Kuro5hin.org: technology and culture, from the trenches
create account | help/FAQ | contact | links | search | IRC | site news
[ Everything | Diaries | Technology | Science | Culture | Politics | Media | News | Internet | Op-Ed | Fiction | Meta | MLP ]
We need your support: buy an ad | premium membership

[P]
2004 Abel Prize Awarded to Atiyah and Singer

By flo in Science
Wed Mar 31, 2004 at 11:06:31 AM EST
Tags: News (all tags)
News

The 2004 Abel Prize has been awarded jointly to Sir Michael F. Atiyah of the University of Edinburgh, and Isadore M. Singer of the Massachusetts Institute of Technology,

``for their discovery and proof of the index theorem, bringing together topology, geometry and analysis, and their outstanding role in building new bridges between mathematics and theoretical physics.''


The Index Theorem

The Atiyah-Singer Index Theorem is considered one of the greatest achievements of 20th century mathematics. Unfortunately, the exact result is impossible to state in layman's terms, but very roughly it provides a link between differential equations and topology. (See this paper for a professional introduction, and this comment for a lighter treatment). This result is the culmination of a long chain of ideas, starting with Stokes Theorem, which some readers may have encountered in undergraduate math courses, and passing through Hodge theory and the Hirzebruch-Riemann-Roch Theorem. Surprisingly, the Index Theorem has numerous applications, not just in pure mathematics, but also in theoretical physics. Indeed, this result, as well the tireless efforts of Atiyah and Singer in general, have lead to a highly fruitful cross-fertilization between mathematics and theoretical physics, which has left a profound impact on both disciplines.

The Abel Prize

The Abel Prize was created on the occasion the of the 200th birthday of Niels Henrik Abel, a brilliant Norwegian mathematician who tragically died at a young age just as his achievements were starting to receive due recognition. The prize, worth NOK 6,000,000 (USD 875,000, EUR 710,000), is awarded once per year by the Norwegian Academy of Science and Letters to acknowledge outstanding accomplishments in the field of mathematics. It was awarded for the first time last year, when it went to Jean-Pierre Serre.

As is well known, there is no Nobel Prize in Mathematics. Instead, most people have looked to the Fields Medal as the most prestigious award for mathematical achievement. The Fields Medal, however, differs significantly from the Nobel Prize in that it is only awarded every four years (at each meeting of the International Congress of Mathematicians), to two, three or four recipients. Moreover, it is only awarded to mathematicians not older than 40, in order to encourage further achievements. This also has the pleasant side effect of rewarding recent work, rather than work done half a century earlier, as is too often the case with Nobel Prizes. Tellingly, both Atiyah (1966) and Serre (1954) have also won Fields Medals.

The Abel Prize, by contrast, seems to resemble the Nobel Prize more closely. It is awarded annually by a Scandinavian learned society (the Nobel Prize is awarded by the Swedish Academy of Sciences), it is worth a considerable amount of money, and so far all three recipients are over seventy years old. Happily, this allows rewarding those (relatively few) mathematicians who missed the Fields Medals due to their age. In particular, by 1966 Singer was already over the age limit (he was born in 1924), and only Atiyah was awarded the Fields medal for the Atiyah-Singer Index Theorem.

Sponsors

Voxel dot net
o Managed Hosting
o VoxCAST Content Delivery
o Raw Infrastructure

Login

Poll
Who's up next (2005)?
o John H. Conway 8%
o Pierre Deligne 2%
o Israil M. Gelfand 4%
o Misha Gromov 0%
o Alexander Grothendieck 0%
o Friedrich Hirzebruch 0%
o Barry Mazur 0%
o Goro Shimura 4%
o John Tate 1%
o William Thurston 1%
o Andrew Wiles 8%
o See comment 1%
o Who the hell are all these people? 68%

Votes: 92
Results | Other Polls

Related Links
o 2004 Abel Prize
o Sir Michael F. Atiyah
o Isadore M. Singer
o Atiyah-Singer Index Theorem
o this paper
o this comment
o Stokes Theorem
o Niels Henrik Abel
o awarded
o first time
o Jean-Pierre Serre
o Nobel Prize
o Fields Medal
o Also by flo


Display: Sort:
2004 Abel Prize Awarded to Atiyah and Singer | 129 comments (76 topical, 53 editorial, 2 hidden)
+1, science/technology (1.21 / 14) (#11)
by Hide The Hamster on Tue Mar 30, 2004 at 05:35:59 PM EST




Free spirits are a liability.

August 8, 2004: "it certainly is" and I had engaged in a homosexual tryst.

The Atiyah-Singer Index Theorem (3.00 / 40) (#27)
by the on Tue Mar 30, 2004 at 07:24:35 PM EST

OK. Here comes a quick summary.

A manifold is a smooth surface. Like the surface of a donut or sphere or the whole of 3D (or even N-dimensional) space.

If you're a topologist you tend to think of manifolds a bit like rubber sheets. One donut is pretty much the same as another even if one is twice as big as the other or if one is deformed out of shape. As long as it still has one hole in the middle a topologist thinks it's the same as any other donut.

Differential geometers, on the other hand, do care about the shape of their manifolds. A field on a manifold is something that takes different values at different points on a manifold. A simple example is the temperature on the surface of the Earth. The Earth's surface is a 2D manifold and at every point on it there is a defined value of temperature. Among other things differential geometers they like to study fields that satisfy partial differential equations on manifold. A partial differential equation is an equation describing the properties of a field in an infinitesimal area. For example a well known differential equation is the Laplace equation and if a field satisfies that then in any small spherical volume (say) the value of the field at its center equals the average over that volume.

Given a certain type of partial differential equation (those arising from what are known as elliptic differrential operators) there is a thing you can compute called its index. (I won't go into details but it comes from looking at how big the sets of solutions to these equations are). Now almost everything you might want to know about a partial differential equation's solution comes from knowing specific details about the exact shape of the manifold you're on. For example the electromagnetic field in a vacuum satisfies a partial differential equation (the Laplace equation no less). We know that how far you are away from a charge affects how strongly you feel the field produced by it. These are the kinds of specific details I'm talking about - things like distances matter when working with these equations. But topologists can't see distance because to them everything is made of infinitely stretchable rubber. So topologists can't see most of the information you need to work with these equations.

