Wednesday, April 28, 2010

Perfect Rigor

Some time ago there was this Science cover plainly announcing that one of the famous and long standing problems - Poincare conjecture is solved. First there is a feeling of joy that human mind has conquered another victory. This problem became another type of problem after the original one was solved. The reason? There were $1,000,000 prize attached to it. Mathematicians almost never come up nowadays with purely original solutions - they are using ideas and methods which they learn from others, even mistakes are useful because they show where not to go in your thinking. 
No surprise there came a controversy - who should receive money - just this one strange guy from St. Petersburg who even did not care about a requirement to publish his proof in "peer reviewed scientific journal" - he simply deposited his proof in - open access to 600,550 e-prints in Physics, Mathematics, Computer Science, Quantitative Biology, Quantitative Finance and Statistics hosted by Cornell University Library on 11 November 2002 with a title: The entropy formula for the Ricci flow and its geometric applications , after this post over the period of 8 months two more posts followed completing the proof.

After this submission mathematicians carefully started to check the proof to make sure that this indeed is the proof - it was not the first attempt to prove geometrization conjecture from which Poincare conjecture would follow but all previous proofs failed to give a complete answer. In the abstract to his paper Perelman already said that he is giving a sketch of the proof - would this should count as a complete proof? The following events made a story to appear in The New Yoker Manifold Destiny - it was prompted by Perelman refusing to accept Fields Medal - highest honor any mathematician could dream off.
Sylvia Nasar, who already had written a book about a strange mathematician John Nash, went to St.Petersburg and tried to interview Grisha Perelman, who was and still is avoiding to talk to press or generally stopped communicating with people except his mother.
This year finally Clay Institute made a decision that the prize $1,000,000 should be awarded to Perelman... but he refused to accept money.

Of course, such story is worth an investigation - as predicting this outcome(not too hard task in his case!) Masha Gessen has written a book /Perfect Rigor: A Genius + The Mathematical Breakthrough of the Century

No, she did not get to talk to Perelman himself or to his mother but she has interviewed number of people who knew Grisha Perelman since he was a child and first exhibited his extraordinary skills in math problem solving. These interviews and author's personal experience paints a picture not only of the strange mathematician but also about the way Soviet mathematical world worked.
This is a book I am reading now. Already on a second page I read a description why young people chose to study mathematics - Gessen writes:
"Mathematics was antithetical to the Soviet way of everything. It promoted argument; it studied patterns in a country that controlled its citizens by forcing them to inhabit a shifting, unpredictable reality; it placed a premium on logic and consistency in a culture that thrived on rhetoric and fear; it required highly specialized knowledge to understand, making the mathematical conversation a code that was indecipherable to an outsider; and worst of all, mathematics laid claim to singular and knowable truths when the regime had staked its legitimacy on its own singular truth. All of this is what made mathematics in the Soviet Union uniquely appealing to those whose minds demanded consistency and logic, unattainable in virtually any other area of study."

Yes, this is exactly why I chose to study mathematics.

