Swedish Student Partly Solves 16th Hilbert Problem

An anonymous reader writes "Swedish media report that 22-year-old Elin Oxenhielm, a student at Stockholm University, has solved a chunk of one of the major problems posed to 20th century mathematics, Hilbert's 16th problem. Norwegian Aftenposten has an English version of the reports."

Comments

Where I went to school

[Anonymous Coward]

You solved the whole thing or you got an F.

I'd hit it!

[dewie]

Uh, sorry. Thought I was on fark for a second.

Seriosly though, a hot Swedish mathematician? That's so much like my dreams it's scary.

Re:Wow he's good

[Flarenet]

I know you intended to be funny, but if you had loaded the article, you would have noticed that "he" is actually a "she" (and a fairly good looking she at that. :) But this is slashdot, and reading the article should not get in the way of a good joke!

Re:I remember

[mc_barron]

Yeah, and he had this group of construction worker buddies he would hand out in bars with. He had a great mind, but he was abused as a child and couldn't express intimate emotions. He solves this problem on the board, and the next hting he knows the math professor really wants him to work on problems together. Then Robin Williams shows up and...oh, wait a minute.

Mathematicians cheering in the aisles

[Frisky070802]

Two of the last three headlines I see on slashdot are about math (this one and Robin Milner). Timothy, the rest of us submit stories too!

Just kidding ... these are perfectly reasonable stories. But I'm still a bit surprised. But then, slashdot readers don't disappoint. They immediately honed in on Turing's sexuality and the student's physical attributes. Math, what math?

hmmmmm

[JeanBaptiste]

the caption below the photo says "Elin Oxenhielm pointing to the second part of Hilbert's 16th problem on her web page"

looks like a chalkboard to me...

oh well.

Re:I remember

[CSharpMinor]

Click.

Legend: A student arrives late to math class and finds two problems written on the chalkboard. Assuming they're homework problems, he jots them down in his notebook and works on the equations over the next few days before turning his solutions in to the instructor. Several weeks later, the professor turns up at the student's door with the student's work written up for publication. The two problems were not a homework assignment; they were problems previously thought to be unsolvable which the instructor had used as examples in his lecture that day.

Origins: This has to be one of the ultimate academic wish-fulfillment fantasies: a student not only proves himself the smartest one in his class, but also bests his professor and every other scholar in his field of study.

As far as we know, this legend is based upon a true incident. (That is, a version of this legend that antedates a known true incident has not yet been discovered). George B. Dantzig, then a graduate student at the University of California, Berkeley, arrived late for a statistics class one day and found two problems written on the board. Not knowing they were examples of "unsolvable" statistics problems, he solved them as a homework assignment. Dantzig, who later became a staff mathematician at Stanford University, recounted his solving two "unsolvable" problems in a 1986 interview for College Mathematics Journal, and his solutions to the two problems can be found in the journal articles listed in the Sources section below.

Hot sweedish chicks

[Andreas(R)]

I'm impressed by the sweedish girls at Stockholm University.

One

Two

Three

Four :)

Enjoy :)

problem description

[combinatorics]

Here's a description of the problem from
http://aleph0.clarku.edu/~djoyce/hilbert/toc.html
snip...A thorough investigation of the relative position of the separate branches when their number is the maximum seems to me to be of very great interest, and not less so the corresponding investigation as to the number, form, and position of the sheets of an algebraic surface in space...

Can someone please post graphical, dumbed down representation of this problem so we can better understand it?

Re:It's funny that college kids....

[orthogonal]

It's a chick who solved it

Math chicks always get me hot. And she is one hot math chick.

I'd love to estimate the area under her curves.

Re:Heh

[DigiShaman]

Based on the photo alone. I would say she is engaged or even *gasp* married. Yup, when your single and on the prowl...the "ring finger" is the first thing you look at. Why bother wasting hers and your time?

Re:It's funny that college kids....

[freeweed]

To hell with estimating, I'd rather have a firm grasp on the number.

Re:It's funny that college kids....

[Lovepump]

I believe it'll remain imaginary for you...

Re:It's funny that college kids....

[hurtstotouchfire]

Fermat had a full-time job as a respected jurist, and he was an extremely prolific mathematician.

However, Andrew Wiles, who solved Fermat's last theorem, spent seven years in his attic to do so.

I guess broad generalizations don't work so well, eh?

I wonder how many people

[Dlugar]

I wonder how many people read the article only because of this post here.

I know I did.

SwedishHot at SlashDot

[Get Behind the Mule]

I wanted to read the responses to this article because I thought that maybe one Slashdotter could give a qualified explanation of Hilbert's 16th problem, and maybe even explain something about the partial solution. That was possible back when Andrew Wiles proved his theorem, you know.

And look at this, not a single post even gets started on the subject! At least not when you browse at +2, like I do. But we're all standing around slobbering over the thought of a hot Swedish math babe! And so am I!

Hey Taco, can we get this gal for an Ask Slashdot interview? She could explain her theorem, and tell us something about her lingerie.