But here's the weird thing: even though a topologist can tell you next to nothing about the a partial differential equation, they can tell you its index. In fact, topologists have a kind of index that they can compute and it turns out to be equal to the other kind of index. And that's the content of the Atiyah-Singer Index theorem. Knowing only topological information about a manifold is enough to draw conclusions about the set of solutions to partial differential equations. That's a pretty remarkable result. Actually, what I've described applies equally well to a number of different theorems, for example the De Rham theorem and the Hodge Theorem. The Atiyah-Singer Index theorem is just one particular theorem of this type.

--
The Definite Article

walks (2.80 / 5) (#47)
by martingale on Wed Mar 31, 2004 at 04:38:50 AM EST

One of the nice things about second order elliptic operators is that they arise in the study of random walks (more precisely, Brownian motions). Parabolic operators based on such elliptic ones arise too.

Now, from a probabilistic point of view, the solutions to equations arising from these operators have meaning. Usually, the meanings are things like the probability of ending up on some particular part of the manifold from somewhere else.

Now if you think about such kinds of solutions, then it's natural that topology should play a role, because it codifies the possible paths the random walk could take to end up in a particular place.

For example, think of a donut and pick two points on it. To get from one point to the other, you can walk along several different parts of the donut, some shorter than others. All these possible ways contribute to the solution.

What's interesting is that the possible paths which contribute to the solution don't really depend much on the exact shape of the donut. A donut has a characteristic number of different types of paths. But the exact solution solves the operator equation, and that depends on the exact shape of the manifold.

[ Parent ]

I spotted a stochastic proof of the Atiyah-Index.. (none / 0) (#86)
by the on Wed Mar 31, 2004 at 09:34:39 PM EST

...theorem which looked interesting.

I guess one way to look at path counting is this: with the right topology the set of paths from A to B can be viewed as the disjoint union of disconnected components so in some sense the final count is a discrete sum of a bunch of integrals. The set of separate integrals is then determined only by the topology of the manifold. And of course the connection with Feynman path integrals, and hence quantum mechanics, is pretty clear from this point of view.

--
The Definite Article
[ Parent ]

Topography (none / 0) (#83)
by lens flare on Wed Mar 31, 2004 at 08:56:12 PM EST

You explain the subject well. I know already about EM equations, and other field equations, such as gravity. I would appreciate it if you could explain topography from a Mathematical point of view more please. I've been looking at the dictionary, and it says a manifold is a bijectable surface. This means that its function would be one-one. But in the dictionary, it also says that donuts and picture frames are topogically equivalent. Because they have a hole in them and a continuous 'ring'? I don't quite understand it... I would write more of my thoughts, but to be honest they're not very useful, and sound wrong. Surely topography isn't that hard to grasp? When would you use their theorem?

[ Parent ]
Topology (none / 3) (#97)
by flo on Thu Apr 01, 2004 at 04:38:34 AM EST

First off, topology and Topography are not the same thing. Topography concerns exact measurements of a surface (used in geography), while topology totally ignores distances.

Intuitively, topology is the study of how bits of things connect to each other, without bothering with distances or angles. A simple example is an "undirected graph", which is just a set of nodes (called vertices), and connections between them (called edges). So the only information such an object encodes is which vertices are connected to which other vertices. The vertices could represent points in 3D space, with the edges straight line segments between the points. Or the vertices could represent people, with edges representing which of them have had sex with each other. All that matters is which things are connected to which other things. From that one derive other properties, for example some edges might form a closed loop: (A,B), (B,C), (C,A).

Another example is "closed surfaces", such as the surface of a ball, or of a doughnut. Two surfaces are topologically "the same" (we say homeomorphic) of the one can be deformed into the other by stretching and bending, but without tearing, cutting, or glueing any of it. Try as you might, you cannot deform a ball into a doughnut. But you can deform a doughnut into a picture frame (just pinch it a bit to make it squarerish), or into a coffeecup, by flattening part of the doughnut, then bending and stretchin the flat part upwards to form a bowl with the remaining loop forming a handle.
---------
"Look upon my works, ye mighty, and despair!"
[ Parent ]
Hmm... (none / 0) (#113)
by lens flare on Thu Apr 01, 2004 at 06:14:58 PM EST

That makes topology a lot clearer, thanks. So really they can represent lots of different things topologically - but most usefully as a model for real world processes? You didn't give an example of how the theorem could work - but something just popped into my mind, which is probably wrong though. Charge on a set of particles is something which could be modelled using topology? You have a threshold value of e*10^(-10) of the charge experienced on average by the particle or something. Then you can use topology to show they affect each other? Which ones affect each other, and which don't. So then you want to know actually based on that data and the threshold value, what the actual charges involved are - you use something like that theorem?

[ Parent ]
Possible analogy (none / 1) (#122)
by scheme on Fri Apr 02, 2004 at 01:33:48 AM EST

I believe the Atiyah-Singer Index Theorem is a generalization (to put it mildly) of the Stokes Theorem. Basically the Stokes Theorem lets you figure things out about the curl field over a surface by looking at the field over the boundary of the surface.

The Atiyah-Singer Index Theorem lets you take this a lot further and figure things out about the behavior of a differential operator over a manifold by looking at the "shape" of the manifold.

Just like the Stokes Theorem makes certain types of calculations ridiculously easy since you can replace certain types of calculations with easier calculations, the Atiyah-Singer Index Theorem has useful applications in certain areas of theoretical physics.


"Put your hand on a hot stove for a minute, and it seems like an hour. Sit with a pretty girl for an hour, and it seems like a minute. THAT'S relativity." --Albert Einstein


[ Parent ]
This POST deserves an Abel prize. nt (none / 0) (#89)
by Russell Dovey on Wed Mar 31, 2004 at 10:31:46 PM EST


"Blessed are the cracked, for they let in the light." - Spike Milligan
[ Parent ]

Conway deserves one of these awards (2.60 / 5) (#28)
by the on Tue Mar 30, 2004 at 07:32:51 PM EST

Not for anything specific, but for his immense and diverse contribution to mathematics overall. He gave us the Game of Life, much of the theory of regular algebras and machines (regular expressions and so on), the Monstrous Moonshine conjectures, the theory of surreal Numbers (as Knuth calls them) and games, some great work on sphere packings, a large part of the classification of finite simple groups, contributions to fields ranging from knot theory to quadratic forms, and a ton of 'recreational' mathematics.