I have made it approximately half of the book so far. I have some different views of the picture of Soviet mathematics painted in this book. Author at the beginning (p.12) says that "..and then there were those who almost never became members of the establishment: those who happened to be born Jewish or female..."
Actually it was more than that. For example, I could not defend my PhD thesis in Riga because it was not allowed by Soviet authorities that there would be appropriate scientific board allowed to award scientific degrees. Anybody who did graduate work in mathematics or computer science in Latvia up to 1992 had to find a scientific board which would accept thesis for the defense. Then one should find an opponent - a person who would read carefully your thesis, will try hard to find any mistakes, gaps in your proofs etc. To find an opponent usually was up to the thesis advisor who had stronger connections with respectable scientists in your area of research. It took 3 years(1987-1990) for my thesis advisor to find the scientific board which would accept my thesis for a defense. It was nothing to do with my work itself - it was a time when everything was cracking in Soviet Union, you may find the place you could defend your thesis but then next day this board could be dissolved and your thesis defense is no more valid. That is why I ended up with a degree from a place which I had actually visited twice - once to give a talk in the seminar when they decided to accept my work and the second time during the actual event. But officially my degree states that it is awarded by the Institute of Mathematics of the Academy of Sciences of Belorussia. I had as an original opponent Dima Grigoriev - he was the head of Laboratory of algorithmic methods Leningrad Department of the Steklov Mathematical Institute - form the same Institute Steklova that is mentioned in the book about Perelman. The day before I had to leave for Minsk my advisor Prof. Freivalds called Grigoriev to St. Petersburg to ask in which hotel in Minsk he will be staying. Grigoriev had just returned back from France (he was at that time already privileged to have international contacts and was well known, since 1998 he is the Research Director at CNRS, Lille, France). My case was so unimportant for him (understandably - it is just one unknown graduate student - who cares!), that he had forgotten all about it and said that he is too tired to go to Minsk, and he suggests to postpone it till September, not to mess up nice White Nights in St. Petersburg in June. I was at that time expecting my second child in September, so September was not an option for me. I was standing there, in the office of my advisor - pale and and almost fainting. He had to rush to a meeting, so he told me to sit down, gave me bad quality photocopy (brought from Moscow, the only library you could get access to at least some publications in English) of something to read and told to wait for him and he will sort things out. And then he left, locking the door behind him. Years later he told me that he was afraid I might go out and jump into the river (I did not have such thoughts).

What he gave me to read were extracts from Feynman's "Surely you are joking, Mr. Feynman?. When he returned about two hours later he had found another professor who had agreed to save me and to be my opponent. My thesis were saved.

My daughter is now 19 and likes Feynman's books very much.
Last Saturday I told her what happened just before she was born.

Monday, April 19, 2010

The talk I missed

It was last Tuesday when there was an interesting sounding talk "Building  Rome in a Day" in the afternoon. I wanted to go but instead that morning I was in a surgical care prepared for an operation - you can see me with surgeons initials on my cheek after we both agreed that it is really my right side and he will be performing surgery in that side of my nose. Somebody told me that it is really assuring that doctors are signing there work. Since it is inside I cannot tell if it is a masterpiece or not, I certainly still do not feel all benefits we hoped for. But a question still remains - is colonoscopy signed also? The surgeon said he will be done in 15 minutes and the anesthesiologist said I will be awake 15 minutes later. I took a proper nap afterwards and woke up only good two hours after this surgery. So I may say that I overslept the talk I was interested in.
Still I looked up what it is about "Building Rome in a Day". The talk was presented by Noah Snavely, assistant professor in Computer Science Department, Cornell University. His research interests are in computer vision and computer graphics, and in particular in recovering 3D structure from large community photo collections for use in graphics and visualization. 
The project he was giving a talk about is being developed by the team whose other members are Sameer Agarwal, Ian Simon, Steven Seitz and Richard Szeliski.
There are so many people taking pictures of Rome. If one clicks search term Rome in Flickr, then search will give about two million answers. Quite a choice! And I never entered my photos of Rome in Flickr. This actually reminded me that I never sorted them after downloading on my laptop. I looked them up and choose some for posting here. What these guys are doing is harvesting photos on web and then making 3D reconstruction of entire city.
This is how I saw Coliseum, thousands of other people have pictures of it from a thousand different angles.

 When all these pictures are combined then computer can do a ">3D reconstruction of Coliseum.

This new system is called Photo City. The original idea got dubbed as Photo Tourism and now it is commercialized by Microsoft. (This makes me a little sorry that hyperbolic plane can be crocheted but not commercialized on computers.) Researchers actually are trying to make this system to become a social game when teams can compete. Actually the first game is ending tomorrow between Cornell University and University of Washington about reconstructing campuses.
So far these reconstructions are more like a drawings (at least the ones I looked up in connection with their paper Building Rome in a Day

May be some day this social game will extend to add mood to these 3D pictures - whether it is walking across a bridge at night, seeing Campo di Fiore on a very rainy day or to capture finally some sunshine in Rome.
So far computers have not reached that amazing ability shown by Stephen Wiltshire who draw a 16-foot panoramic view of Rome after taking one helicopter ride above the city. I was in Rome at the time when I was writing my book, so some of pictures taken there were used to show some mathematics in architecture of famous buildings like this double spiral staircase in Vatican museum.

