Swedish Student Partly Solves 16th Hilbert Problem 471
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."
Link for the lazy to her website (Score:3, Informative)
Re:Wow he's good (Score:5, Informative)
Re:Wow he's good (Score:1, Informative)
(heh, that ought to get people to rta)
Re:I remember (Score:4, Informative)
As far as we know, this legend is based upon a true incident.
Useful Links / Karma Whoring: (Score:4, Informative)
The abstract for her paper is here [sciencedirect.com].
Re:I'd hit it! (Score:3, Informative)
Re:Hilbert? (Score:1, Informative)
Yep. But he's likely much more famous for the 23 problems he presented in 1900. Proof of closure on C and R with the vector stuff implied by Hilbert is probably pretty cool, I just wish I understood it... I've only gotten as far as Riemann integration, and only on elementary functions.
Re:It's funny that college kids.... (Score:3, Informative)
Proof?
If a 45 year old college professor solved it, would this be news?
I think it's pretty well-known that among mathematicians, the older you get, the less likely you are to do anything really important. In other words it's not really "funny" that a college kid would solve this; it's pretty much the norm.
There's a PBS documentary about John Nash that I recently saw where this is talked about a bit; the commentators liken mathematicians to ballerinas, and Nash himself said he felt his best years were behind him at age 30 (and not because of his mental illness - in fact, his mental illness may have in part been due to the stress he was feeling). It's on DVD if you want to look for it - A Brilliant Madness was the title, I believe.
In fact, you're in luck - I just Google'd it for you and there's a web site here [pbs.org] that includes a transcript of the program.
Re:I remember (Score:5, Informative)
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.
problem description (Score:5, Informative)
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.... (Score:1, Informative)
What? They didn't solve any unsolved problems. They just made a lot of money.
Re:It's funny that college kids.... (Score:5, Informative)
However, Andrew Wiles, who solved Fermat's last theorem [st-and.ac.uk], spent seven years in his attic to do so.
I guess broad generalizations don't work so well, eh?
Re:hmmmmm (Score:2, Informative)
Re:problem description (Score:3, Informative)
But in brief, it appears to be a problem about the "topology of real algebraic curves"
"Topology" is all about the shape of things. e.g a donut and coffee cup are the same from a topological viewpoint because you can transform one to the other without tearing the donut or coffee cup. There is probably lots of good introductions on the web.
As to "real algebraic curves", here is a link: [uiuc.edu]
I quote:
Curves that can be given in implicit form as f(x,y)=0, where f is a polynomial, are called algebraic. The degree of f is called the degree or order of the curve. Thus conics (Section 7) are algebraic curves of degree two. Curves of degree three already have a great variety of shapes, and only a few common ones will be given here.
Basically polynomials of several variables is what they are, as far as I can tell. y = x^2 (which is a parabola) is a simple example.
So Hilbert was asking about the "shape" of algebraic curves (I think).
Now that was just the first part! I am not really sure the second part is about ...
The link again is
here [clarku.edu]
I welcome corrections from anyone with more math knowledge.
Yes, really (Score:1, Informative)
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.
Re:I'd hit it! (Score:3, Informative)
Re:Way to go lady ! (Score:2, Informative)
I tend to not like differential equations let alone these ones !
The textbook at Uni was only an inch thick and was titled "Elementary Differential Equations"
Better titled "your worst nightmare"
Let us award genius when it is due... she deserves it.
TG
Except... (Score:2, Informative)
In Sweden, the telltale sign of an engagement ring is an _absence_ of any stone. It's a nondescript gold ring. It looks pretty much like The One Ring but without the elvish runes. On the inside of the ring, though, date and names are engraved.
I'd say this particular ring is either a family heirloom, or that she's extremely Americanized. My guess at odds for the two options would be about 90/10.
Context (Score:5, Informative)
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 [wolfram.com].
Then, a link to a text of Hilbert's original lecture in Paris in 1900 [clarku.edu].
Next, a quote of the 16-th problem as laid out by Hilbert. (Sorry, no fancy LaTeX here.)
Finally, I'll quote the abstract from Miss Elin Oxenhielm's article On the second part of Hilbert's 16th problem :
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 [math.su.se] 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:problem description (Score:2, Informative)
So Hilbert was asking about the "shape" of algebraic curves (I think).
Yeah, that's pretty much the first part of Hilbert's 16th problem. For example, just by using quadratic equations, one can obtain a single infinite curve (like a parabola y = x^2), a single closed curve (like a circle x^2 + y^2 = 1), two disjoint infinite curves (like a hyperbola x^2 - y^2 = 1), or two intersecting infinite curves (like the pair of lines x^2 = y^2). That's about all you can do with quadratics. With cubics, you can get more complicated things (like a closed loop together with another infinite curve - look up "elliptic curve" for some examples), and so on and so forth. The first part of Hilbert's 16th is to classify all the possible numbers of loops and infinite curves that an algebraic equation of a fixed degree can generate, as well as their relative position (if an equation generates two loops, are they disjoint, or intersecting, or is one contained inside the other? etc.)
The second half of Hilbert's 16th is similar, but deals with curves that are not solved by algebraic equations, but rather by differential equations. For instance, the differential equation dy/dx = y gives exponential curves such as y = e^x. Sometimes these curves converge to a periodic loop known as a "limit cycle"; Hilbert's problem is then to count how many limit cycles there are and how they are positioned.
Oxenhielm's paper can be found here [sciencedirect.com]. It seems that she hasn't solved Hilbert's 16th problem for all differential equations, which would be absolutely amazing, but only for a specific class of such equations, although this does still seem to be a substantial achievement.
Terry
(Attempted) explanation of Hilbert's 16th (Score:5, Informative)
The first part of Hilbert's 16th problem asks about the relative number and position of the components of a curve of order n. In other words, if we look at the graph of an equation of nth degree in the plane, what might the graph look like? We can describe it fairly easily for small n.
If n=1, the first order equations are precisely the linear ones, so the curve always consists of a single unbounded component (the straight line).
If n=2, the general equation of the 2nd order is Ax^2+Bxy+Cy^2+Dx+Ey+F=0, also known as the equation of a conic section. Depending on the coefficients, the graph will be a point, a line, a parabola, two intersecting lines, an ellipse, or a hyperbola. Geometrically, all of the cases but the last are only a single component. Therefore an equation of the second order has at most two branches. When there are two branches, they both are unbounded.
The case n=3 is much more complicated, and involves the study of what are known as elliptic curves. Beyond that, it just gets worse.
What Hilbert wished to have investigated was the geometry of the branches in the case of the curves with the most branches. As it turns out, you can't just have any orientation. If n=6, for example, the greatest number of branches is 11, but if the curve has 11 branches then one of the branches will always lie completely inside another branch. The 16th problem asks what similar restrictions are required for other n, and what happens if we look in higher dimensions than the plane.
A related problem that Hilbert referred to in his problem was that of curves defined by differential equations instead of polynomials. Here the objects of interest are boundary cycles of first order (featuring no derivatives higher than the first) differential equations. I have not encountered this term before, but if I had to guess I would say a boundary cycle was a closed, limiting path of a function satisfying the differential equation (so, for example, a boundary cycle of the second-order differential equation given by gravitation would be a planet's orbit after it is sucked in the system). The same sort of question is asked: how could these cycles be placed relative to one another in the plane? It is this question that may have been answered by the student in the article.