--
The Definite Article
A lot of people deserve to (2.60 / 5) (#29)
by khallow on Tue Mar 30, 2004 at 08:07:02 PM EST

This is the second year of this prize. In a fairer world, the prize would have been given out since 1900 or so, and Conway would have won one by now.

Stating the obvious since 1969.
[ Parent ]

You know, I've seen Conway talk (none / 2) (#31)
by ninja rmg on Tue Mar 30, 2004 at 10:05:59 PM EST

And he's a hell of a guy, but I don't think he's quite on the level of some of these other guys out there. He's done a lot of work that is accessable to layman and a lot of it is kind of fun stuff, but that doesn't make put him on the level of Serre or Atiyah or a lot of these other guys. The Atiyah-Singer index theorem really is quite a big deal in mathematics and shows up in various guises in many places.

Clearly, Conway's a genius and his work on game theory and especially group theory is very important and his reformulation of the Alexander polynomial in knot theory was revolutionary (in the field of knot theory), but Atiyah is such a huge figure in topology -- he invented K-theory, for example... I don't think Conway's comparable.



[ Parent ]

I agree (none / 1) (#51)
by martingale on Wed Mar 31, 2004 at 04:58:49 AM EST

Conway is further down the list. On a related note, I inherited his old office for a year. Although I shan't tell which one.

[ Parent ]
Yeah, if I had to choose the next Abel winner, (none / 2) (#58)
by ninja rmg on Wed Mar 31, 2004 at 11:02:59 AM EST

I'd probably have to go with Thurston, myself. Of course, I'm just a lowly advanced undergraduate.



[ Parent ]
I hope you cleaned it up first (none / 2) (#65)
by GenerationY on Wed Mar 31, 2004 at 12:38:53 PM EST

with a heavy bleach and a stout mop. Move the rug and out crawl a load of gliders, blinkers and beehives wriggling away. And the odd confused looking cockroach. Yuck.

[ Parent ]
there was a cupboard (none / 0) (#77)
by martingale on Wed Mar 31, 2004 at 07:07:11 PM EST

The cupboard contained all sorts of paper junk dating back twenty years, but there was no telling if some of it was his, or the next occupant's before I got it. It was left there, as the space wasn't needed.

[ Parent ]
Conway's a pretty big figure (none / 2) (#67)
by the on Wed Mar 31, 2004 at 12:49:49 PM EST

The classification of all finite simple groups was one of the big results of the 20th century. The discovery of the more esoteric sporadic simple groups, in which Conway had a big hand, was one of the big surprises of Mathematics. In addition, the connection with Monstrous Moonshine is one of the most profound results in recent years and the only reason it currently doesn't get as much publicitly as it does is that nobody has quite worked out what it all means yet. The implications are far reaching, playing an important role in String Theory as well as making new and curious connections between Complex Analysis and Group Theory. This is all in addition to his more accessible stuff.

I have to admit that I never really got all that turned on by K-theory, though Bott perodicity, as it appears in K theory, is cool and I'd put money on it being intimately tied up with Monstrous Moonshine (see here). But really I ought to be supporting the master of my old college.

--
The Definite Article
[ Parent ]

To my shame, (none / 0) (#74)
by ninja rmg on Wed Mar 31, 2004 at 04:17:42 PM EST

I have not heard about this Moonshine thing. What's it all about? Is it good or is it wack?



[ Parent ]
Monstrous Moonshine (none / 3) (#75)
by the on Wed Mar 31, 2004 at 06:15:33 PM EST

I don't know what your background is. But...

You may know the j-function j(z). Basically the j function gives a way to parameterise tori that are formed from the quotient C/(ZZ). ZZ is a 2D lattice in C and so the quotient folds up the complex plane into a donut. But it's easy to show that τ and (aτ+b)/(cτ+d) (for integer a,b,c,d) give conformally equivalent tori so you'd like to have a function that parameterises tori in such a way that equivalent tori get the same τ. Up to some trivial changes the most natural such function turns out to be the j-function. So j((aτ+b)/(cτ+d))=j(τ) (a function satisfying this property is said to be modular).

Here's the start Taylor series of j(q) where q=exp(2πiτ):

q-1+744+196884q+21493760q2+864299970q3+...
Meanwhile, the largest sporadic group is the monster group. The dimensions of some irreducible linear representations of this group are 1, 196833, 21493760, 86429970, ..

196884 = 196833+1
21493760 = 21296768+196833+1
864299970 = 842609324+21296786+2*196883+2*1

In fact all of the power series of j(q) comes from linear combinations of the dimensions of irreps of the monster in a relatively simple way and this conjecture forms part of the Moonshine Conjectures.

The fun has only just started. The densest known lattice (and sphere packing) in 24 dimensions is the Leech lattice. The number of nearest neighbors a point in the Leech has is 196560. 196884=196560+1+2+3+...+24+24. That's no coincidence as it is known modular functions and modular forms (a generalization of modular functions) count neighbors in lattices. But what's extra cool is that this 24 is the same 24 that appears in the simplest String Theory theory which is 26 dimensional (but really has only 24 degrees of freedom so mathematically its 24D even though it's set in a 26D universe). And Borcherds received the Fields medal for proving the Moonshine conjectures by 'stealing' a bunch of mathematics out from String Theory.

Conway lurks in the background of a lot of this work. He wasn't the heaviest hitter in this stuff - but I think he deserves an award for all the other fun stuff he's done too.

Oh yeah...here's the easiest way to construct the Leech lattice: 12+22+32+...+242=702. (BTW This is the only non-trivial example of an equality of the form 12+...+n2=m2.) This means that if you put a Lorentzian pseudo-metric on R26 then x=(70,0,1,2,3,...,24) is lightlike i.e. x.x=0. Now Let T be the 1D lattice generated by this vector in Z26 and S be the lattice of points perpendicular to x (which clearly includes T as x.x=0). Then S/T is the Leech lattice. So here we have a bizarre mix of number theory, group theory, String theory, complex analysis (and even information theory as the Leech lattice gives rise to the 'perfect' error correcting Golay code) and nobody really has any clue what it's all about. For example Borcherds proves the conjectures but it's one of the least enlightening proofs out there.

I'd give links but it's easy to google for this stuff.

--
The Definite Article
[ Parent ]

Sounds in interesting, (none / 0) (#90)
by ninja rmg on Wed Mar 31, 2004 at 11:03:56 PM EST

But if Conway didn't prove it himself, I don't think you can give him much credit for it. Seems to me that the only reason he'd be mentioned in the discussion would be his existing notoriety.