     But some places from that trip I even do not want to see reconstructed like the enormous summer residence of Roman emperor Hadrian - Villa Adriana. I like it as it is - this reminder that all your wealth and power one day will be only ruins. May be he had that thought himself when shortly before his death Hadrian wrote this poem (in Latin but I will quote its version in English):

Roving amiable little soul,
Body's companion and guest,
Now descending for parts
Colourless, unbending, and bare
Your usual distractions no more shall be there...

Sunday, April 11, 2010

Math and Design meets again!

This is my favorite time to walk on Cornell campus. Cherries on a path by Olin Library do not stay in bloom for long, I always try not to miss those couple days. Yesterday was a perfect day to see them.

 The reason to go to campus on Saturday was Mathematics and Design workshop, organized by Susan Ashdown and Van Dyk Lewis and led by Bill Thurston.

First we had a little glimpse "behind the scenes" of Paris fashion where geometry was taken on catwalk. We had a chance to see some pictures from the show and the reception in Dai Fujiwara's studio. Bill Thurston was wearing the famous jacket designed for him by Fujiwara. The jacket was noted by all journalists when they reported about the Paris fashion show. Susan Ashdown pointed out an interesting detail about orange peels - Bill Thurston often in his classes uses orange peels or cuts the edge of a leaf in order to better explain a concept of surface curvature for those who learn about it for the first time.
When Thurston and Fujiwara started to collaborate, Fujiwara told that orange peels are his favorite tool also -  for design classes! He used them to show a concept that a dress can be made all in one piece - if you carefully peel an orange, you get one long peel that is almost flat. Of course, mathematicians can see that as an example that a sphere which has positive curvature locally can be approximated with a plane. But such verbal explanation for those unfamiliar with differential geometry language sounds intimidating while with the orange peel concepts become clear visually.
Bill Thurston had brought with him a lot of things - as he puts it - "toys to play with". He said:
"Mathematics is all around us, and we are using some mathematically complicated things not noticing it. When we are putting on T-shirt, we are not thinking how difficult it could be to describe this action mathematically, we just take it for granted. Mathematics is to stimulate your imagination. Unfortunately it is missing in most of math classes." Susan Ashdown sadly intercepted: "Sometimes it is missing from design classes too."

Bill Thurston passed out templates to cut and explained how to assemble tools to measure curvature. He said that he was excited finding connections between design ideas and mathematics, and finding that concepts are not really so far apart as people might think:
"I had a lot of fun thinking about these designs and playing with different forms." I have to admit that we all had fun too in this workshop and hopefully students will come up with some new unexpected design ideas.
Bill Thurston explaining how to measure curvature on a body.

Susan Ashdown and Bill Thurston had done some homework earlier by constructing "one piece dress "with seems in unusual places.
Here it is when taken off from a dummy. It is clear that shape is so unusual and non-recognizable that in order to be able to put it back, one has carefully mark where is neck, where is armhole etc.

Then it was time for hands-on part - each group had to figure out how to fit geometric forms (which were made movable) on a body. This was really figuring out about different curvatures - where it is flat, where positive, where negative, how to measure it.

Just a proof - I was there too to play :-)    
In every real man a child is hidden that wants to play.  ~Friedrich Nietzsche

My childhood may be over, but that doesn't mean playtime is.  ~Ron Olson

He who can no longer pause to wonder and stand rapt in awe is as good as dead; his eyes are closed.  ~Albert Einstein

Thursday, April 8, 2010

How to make simple things complicated

Today a friend sent me an article from The Mathematics Magazine, April issue with a note saying that there are still people who never heard about crocheted hyperbolic planes.
Here is an abstract:
Drawing a Triangle on the Thurston Model of Hyperbolic Space 
by Curtis D. Bennett, Blake Mellor, and Patrick D. Shanahan
pp. 83–99
In looking at a common physical model of the hyperbolic plane, the authors encountered surprising difficulties in drawing a large triangle. Understanding these difficulties leads to an intriguing exploration of the geometry of the Thurston model of the hyperbolic plane. In this exploration we encounter topics ranging from combinatorics and Pick’s Theorem to differential geometry and the Gauss-Bonnet Theorem.

