]

If you are interested in this set of posts, try to read it, and work it out on paper. try to confirm the statements made there. if you do not succeed after a while, take it on faith, and proceed to the next statement. this is the way to learn math. if a statement seems hard, try to verify it in dimension one. then try to go up.

 

[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!