[ Parent ]
Oh no... (none / 1) (#101)
by the on Thu Apr 01, 2004 at 10:26:44 AM EST

I'm not promoting Moonshine just to promote Conway. Moonshine is totally cool in its own right and the work of lots of people. I get carried away talking about Moonshine sometimes...

--
The Definite Article
[ Parent ]
don't see why... (none / 2) (#68)
by khallow on Wed Mar 31, 2004 at 12:50:21 PM EST

Clearly, Conway's a genius and his work on game theory and especially group theory is very important and his reformulation of the Alexander polynomial in knot theory was revolutionary (in the field of knot theory), but Atiyah is such a huge figure in topology -- he invented K-theory, for example... I don't think Conway's comparable.

I disagree here. The problem is that you're thinking of Conway as another specialist which he isn't. Conway is probably the best generalist out there these days. I've seen his work in many different fields, and the fact that so much of it is accessible to laymen is IMHO a sign of his genius. If only those who specialize in a deep, difficult subject should count, then yes, that rules out Conway.

Stating the obvious since 1969.
[ Parent ]

Not at all. (none / 0) (#73)
by ninja rmg on Wed Mar 31, 2004 at 04:15:51 PM EST

I'm just saying that if you're talking about deep results with wide reaching impact, the monster is about the best you can come up with (which is not too shabby of course). I'm just saying he's no Serre.

And let's get real here, Conway's most important stuff (which I'd say would be his group theory) is far from accessible to the layman. Mathematics is just not a spectator sport, no matter how fun people might think it is to watch games of life or whatever.



[ Parent ]

stubborn am I (none / 2) (#92)
by khallow on Wed Mar 31, 2004 at 11:58:00 PM EST

I consider the Fifteen Theorem (all quadratic forms with integer coefficients need only be checked through fifteen to determine if their image covers the natural numbers) to be a good example of an accessible yet deep result and comparable to his work on the Leech lattice and the Monster group (though obviously the mathematical community disagrees with me on that). And IMHO it will be among his most important work yet bizarrely published only through his graduate students.

When Conway created the game of "Life" he created a new field (cellular automata). I tell you, when you look at the breadth and quality of his work it compares well with the greatest of mathematicians living or dead.

Stating the obvious since 1969.
[ Parent ]

The exact result is impossible to state? (1.88 / 9) (#54)
by 87C751 on Wed Mar 31, 2004 at 10:01:46 AM EST

Unfortunately, no one can be told what the index is... you have to calculate it for yourself.

My ranting place.

Alright then buckaroos, (2.66 / 21) (#59)
by ninja rmg on Wed Mar 31, 2004 at 11:40:45 AM EST

We'll take this real slow-like. Mr. Bigballs wanted a layman's layman's explanation and he's gonna get it, but that city slicker ain't learned ya don't ever post a editorial if yer a lookin' fer fish.

Now about that thar theorem:

Ya see you gots these here shapes that are real smooth-like. Then you take these big sheets a paper called vector spaces and ya glue them things on there to make what folks call a vector bundle. Then ya take all them different ways to glue them pieces a paper on that curvy little shape just like you do when yer paintin' up yer hotrod and ya add 'em up and multiply 'em just like bunch a apples. Now ya take them apples and ya make an short exact sequence out of 'em, now that's just like a threesome -- you put the one thing in back, one in the front and you make sure you don't get none of the two on the outsides in each other (if you do, that's called homology, it's for the queers). Now you got a threesome with your bunch of apples and now you balance them off just like the livestock scale at the county fair and now you got yerself what them yankees call a topological index.

Now you got to get yerself another index. First thing ya do is you take them papers on yer curvy little shape and ya put little rulers on em. Now yer gonna take this little ol' elliptic operator, which is just a fancy word fer a way to figure out how much hotter one part of your hotrod is compared to one next to it. Now there are a bunch a ways you can heat up your little shape, just like there's different weather in the summer and the winter. Now ya call your operator to tell you what the weather's like on yer manifold and some days it'll tell you that shape's ice cold. Now if you take the days it's cold and take off all the days devided by the days it ain't, you'll get yerself another index.

And damn if them two indexes ain't the same thing! HOOWEE!

A note from the editor: Do not harass Mr. rmg with questions like "DO YOU EVEN KNOW WHAT HOMOLOGY IS?" The answer is yes (though I admit I don't know a hell of a lot about K-theory). Deviations were made to add color. Those who do not appreciate this will be dealt with severely.



DO YOU EVEN KNOW WHAT HOMOLOGY IS? (2.00 / 5) (#63)
by The Amazing Idiot on Wed Mar 31, 2004 at 12:33:16 PM EST

Notice: Ignored fine print, as I always do ;-)

[ Parent ]
And this, folks (none / 1) (#69)
by ZorbaTHut on Wed Mar 31, 2004 at 02:30:35 PM EST

is how to make mathematics appealing to the masses.

[ Parent ]
now we're gettin somewhere (none / 0) (#125)
by tincat2 on Fri Apr 02, 2004 at 05:54:15 PM EST

nice

[ Parent ]
heh.. (1.83 / 6) (#60)
by Work on Wed Mar 31, 2004 at 11:45:18 AM EST

i keep reading that as "2004 abel prize awarded to aliyah the singer"

Write-in vote (2.22 / 9) (#61)
by IHCOYC on Wed Mar 31, 2004 at 12:05:55 PM EST

Gene Ray, discoverer of the Time Cube.
--
Nisi mecum concubueris, phobistę vicerint.
   --- Catullus
heh heh heh (none / 2) (#62)
by The Amazing Idiot on Wed Mar 31, 2004 at 12:32:05 PM EST

THAT guy.

He forgot (cold chill) COLD FUSION!!!

[ Parent ]

the Time Cube rocks! (none / 2) (#71)
by Mindcrym on Wed Mar 31, 2004 at 03:12:01 PM EST

All of those educated stupid scientists in the article above have got nothing on that guy.

  -Mindcrym

[ Parent ]

superb (none / 1) (#99)
by GenerationY on Thu Apr 01, 2004 at 07:27:37 AM EST

You Word-Murder your children,
You've been educated Stupid & Evil,
and All Clock Faces are wrong!