Well, I was surprised about their difficulties since they have a photo with an actual paper model. This model is made gluing together 7 equilateral triangles at each vertex. It is one of the possible approximations of the hyperbolic plane, not the most convenient one to make and with some roughness, but it give an idea how hyperbolic plane looks. The authors now say that they had difficulties drawing straight line. What is difficult about that? Why don't you just fold your model and you have a geodesic or straight line in the hyperbolic plane?

It is true that this not the way most mathematicians like to describe things, then some artificial,complicated theories are being invented. I have to confess that I did not read the whole article very carefully and I apologize for my ignorance.

 It was not because I think the method I use to explain the straight lines in hyperbolic planes is the best or only one, but because I have lost interest in most of papers in mathematics ( I guess I never had...) published in specialized magazines which are read  (if at all) by some highly specialized others. The Mathematics Magazine is a publication of MAA and is aimed to be more expository type, and this particular paper is aimed to the students. May be students will enjoy it or they will think that simple solutions (like folding a paper to construct a straight line) are not mathematical enough, explanations should be lengthy and with the greatest number of possible theories involved...
Or may be true mathematicians are so appalled that some notions can be explained using crochet that they have decided finally "to put the theory on a strong basis"?

I am happy that I do not have to write any mathematical papers anymore. Once I was sitting with two mathematicians in the famous Tea Room in the Institute for Advanced Studies in Princeton having afternoon tea. Both men are really good mathematicians and I think about them really highly. They both had collaborated on some mathematics, and this collaboration had produced some new result. They both energetically started to discuss in how many different ways they can present it in order to publish in as much publications as possible because each of these publications would have a limited number of readers anyway. I innocently asked - why do you care about the number of publications? why don't you put all of your results together (and they do have a lot) and publish a book where it all will be in perspective and will show all connections. They looked at me like I had sinned in sacred walls of the Institute, and then one of them said: 'A book will count only as one publication, while this way the number will grow significantly. Imagine how big this number will be in my obituary!"
I don't think that at that point I will care what the number is...

Today I was working on a lecture I have to present in June, and found among some notes these quotes from Jacques Derrida:

         I have always had school sickness, as others have seasickness. I cried when it was time to go back to school long after      I was old enough to be ashamed of such behavior.

         Still today, I cannot cross the threshold of a teaching institution without physical symptoms, in my chest and my stomach, of discomfort or anxiety. And yet I have never left school.

Isn't this a normal reaction on too much formalism? Did these experiences made Derrida think of destructivism?

Sunday, April 4, 2010

Nature by numbers

This is a still from a short movie Nature by Numbers - the topic is nothing new but what I found interesting is that  on the webpage there are links to the process of creating these animations.

I found another movie Geometry of Nature that is quite different - it talks about Henri PoincarĂ©'s mistake that led to the discovery of chaos. This movie is produced by BBC Two in connection with Open University.

Here is a little PBS podcast about Nature's Hidden Geometry - I liked there two things -

  • other people than me also saying that  formal geometry can be used only to the things humans have made, nature uses different geometry, and
  • mathematics and art is not so far from each other.
If this intrigues you, the whole program is called Fractals and can be seen in 5 parts.

Still I do not know any mathematics that would explain volcano eruption. I keep following the recent eruption in Iceland - volcano erupted the day after we left... I used some photos from that trip in previous blog entries. Now I can only imagine myself seeing scenes like this:
These are pictures from IcelandReview online (where you can find also amazing videos and other pictures) and used by permission.

Thursday, April 1, 2010

Spring at last!

Happy Spring! First time this year I could run out after my exercise hour in pool with wet hair and come home without worries that I may catch cold or sinus infection. The soil in the garden still is too cold and too wet to do any significant gardening, so I used the sunlight to take pictures of the project I finally finished yesterday - my long vest. I started it last fall after Kaffe Fassett talk in Cornell. It was amazing coincidence - he gave a talk the same day as Dai Fujiwara. It was very inspirational design day indeed. The local yarn company Schaefer Yarn was selling yarns specially designed in colors of Bloomsbury group (their exhibit was in Johnson Art Museum). The yarn was quite fine (designed for socks), so it took me all this time to finish the project between crocheting, of course. I always need some knitting project to balance crocheting.