Re:SECKS

[DoctorHibbert]

What are you an idiot?!? Haven't you seen any teen love movies? Geek chicks always turn out super hot!! All you need to do is take of the glasses, let down her hair and unbutton her shirt a little.

Glasses? check
Long hair in bun? check check
Dowdy, boyish outfit? check check eheck!!!!

She is the trifecta! MAN SHE IS RIPE FOR THE TAKING!!!!

If you can't see that, well, then that's just sad.

Context

[ixache]

I know that this is Slashdot and that around here the looks of a mathematician are more important than her work, but if anyone is interested, here are a few pointers to get to know more.

First, a short description of Hilbert's problems at Wolfram: Hilbert's Problems -- from MathWorld.

Then, a link to a text of Hilbert's original lecture in Paris in 1900.

Next, a quote of the 16-th problem as laid out by Hilbert. (Sorry, no fancy LaTeX here.)

16. Problem of the topology of algebraic curves and surfaces

The maximum number of closed and separate branches which a plane algebraic curve of the n-th order can have has been determined by Harnack. There arises the further question as to the relative position of the branches in the plane. As to curves of the 6-th order, I have satisfied myself--by a complicated process, it is true--that of the eleven branches which they can have according to Harnack, by no means all can lie external to one another, but that one branch must exist in whose interior one branch and in whose exterior nine branches lie, or inversely. A thorough investigation of the relative position of the separate branches when their number is the maximum seems to me to be of very great interest, and not less so the corresponding investigation as to the number, form, and position of the sheets of an algebraic surface in space. Till now, indeed, it is not even known what is the maxi mum number of sheets which a surface of the 4-th order in three dimensional space can really have.

In connection with this purely algebraic problem, I wish to bring forward a question which, it seems to me, may be attacked by the same method of continuous variation of coefficients, and whose answer is of corresponding value for the topology of families of curves defined by differential equations. This is the question as to the maximum number and position of Poincare's boundary cycles (cycles limites) for a differential equation of the first order and degree of the form dy/dx = Y/X where X and Y are rational integral functions of the n-th degree in x and y. Written homogeneously, this is X(y dz/dt - z dy/dt) + Y(z dx/dt - x dz/dt) + Z(x dy/dt - y dx/dt) = 0, where X, Y, and Z are rational integral homogeneous functions of the n-th degree in x, y, z, and the latter are to be determined as functions of the parameter t.

Finally, I'll quote the abstract from Miss Elin Oxenhielm's article On the second part of Hilbert's 16th problem :

Let k be an integer such that k is larger than or equal to zero, and let H be the Hilbert number. In this paper, we use the method of describing functions to prove that in the Lienard equation, the upper bound for H(2k+1) is k. By applying this method to any planar polynomial vector field, it is possible to completely solve the second part of Hilbert's 16th problem.

Author Keywords: Second part of Hilbert's 16th problem; Hilbert number; Lienard equation; Describing function; Limit cycle; Polynomial vector field

To get the full text of the article you must apparently have a subscription of pay a $30 fee. It is easily available if you follow the directions from the author's page as I did.

Hope this helps

Now allow me for a few comments: solving one of Hilbert's problem is a huge achievement, even it's only part of one. What is even more stricking is that it's coming from a woman. Don't get me wrong, I'm no sexist, quite the contrary. What I mean is that only very few women made it to be recorded in the history of the mathematical science at large: other than Hypatia of Alexandria; Maria Gaetana Agnesi; Sophie Germain; Ada Byron, Lady Lovelace; Sofia Kovalevskaya; Emmy Noether, not many names come to mind. It would be really nice to add another one, to begin, and then work up from there.

Xavier

Re:Context

[Anonymous Coward]

I've taken a look at her article (downloaded it via an institutional subscription). It's eight pages long, with a lot of figures, and is short and easy to read. It's also categorically not an important theoretical contribution to Hilbert's 16th problem.

The author tries to determine the number of limit cycles for the Lienard equation. This would not solve the full 16th problem, but it would deal with an interesting special case, and it would likely take powerful new techniques to solve even this case. She tries to do so as follows:

She notes that numerical calculations show that the solution is well approximated by a simple trig function. (The figures are evidence in support of this assertion.) She then bounds the number of limit cycles, under this approximation, in a straightforward and elementary way. I have not carefully checked this bound, but I see no reason to doubt it (or to believe there's anything novel about it, for that matter). However, there is no attempt whatsoever at a rigorous justification of the approximation, or even a rigorous formulation of it. Therefore this simply does not constitute a full proof, although the article refers to it as a proof. Hilbert's 16th problem is already well understood in simple cases, and any attempt to reduce the more complex cases to simple cases must justify all approximations.

Incidentally, if this were an important theoretical paper on Hilbert's 16th problem, the journal "Nonlinear analysis" would be a strange place for it (it's more interdisciplinary, and is not a mainstream outlet for theoretical mathematics). That's no reason it couldn't be true, but it's some cause for initial suspicion as well as explanation for why the article was accepted. Probably the editors and referees were applied scientists unfamiliar with the problem, who were perfectly happy to accept an approximation justified by some numerical data.