Heh. I couldnt' get any further, its a nonsmoking website apparently...!

[ Parent ]

A few (2.25 / 4) (#64)
by khallow on Wed Mar 31, 2004 at 12:34:17 PM EST

I'll spew out a bunch of names. Had a real nice and linked article all set up, but I closed that window. Should have done it in an editor.
  • Edward Witten, perhaps grouped with the appropriate Schwarz and Green.
  • Langlands of the "Langlands program"
  • Peter Lax
  • Kiyosi Ito
  • V.I. Arnold
  • maybe I. G. MacDonald
  • Rodney J. Baxter - my favorite dark horse.
Think I'll stop there.

Stating the obvious since 1969.

They say Ed Witten never proved a theorem... (none / 1) (#85)
by the on Wed Mar 31, 2004 at 09:27:59 PM EST

...in his life. There was a bit of resentment that he received a Fields medal. He is primarily a physicist and a lot of his work is best described by mathematicians as 'conjectures'.

--
The Definite Article
[ Parent ]
good work if you can get it (none / 3) (#93)
by khallow on Thu Apr 01, 2004 at 12:09:10 AM EST

They say Ed Witten never proved a theorem... ...in his life. There was a bit of resentment that he received a Fields medal. He is primarily a physicist and a lot of his work is best described by mathematicians as 'conjectures'.

Just a counterexample to the claim that rigor is the only way to go. I imagine string theory will eventually run into the problems (maybe ten or twenty years ago :-) that the Italian school of algebraic geometry supposedly faced before the Second World War. Namely, that the status of some results as "proven" was disputed until eventually new mathematics was established up in the 50's or so.

Stating the obvious since 1969.
[ Parent ]

Ed Witten even says new tools are needed (none / 0) (#108)
by modmans2ndcoming on Thu Apr 01, 2004 at 02:31:25 PM EST

he has even been quoted as saying that M-Theory cannot be fully understood until we have new tools in Mathematics developed.

basically he is saying "hey all you mathematicians.... us Physicists need some more tools...get crack'n

[ Parent ]

Langlands (none / 0) (#96)
by flo on Thu Apr 01, 2004 at 04:15:35 AM EST

You're right, I really should have included him in the poll.
---------
"Look upon my works, ye mighty, and despair!"
[ Parent ]
don't know any of these people, but... (none / 3) (#70)
by Mindcrym on Wed Mar 31, 2004 at 03:01:12 PM EST

Is John Tate of any relation to Geoff Tate?

  -Mindcrym

An Anecdote on Mathematical Maturity (3.00 / 8) (#72)
by teece on Wed Mar 31, 2004 at 03:56:19 PM EST

There is so much mathematical knowledge.

I like to delude myself into thinking that I am reasonably mathematically mature.

I have essentially completed a BS in mathematics (I am still finishing the degree, but all of my work left is for my other major, physics). I know all the standard differential and integral calculus, I can do a lot of ordinary differential equations on the spot, I understand some of the more common partial differential equations (Heat equation, Laplace's and Poisson's equations, Schroedinger's equation, the Wave equation). I know the standard introductory material of statistics, and a good deal of probability theory. I have a firm grasp on linear algebra and vector spaces, and even a very rudimentary understanding of tensor analysis. I am currently trying to get better at topology and complex analysis, as well as solidify my understanding of vector analysis and higher algebra.

But wait! Flo comes along on K5 talking about Atiyah-Singer, and I have never heard of this theorem. What am I, an idiot?

So I just started trying to get the gist of Geometry of Physics by Frankel, and this seems right up Frankel's alley. So I go to the index, sure enough, Atiyah-Singer is on page 465 of a 600 page book. A book which constitutes a year's worth of study of differential geometry at the graduate level. And Frankel has this to say about the theorem:

The Atiyah-Singer index theorem is a vast generalization of (17.27)[Gauss-Bonnet theorem] replacing the tangent bundle by other bundles ..., the Gauss curvature by higher-dimensional curvature forms ..., and replacing the Laplacian by other elliptic differential operators associated with the bundle in question. The Atiyah-Singer theorem must be considered a high point of geometrical analysis of the twentieth century, but it is far too complicated to be considered in this book.
(Emphasis Frankel's).

My understanding of the concepts expressed is vague, but it seems clear that there is much more studying to be done (and this near the end of a graduate course on the subject!) to understand Atiyah-Singer.

At what point does the body of mathematical knowledge become so large as to make it very difficult to further? A BS in mathematics teaches one, almost without exception, math that is all hundreds of years old. And the average BS in mathematics has mathematical analytical skills that really dwarf what the common person has. Yet, said student is a couple hundred years behind the state of the art.

-- Hello_World.c, 17 Errors, 31 Warnings...

It's true. (none / 1) (#76)
by bjlhct on Wed Mar 31, 2004 at 06:55:30 PM EST

The obvious result of this is the increasing specialization we see in mathematics. Mature is relative. You surely know far more math than average, and we all surely know far less math than has been done. Even the people creating that kind of advanced stuff know only a small amount of relevant math. I wouldn't feel too bad unless I thought I was Gauss. After all, this real advanced math will not be applied until you're dead, and might never be applied. Know too that the great geniuses of physics did their most powerful work conceptually rather than mathematically. I for one would rather do 4d visualization than the kind of math the string theory folks do.

*
[kur0(or)5hin http://www.kuro5hin.org/intelligence] - drowning your sorrows in intellectualism
[ Parent ]
phd (none / 2) (#79)
by martingale on Wed Mar 31, 2004 at 07:28:50 PM EST

It's not as bad as you think. Normally, you can cover the remaining hundred odd years if you do a phd. You can learn at your own rythm, unimpeded by lecture schedules. Of course, you'll never get an overview of everything, but you can become an expert on one small part of the state of the art. The rest, you learn over time by osmosis, if you keep an eye out. That's sort of how it's supposed to be.

Of course, then there are the geniuses who achieve insight without lengthy formal training. You can never compete with them.

[ Parent ]

A somewhat different perspective (none / 2) (#80)
by sheafification on Wed Mar 31, 2004 at 07:36:29 PM EST

At what point does the body of mathematical knowledge become so large as to make it very difficult to further? A BS in mathematics teaches one, almost without exception, math that is all hundreds of years old.
I think this view is largely a result of your own personal experiences. There is no reason that current topics could not be covered in undergraduate mathematics courses, and there are programs which do this. Of course it would be rather difficult to present topics like the Atiyah-Singer index theorem, but there is still plenty of interesting material from the past fifty years or so which undergraduates should be able to handle.

