Swedish Mathematician Lennart Carleson Wins Abel

William Robinson writes "Sci Tech is reporting that Swedish mathematician Lennart Carleson has won the Abel Prize on Thursday for proving a 19th century theorem on harmonic analysis. His theorems have been helpful in creating iPod. Prof Carleson's major contributions have come in two fields - the first has subsequently been used in the components of sound systems and the second helps to predict how markets and weather systems respond to change. One of Carleson's many triumphs was settling a conjecture that had remained unsolved for over 150 years. He showed that every continuous function (one with a connected graph) is equal to the sum of its Fourier series except perhaps at some negligible points."

Comments

Re:Young people today

[LarsWestergren]

I'm 31 and have recently started doing a lot of maths in my spare time so that I can get a real computer science and engineering degree one day (I have a degree, but it is CS light... now that I work as a programmer I know how much I'm really lacking), so it is nice to see that at least for some people the old saying by Hardy, "mathematics is a young man's game" isn't true. Carleson is 78 today, and around 40 back when he did the main breakthroughs he is honored for today.

Hardy's saying is a bit of slight against all female mathematicians too, come to think of it...

Re:Indeed

[The Cow of Pain]

This is sometimes mis-stated as 'you can draw the graph without taking your pen off the paper'.

That's not a mis-statement in the case of a real function of a real variable. It's not that informative, but definitely correct in the sense that a function (real etc.) is continuous iff the graph is path-connected (i.e. every two points on the graph can be connected by a continuous path (and by saying 'continuous path' I have of course made the definition self-referential and thus silly, but it is still true)).

Re:Indeed

[Bromskloss]

I recall correctly: a continuous function from one space to another is a function such that the inverse images of open sets are open.>

Yes, you do recall correctly! ;-) Your version is a more general one. In a metric space, where your parent's version is applicable, the two are equivalent. Sorry, I probably shouldn't tell you this, you certainly knew it already. However, someone else might be interested.

Re:Except at some negible points?

[F�an�ro]

Isn't the set of all computable numbers also countable infinite?
So you could have a continuous function which diverges from the sum of its Fourier series in all computable points?

What is convergence in a non metric space?

[Bromskloss]

This is interesting, please give us more!

In all topological spaces you have a sense of convergence of a sequence

I must admit I didn't know of any way to speak of convergence without the notion of a metric. How is that possible?

For even more general topological spaces you need the concept of a net

More general than what? And do you mean we need the "net" to replace the sequence? If you say so I'll believe you. However, one must still be able to define a sequence (a function from "the set of all natural numbers" to "the topological space in question"), since it doesn't really require much of the space, right?

(We need more maths on Slashdot!)