Sunday, September 2, 2012
William Thurston passed away recently - August 21, 2012
I thought I'd seen signs in a vidoe lecture of the Poincare conjecture at the Clay mathematics institute; but, I never confirmed that he was having health problems.
William Thurston's work formed the basis of the great "Not Knot" video found in the farely beginning of my blog(and of course, you can find it easily be typing in non knot at youtube).
I remember when in junior high, I elected to go to the library during lunch instead of mess around. One article in a "Science News" magazine was about turning a sphere inside out; William Thurston had a big hand in that; mathematicians went on to show you can turn a tube inside out and even more exotic shapes as well!
In the 1930s, mathematicians had solved a classification problem for two dimensional surfaces William thurston solved the third dimension. That's one way of saying his mathematical importance. Another would be to say that his work went a long way towards solving the Poincare conjecture recently - officially by Perelman. A lot of his work seems to me to be analysing geometry and topology by means of cartesian products - actually, geometric generalisations of the cartesian product. A cartesian product is like a is a A, and b is a B. Relations are subsets of cartesian products; functions are special cases of relations. Cartesian products can be viewed as cartesian coordinates. Foliations are a vast generalistion of this it appears to me.
I've literaly just about finished reading his Geometry and Topology of 3-manifolds. I'm two pages to go. I just wanted to read it; and really, the book is turning out to be every easy example of the mathematics he's created. It's kind of a modern day "Geometry and the Imagination" from David Hilbert. I wanted to at least read it because I figured that reading it gives me the best account of the implications of the Poincare conjecture. Seems to me that the Poincare conjecture can generalise lie theory is some ways(Lie theory can be generalised in many ways). Lie groups were used by those working on the classification of simple groups. Lie algebra of E8 was one recent major accomplishment. Both the classification of simple groups and E8 are hugh deductive structures. These are amongst the great achievements of mankind right now. One can see the abundance and remarkable connections between all mathematics with William Thurston's work.
There's so much to say! Mathematicians really back in Poincare's day and maybe before him had started to see that hyperbolic geometry is the ideal viewpoint from which to do mathematics. William Thurston's work established that at least for the third dimension.
Some of the connections with other mathematics are that of the uniformatisation theorem that Poincare did a good amount of work to solve with his automorphic functions(a group theoretic generalisation of elliptic functions.) The elliptic functions were used to solve galois theory and number theory. So, the solving of the Poincare conjecture points to a vast generalisation of the algebraic geometric relation between elliptic functions, Riemann surfaces.
To say the least, I havn't describe algebraic number theory(much less David Hilbert's algebraic field theory) and it's relations to algebraic geometry. And, there's plenty more mathematics to describe or at least give some hint at the full scope of twentieth and now twenty first century mathematics.
------------------------------------crazy science/technology extra for the day
I thought I'd give some more nanotech news because the nanomanufacturing revolution is really picking up and it's disappointing that William Thurston couldn't make it(nanomanufacturing could dramatically extend human lifespans).
"The combination of computational design and molecular biology "leads to a catalyst for whatever reaction is needed, if we can get this all to work properly," Houk said."
"Or maybe to design catalysts to attach to specific locations on a DNA origami lattice to accomplish a multistep reaction to synthesize some complex molecular building block?
—James Lewis, PhD"
------------------------------------back to William Thurston; really, kind of wild thought on my part;
When I try to think about the extent of mathematics, I often wonder if I can see what's next to do. I try to see if I can dam up nature(see my origin of mathematical knowledge article; third post from the first of this blog if you havn't done so); but, I find that after the initial abstractions of number and geometry, the daming up that creates more mathematics occurs within mathematics. Still, one can see in mathematics that if one wants to create new mathematics, just go to the next degree equation, or the next higher dimension. After Perelman's dotting of the i's for the Poincare conjecture, mathematicians(and really our scientific species, the Homo Sapiens) pretty much settled the third dimension. There's still work to do; but, the major action for mathematicians is the fourth dimension(the mathematics of Donaldson). One could say that we've reached the third dimension and are now working on the fourth dimension.
Now, sometime in the 1950s, Enrico Fermi was sitting around talking with a couple of physics friends, and he asked, "where are they?" Fermi was asking where's the extraterrestrial civilizations? If there was a big bang ten to twenty billion years ago(because of the Hubble Space Telescope, we have this number down to 13.7 billion years now almost precisely), and the earth is only 4.5 billion years . . . factor in that it takes a couple of generations of supernova to create the higher elements/atoms to create Earths . . . and supernova go off in a few hundred million years(for supergiant stars. for stars like ours, they don't go bang like that; and, they take a lot a lot longer to age), there should be five to fifteen billion years for extraterrestrial civilizations to storm the cosmos. Look at all those stars!
The Fermi question has been generalised since then with knowlede of what a.i. coupled with nanomanufacturing can do. Basically, we're seeing that a species only a hundred years older than us must be rediculously more advanced! And, 99.9% of the extraterretrials must be thousands if not millions and maybe billions of years more advanced than we are! What I'm doing is saying, what if we met an extraterretrial civilization sometime. What will we ask them? Perhaps, we could ask them, what degree of mathematics are they working on! Could be meet degree 100 cultures? How about a thousand? Hundred thousand? Million!? Billion would be like meeting you know who! Only, you know they're not gods. They're just gods to us!