As to your question... It seems that mathematics is becoming a collection of specialties, and the field has grown to the point where nobody can be expected to be an expert in all of them. There is, of course, nothing stopping you from becoming a specialist in a narrow field of mathematics. We are also seeing important work which bridges across these specialties. Examples include the Atiyah-Singer index theorem, the Langlands program, motivic cohomology, and Taniyama-Shimura-Weil. This is also the type of work which is most likely to win you one of mathematics' highest prizes, which shows how important mathematicians view it. So mathematics seems to be developing more specialties, but we also have ways of linking them together into something that is cohesive. In my opinion, this is a sustainable way to make progess in the field, at least for while.

Note that your question could also be asked about the physical sciences. So, at what point does our knowledge of the physical world become so large that it is difficult to progress?

[ Parent ]
from a high school student... (none / 1) (#81)
by jdtux on Wed Mar 31, 2004 at 07:41:11 PM EST

This comment reminds me of something my physics teacher(I'm still in high school, btw) said to me. I said to him, mainly jokingly, that he should include some quantum physics into the curriculum, and he said "No, I'm doing good if I can get you up to the 1700's". That was the first time I ever really thought about it, that for the most part, all this "new" stuff we're learning in high school is over 300 years old. We did little research projects on mathematicians for our pre-calculus class, most of them 200+ years old, and we have yet to even get near most of the stuff they did.
 If we're lucky, by the time we finish university, we'll have gotten up to 100 year old stuff.
I'd love to learn more right now, I know I could, but there aren't exactly many good learning resources.

my two cents. *clink clink*

[ Parent ]

you can take a look here. (none / 0) (#102)
by Wah on Thu Apr 01, 2004 at 10:52:33 AM EST

just to get a feel for whats out there.

Doesn't come with books or professors, but it's a start.
--
K5 troll comment rating guidelines....
The Best Troll Comment Evar, really great stuff, trips up a bunch of people, and wastes a day. == 1
Any
[ Parent ]

seen it before.. but will take another look (none / 0) (#119)
by jdtux on Thu Apr 01, 2004 at 09:26:12 PM EST

thanks, I had found that site awhile ago, looked through it but didn't really get any of it then. Take another look, see if I understand any more of it...

[ Parent ]
there are few good reseources... (none / 0) (#107)
by modmans2ndcoming on Thu Apr 01, 2004 at 02:27:19 PM EST

your right...when I was a senior in highschool, I wanted something to take me into calculus taught in college...there was almost nothing to take you from pre-calc trig or even highschool calc (which is basically the first half semester of calc 1) to calculus.

I was very disappointed.

most of the resources are use normal mathematic speak which is not taught in highschool at all because it might interfere with the students ability to understand the concepts (please...concepts are important, but if they can't speak the lingo, do they really understand it?) and this is also a problem in many undergrad programs in universities.... a mathematics major will not normally be exposed to the greek symbols, the logic symbols or set notation until their upper level classes which is a poor way to teach I think.

[ Parent ]

don't despair (none / 0) (#112)
by adiffer on Thu Apr 01, 2004 at 04:14:10 PM EST

Most high school physics teachers will also face problems if they DO try to get you to the 20th century.  Many don't know the material well enough.

What you are running into is the harsh reality that some of this material is hard to learn and your teachers have their limits.  There are two ways around it for you.  One is to self learn from books, but it is the long, hard, frustrating road.  The other is to hunt for a more experienced teacher.  Don't expect public or private school systems to do it for you as they aren't structured to deliver that service.  Think like an apprentice and find a master.  The second approach is how we finish grad school with PhD's.

There aren't many good learning sources.  Few people actually ask for them.
-Dream Big. --Grow Up.
[ Parent ]

HS teachers perspective (none / 1) (#129)
by losang on Sun May 09, 2004 at 01:42:19 PM EST

I completely disagree with what you say and in fact you are wrong. The level of 20th century physics that would be presented to HS students is not beyond the understanding of a HS physics teacher. Because it is presented in such a basic form it is easier than much of the material that forms the required curriculum. In particular, the material on elecric fields, potential, etc.

I designed a 2nd year HS physics course dealing with modern physics. We did many of the proofs and derivations in SR including time dilation, Lorentz contraction, Lorentz transformations, relativistic dynamcis and E=mc^2. This class was more advanced than my freshman college course on modern physics. It was more similar to the 2nd year college class I took on SR for physics majors.

So I don't see how you justifiy your statements. I would say for sure you are not a HS physics teacher and probably don't know much about physics either.

[ Parent ]

limits of knowledge (none / 3) (#84)
by gdanjo on Wed Mar 31, 2004 at 09:13:11 PM EST

The issue you bring up is one often dealt with by philosophers - that is, what are the limits of knowledge?

You bring up one such limit - that of knowledge transfer (or bandwidth). Will our body of knolwedge ever get so great that it will be impossible to transfer this knowledge to a person in their lifetime?

The trend has been to "compress" knowledge into "axiomatical fragments"; that is, the knowledge we have about logic was worked out once, then "axiomatised" such that we need not teach the full process of how it was worked out - you should just accept it and use it as a tool, just like the carpenter need not know the history of the hammer to be able to use it. This process has been sped up recently with computers, where large, complex formulas are embeded in a computer program and you simply must assume that the program is correct to make proper use of it. Just look at the "four colours suffice" problem; it was "proved" by using huge amounts of computer time such that no single person could possibly have done it by hand.

The language of mathematics is designed to be "infinitly maleable" which means that it is possible to describe any process by mathematical means. The problem of knowledge compression, however, is not handled well by mathematics; so much so that every mathematical theorem must also be accompanied by natural language description and justification of the steps involved (that is, you will never see a paper written exclusively in "mathematics", since you need a natural language to compress each mathematical step to be able to actually work with it).

So we can say that mathematics can describe all of reality (since it is infinitely maleable), and natural language can describe all of mathematics, plus some knowledge that mathematics rejects (irrational statements, etc.). The question is, can natural language describe everything, possible and impossible?

