It has already been a couple of years that I stumbled over two youtube-videos of talks by Terence Tao. The fact that I still keep watching those videos time and again, merits a little mention here.
Terence Tao has been awarded the Fields Medal in 2006, which is the highest award that can be reached in mathematics. It is often compared to the Nobel Prize, with the boundary condition: the awardees must be younger than 40 at the time they are honored, and the decoration will only take place once in four years (for four mathematicians at a time). So, Tao can be considered one of the greatest contemporary mathematicians – in a row with Schwartz, Milnor, Grothendieck or Werner.
His working areas are far away from what I’m familar with (for that matter, I should make an entry on Wendelin Werner some day – when I have understood some more than a few sketches concerning holomorphic functions and Brownian Motion). But then again, his interests are obviously very broadly scattered: he is concerned with number theory, particularly the distribution of primes, about functional analysis, Fourier analysis and probability theory. As far as I can tell, he is working on the boundaries of each of these areas, bringing ideas from each area to the other ones – which is quite unusual among those mathematicians that I have gotten to know. I don’t wish to dismiss those, they are highly trained specialists, and in a way so am I (having gotten somewhat rusty in the past years). But Tao is a generalist at the frontier of research.
He, too, has a blog that he updates very regularly. Obviously, he posts his classroom notes for the benefit of his students and others. But recently, there were a number of posts dealing with an approach to the twin prime conjecture (which says there are infinitely many primes that differ by 2). It dealt with a collaboration of several mathematicians from all over the world, communicating via a wiki (a so-called Polymath Project), and Tao being one of them. On this blog, in a way, you could see a famous paper in the writing. We’ll still have to see, if this approach will really get famous, but people have never been so close to a proof of the twin prime conjecture, ever: there are infinitely many primes that differ by 12 only – and that may or may not be the final word on this, coming from the method considered here. The original conjecture is still unproved. But I frankly admit, the proof and even the basic ideas of this are beyond me.
The fine thing about this blog is the very broad scope. One day, when I have a little more time, I shall devote a couple of weeks just scanning through his entries trying to find some gems in there that are 1) beautiful and 2) accessible to me. I at least try to keep my level of accessibility… Just by randomly looking at his posts, I have found some nice treatises on probability theory that I will need to look deeper into, some time.
Another result of Tao’s is an approach to the Goldbach conjecture (every even number larger than 5 can be written as a sum of two primes). He proved that every odd number can be written as a sum of at most five primes.
Now, I’m not so very deep into number theory. But on youtube, you can find a very basic talk that Tao gave a couple of years ago. You don’t have to know so very much about number theory of the Riemann conjecture, or about Tao’s work. He keeps the talk very low-tech, so enjoy this:
I particularly like that last part, where he talks about his own results about arithmetic progressions and what sort of improvement the Riemann conjecture will allow, in case it’s proved 🙂
Another very nice talk is not about mathematics but about astronomy. I don’t really know why he would give a talk on that subject, but it’s one of the best talks I’ve seen so far. It deals with the way in which distances in space can be measured. You first need to know about the size of the earth, then size and distance of the moon, the sun, the planets, stars and, finally, galaxies. Each steps requires a measurement of the previous ones, and Tao talks about the ideas that allowed for these measurements in earlier centuries (for instance how the Greeks could tell the distance of the moon, even before they had an approximation to pi). One of the highlights is Kepler’s and Brahe’s measuring of the distances to the other planets. Enjoy this talk as well!