In other words, is there anything that natural language cannot describe? If the answer is yes, then we do indeed have a limit to knowledge - for if we cannot communicate something, then it's knowledge cannot be transfered to later generations. If the answer is no, then knolwedge is truly infinite and one day we will come up against a "knowledge wall" where we each time we learn something new, something else will be lost (for our capacity for knowledge is not infinite).

If you're interested in this subject matter then you should look into philosophy - a lot of the knowledge you gain from knowing the limits of knowledge can be very useful in other endevours.

Dan ...
"Death - oh! fair and `guiling copesmate Death!
Be not a malais'd beggar; claim this bloody jester!"
-ToT
[ Parent ]

I'm not sure it's a couple of hundred years. (none / 1) (#87)
by the on Wed Mar 31, 2004 at 09:55:33 PM EST

I seem to remember, at undergraduate level, doing a bit of algebraic topology. Poincaré probably had the main results we found figured out already but the formalism we use probably dates from around the 1930s. Cohomology groups were invented in 1949 and I think were part of that course.

Much of the notation taught is relatively recent. 19th century mathematical writing can often be unreadable because of its lack of basic notation like vectors and matrices. Our diagrams to represent functions are heavily influenced by category theory which must be pretty recent.

Also much of representation theory is from the 1920s. We studied the simplex method in a linear programming course and that's from 1947.

On the applied mathematics side the stuff I did was easily 20th century. Quantum mechanics, general relativity and some thermodynamics.

So I'd say that a modern mathematics course brings you up to around the 1940s on average.

--
The Definite Article
[ Parent ]

most courses aren't comprehensive (none / 1) (#94)
by martingale on Thu Apr 01, 2004 at 03:31:41 AM EST

I think hundred years is roughly right.

1) People do specialize in areas, be it analysis, geometry, discrete maths. Towards the end of undergrad study, people are already beginning to be differentiated in the subjects they take, and a hundred plus years gap would apply to the subjects they know least about. For those subjects they study, probably 1950s sounds right.

2) Even old subjects are taught with a "modern" veneer. The Bourbaki method is only 70 years old, but it's revolutionized teaching and exposition. So in a sense, all mathematical methods taught are no more than 70 years old.

3) The other problem is that you need subject experts in all areas in the department, otherwise there won't be courses on advanced topics in particular fields.

[ Parent ]

abstract algebra is pretty recent (none / 0) (#106)
by modmans2ndcoming on Thu Apr 01, 2004 at 02:18:48 PM EST

as is most of the fundamental foundations of calculus(19th century early 20th century)

so, I would say that your senior level classes probably contain many recent(in terms of all of mathematics...2500 years or so) discoveries, but the cutting edge stuff is all in grad school.

hey...you want to make a name for yourself? develop the mathematical tools necessary to fully understand M-theory.

[ Parent ]

Actually... (none / 1) (#114)
by the on Thu Apr 01, 2004 at 06:49:24 PM EST

develop the mathematical tools necessary to fully understand M-theory
Funnily enough, my PhD consisted of ripping off a 'theorem' from a physics paper and re-presenting it as a proper mathematical theorem. I still feel guilty about it because it was already on pretty solid ground though I did take an ever so slightly different path for one part. Still, my examiner was a pretty heavyweight guy and I didn't hide what I had done so I really need to stop this guilt.

The depressing thing about M-theorists is they throw around mathematics that's actually hard for mathematicians. I was pretty amazed by the sophistication of the mathematical tools the physicists around me were using, in addition to having a physical intutition that allowed them to conjecture things where no mathematician would dare. It would be a very hard job for a mathematician to be on top of both the mathematics and the physics.

--
The Definite Article
[ Parent ]

physicists (none / 1) (#115)
by martingale on Thu Apr 01, 2004 at 07:36:06 PM EST

A lot of physisicsts just bullshit their way through. Well, I'm not saying anything new, but this always reminds me of a conversation I had long ago with a friend who's a physicist:

It doesn't matter if the physics results are rigorous or not. Either they are verified experimentally, or disproved. If they are verified, then the theory is good enough to be used for now. If they are disproved, then those results are useless whether they were rigorous or not.



[ Parent ]

right (none / 0) (#118)
by modmans2ndcoming on Thu Apr 01, 2004 at 09:13:21 PM EST

all physicists care about is a symmetrical equation.

after that, the experiments tell you if it is right ow wrong.

[ Parent ]

Many years ago... (none / 1) (#123)
by the on Fri Apr 02, 2004 at 12:56:20 PM EST

...my wife used to work in the admin part of the mathematics department. Occasionally she would have to type up mathematical documents using chiwriter. She had no idea what any of it meant and yet from time to time she would be able to spot typos in the equations just from the symmetry of them.

--
The Definite Article
[ Parent ]
buddy system (none / 0) (#124)
by adiffer on Fri Apr 02, 2004 at 05:15:53 PM EST

A smart theoretical physicist makes friends with a smart mathematician.  Both are bullshitters outside their domains, but together they explain how the universe works.

This lesson was so obvious to people in my field that we weren't considered complete in the research sense without a swim/research buddy.
-Dream Big. --Grow Up.
[ Parent ]

well the fields are diffrent (none / 0) (#117)
by modmans2ndcoming on Thu Apr 01, 2004 at 09:09:06 PM EST

in Physics...you can develop a mathematical system to fit criteria that you think describes the physical word...then work out the non symmetric parts and poof...its good enough to publish.

but in Math, you make a conjecture, and you have to prove it some how....saying "it fits with observable data" is not good enough for the journals (of course you also must face the fact that there usually IS no observable data with theoretical mathematics, you can not observe the distribution of prime numbers for instance)

but then you get into M-Theory, where there is no observable data yet...and you have to wonder "are they just that smart, or just the crazy?"

[ Parent ]

Witten managed to turn the physics... (none / 1) (#120)
by the on Thu Apr 01, 2004 at 09:31:55 PM EST

...around and produce results that were completely mathematical. He'd do a handwavey argument involving completely ill defined Feynman path integrals and get some result. Years later a mathematician would limp along and finally give a rigorous proof. It's neither real mathematics or the physics of any known real physical system, but he still gets correct results!

--
The Definite Article
[ Parent ]
right...so what he has is a (none / 0) (#121)
by modmans2ndcoming on Thu Apr 01, 2004 at 10:13:22 PM EST

mathematical conjecture that looks like it might be right, but is yet unproven that describes a conjecture on the existence of physical elements that seem to logically make sense but are unobservable.

oh my god...it is neither Mathematics or Physics...it is Philosophy!!!! :-)

[ Parent ]

Let me stroke your ego a bit (none / 2) (#95)
by flo on Thu Apr 01, 2004 at 04:11:39 AM EST

I have a PhD in mathematics, and I still don't understand the exact statement of the Atiyah-Singer index theorem.

Of course, had I specialised in differential geometry rather than number theory then I'd probably be using the index theorem regularly in my work. The moral of the story is that today, mathematics is so vast that nobody really understands all fields at once. It is often claimed that the last people to understand essentially all of mathematics were Poincare and Hilbert, in the early 20th century.

As for undergraduate mathematics, it is true that almost all of it is centuries old. The reason is not that more recent stuff is too difficult (though it often is), but rather that the stuff presented at BS level is really important, both for applications, and as a stepping stone towards higher mathematics. As you have to learn this stuff first there is no time to teach you some fun things which are very recent, yet understandable at your level.

It is entirely possible for you, right now, to do some original research. All you need to do is find some problems in mathematics that are as yet untouched, not because they're difficult, but because nobody has bothered with them until now. I did something like that when I was an undergrad, I worked on some elementary number theory called Ducci sequences. It's really easy, but very few people had even heard of them, so there was (and still is) plenty of easy stuff to do there. Of course, though I could publish my results, these would hardly make me famous, and did not appear in any well-respected journals.

On the other hand, there is a program in the USA called Research Experience for Undergraduates, where you can be invloved in a "real" research program. I don't know much about it, but I guess that it is like a summer school where they teach you some more advanced (but accesible) mathematics, and where you can work on some real research problems. Some pretty good results have been found by undergraduates at such programs, but obviously the best ones are much more talented than you (probably) or I (certainly).
---------
"Look upon my works, ye mighty, and despair!"
[ Parent ]
heh...you could have saved it for your PHd (none / 0) (#105)
by modmans2ndcoming on Thu Apr 01, 2004 at 02:15:26 PM EST

that at least would have gotten you through your Thesis..depending on how involved it was. most programs just want some original work, not necessarily monumental work.

[ Parent ]
Not good enough (none / 1) (#110)
by flo on Thu Apr 01, 2004 at 03:35:33 PM EST

The stuff I did as an undergraduate certainly wasn't good enough for a PhD, at least not in Paris.
---------
"Look upon my works, ye mighty, and despair!"
[ Parent ]
well, like I said... (none / 0) (#116)
by modmans2ndcoming on Thu Apr 01, 2004 at 09:02:31 PM EST

some schools might have standards of the level of work.

[ Parent ]
"old" maths is necessary but (none / 1) (#98)
by kobayashi on Thu Apr 01, 2004 at 05:49:11 AM EST

i think that if you got to the end of your degree and had not encountered any fairly recent mathematics you should be feeling a little short changed by your university.

not that you would expect full courses, or in depth proofs, of even complete lectures, but your professors or tutors should at least have mentioned their interests and research when the opportunities arose; in fact i can't imagine them not wanting to. And outside of the formal ciriculum there must have been seminars, at varying levels of, which good undergraduates would have been welcome to attend.



[ Parent ]

your duel Majors is an aid to your mathmatical (none / 0) (#104)
by modmans2ndcoming on Thu Apr 01, 2004 at 02:11:48 PM EST

prowess.

most mathematics undergrads do not necessarily have exposure to the fantastic applications of the theory and function that they are learning, however, with your major in Physics, you get to learn about many of them since Physics is usually one of the first places new mathematic principles and theorems benefit from.

[ Parent ]

I think I feel the same way about everything... (none / 0) (#109)
by gmol on Thu Apr 01, 2004 at 02:58:23 PM EST

but for some reason the lack of math knowledge is particularly humiliating.

I mean, I am doing a grad program in chemistry, but I tend to know a little more math than those around me (not really "hard to understand math", just a superficial understanding of some stuff that other people just wouldn't care about becuase it wouldn't be useful to them (it certainly hasn't been for me)).  But I do feel a great sense of inferiority when I see things some uber brilliant theorem I have never seen and know I will never understand.

[ Parent ]

try this book for algebra/topology (none / 0) (#126)
by senderista on Fri Apr 02, 2004 at 09:31:08 PM EST

I'm currently working thru Robert Geroch's text _Mathematical Physics_ with a coworker of mine. Despite the title, it's actually an early-grad-level overview of pure math, split roughly equally between algebra and topology. The idea is to provide an introduction to pure math for theoretical physicists, all within the framework of category theory. Since physicists are the intended audience, a lot of algebraic structures like semigroups, modules, rings, fields, etc. are skipped in favor of what is commonly used in physics. But I would recommend it to anyone for self-study or as preparation for grad school in math or theoretical physics. The style is very informal, and it's full of pictures, examples, motivation, and intuitive explanations of theorems and proofs. It's one of the best math books I've read, along with _Linear Algebra Done Right_ and _Visual Complex Analysis_.

[ Parent ]
Hey! (none / 0) (#127)
by teece on Sat Apr 03, 2004 at 12:03:46 AM EST

Linear Algebra Done Right (S. Axler?) is a great book. I am going to have to check the other two you mention out. Thanks.

Mathematical physics is my favorite area -- hence the choice of majors.

-- Hello_World.c, 17 Errors, 31 Warnings...
[ Parent ]

Thanks (none / 0) (#128)
by poyoyo on Sat Apr 03, 2004 at 06:17:50 PM EST

This "Visual Complex Analysis" must be pretty hot stuff to be mentioned in the same sentence as "Linear Algebra Done Right". I'll make sure to grab a copy before taking my complex analysis course. Thanks!

[ Parent ]
2004 Abel Prize Awarded to Atiyah and Singer | 129 comments (76 topical, 53 editorial, 2 hidden)
Display: Sort:

kuro5hin.org

[XML]
All trademarks and copyrights on this page are owned by their respective companies. The Rest © 2000 - Present Kuro5hin.org Inc.
See our legalese page for copyright policies. Please also read our Privacy Policy.
Kuro5hin.org is powered by Free Software, including Apache, Perl, and Linux, The Scoop Engine that runs this site is freely available, under the terms of the GPL.
Need some help? Email help@kuro5hin.org.
My heart's the long stairs.

Powered by Scoop create account | help/FAQ | mission | links | search | IRC | YOU choose the stories!