Monthly Archives: August 2017

A 0-1 law for eclipses

A 0-1 law in probability is a result that says in certain situations, for example, when we consider the asymptotic behavior of sums of independent and independently distributed random variables or the short time behavior of Brownian motion, then all events are trivial, i.e., have probability 0 or 1.

Yesterday I learned that law applies to eclipses. For months we have been told that on August 21, 2017 in Durham there would be solar eclipse that will at its peak at 2:45PM it will cover 93% of the sun. That turns out to be about as exciting as being 93% pregnant or have 93% of a proof. The shade of the trees in our front yard seemed a little darker but the sky never did. Turns out that having 7% of the sun exposed is more than enough to be able to see well.

About a month ago I ordered “eclipse glasses” from Amazon, so I could look at the sun without burning my retinas. However, as I learned about a week ago, the glasses were advertised as  ISO 12312-2 certified, but they were not. Amazon was the one who told me and they sent me a refund, but I ended up without glasses. On the big day I made myself a pinhole viewer by sticking the point of a pencil through a note card. When I held it out I did see a light spot on the ground that looked like a circle with a piece missing, but then I wondered if it was due to the fact that my hole was not round. However, soon after that moment of doubt, I noticed that there were a large number of crescent shaped light objects on the ground. In short, the overlaps between leaves in the trees that made hundreds of pinhole cameras. For the long version see:

For a few minutes I wandered around looking at the light shapes on my driveway and in the street in front of my house before I got bored and went in, leaving my neighbors to wonder no doubt what I was doing wandering around in the street holding my smart phone.

In summary, when 2024 rolls around and the eclipse goes from Texas to Maine, either get yourself to where the eclipse is total or take off for Myrtle Beach where the moon will not block the sun and hotel rooms will be discounted.

Rereading Thurston: What is a proof?

As regular readers of this blog can guess, the inspiration for writing this column came from yet another referee’s report which complained that in my paper “the style of writing is too informal.” Fortunately for you, the incident reminded me of an old article written by Bill Thurston and published in the Bulletin of the AMS in April 1994 (volume 30, pages 161-177) and that will be my main topic.

The background to our story begins with the conjecture Poincare made in 1900, which states that every simply connected, closed 3-manifold is homeomorphic to the 3-sphere (i.e., the boundary of the ball in four dimensional space). As many of you already know, after nearly a century of effort by mathematicians, Grigori Perelman presented a proof of the conjecture in three papers made available in 2002 and 2003 on arXiv. He later turned down a Field’s medal in 2006 and a $1,000,000 prize from the Clay Mathematics Institute in 2010.

Twenty years before the events in the last paragraph Thurston’s stated his geometrization conjecture. It is an analogue of the uniformization theorem for two-dimensional surfaces, which states that every connected Riemann surface can be given one of three geometries (Euclidean, spherical, or hyperbolic). Roughly, the geometrization conjecture states that every closed three manifold can be decomposed in a canonical way into pieces that each have one of eight types of geometric structure.

In the 1980s Thurston published a proof in the special case of “Haken manifolds.” In a July 1993 article in the Bulletin of the AMS (volume 29, pages 1-13)  Arthur Jaffe and Frank Quinn criticized his work as “A grand insight delivered with beautiful but insufficient hints. The proof was never fully published. For many investigators this unredeemed claim became a roadblock rather than an inspiration.”

This verbal salvo was launched in the middle of an article that asked the question “Is speculative mathematics dangerous? Recent interactions between physics and mathematics pose the question with some force: traditional mathematical norms discourage speculation, but it is the fabric of theoretical physics.” They go on to criticize work being done in string theory, conformal field theory, topological quantum field theory, and quantum gravity.  It seems to me that some of these subjects saw spectacular successes in the 21st century, but pursuing that further would take me away from my main topic.

In what follows I will generally use Thurston’s own words but will edit them for the sake of brevity. His 17 page article is definitely worth reading in full. He begins his article by saying “It would NOT be good to start with the question

How do mathematicians prove theorems?

To start with this would be to project two hidden assumptions: (1) that there is uniform, objective and firmly established theory and practice of mathematical proof, and (2) that progress made by mathematicians consists of proving theorems.”

Thurston goes on to say “I prefer: How do mathematicians advance human understanding of mathematics?”

“Mathematical knowledge can be transmitted amazingly fast within a sub-field. When a significant theory is proved, it often (but not always) happens that the solution can be communicated in a matter of minutes from one person to another. The same proof would be communicated and generally understood in an hour talk. It would be the subject of a 15 or 20 page paper which could be read and understood in a few hours or a day.

Why is there such a big expansion from the informal discussion to the talk to the paper. One-on-one people use gestures, draw pictures and make sound effects. In talks people are more inhibited and more formal. In papers people are still more formal. Writers translate their ideas into symbols and logic, and readers try to translate back.

People familiar with ways of doing things in a subfield recognize various patterns of statements or formulas as idioms or circumlocution for certain concepts or mental images. But to people not already familiar with what’s going on, the same patterns are not very illuminating; they are often even misleading”

Turning to the topic in our title, Section 4 is called what is a proof? Thurston’s philosophy here is much different from what I was taught in college. At Emory you are not allowed to quote a result unless you understand its proof.

“When I started as a graduate student at Berkeley … I didn’t really understand what a proof was. By going to seminars, reading papers and talking to other graduate students I gradually began to catch on. Within any field there are certain theorems and certain techniques that are generally known and generally accepted. When you write a paper you refer to these without proof. You look at other papers and see what facts they quote without proof, and what they cite in their bibliography. Then you are free to quote the same theorem and cite the same references. Many of the things that are generally known are things for which there may be no written source. As long as people in the field are comfortable the idea works, it doesn’t need to have a formal written source.”

“At first I was highly suspicious of this process. I would doubt whether a certain idea was really established. But I found I could ask people, and they could produce explanations or proofs, or else refer me to other people or two written sources. When people are doing mathematics, the flow of ideas and the social standard of validity is much more reliable than formal documents. People are not very good in checking formal correctness of proofs, but they are quite good at detecting potential weaknesses or flaws in proofs.”

There is much more interesting philosophy in the paper but I’ll skip ahead to Section 6 on “Personal Experiences.” There Thurston recounts his work on the theory of foliations. He says that the results he proved were documented in a conventional formidable mathematician’s style but they depended heavily on readers who shared certain background and certain insights. This created a high entry barrier. Many graduate students and mathematicians were discouraged that it was hard to learn and understand the proofs of key theorems.

Turning to the geometrization theorem: “I’d like to spell out more what I mean when I say I proved the theorem. I meant that I had a clear and complete flow of ideas, including details, that withstood a great deal of scrutiny by myself and others. My proofs have turned out to be quite reliable. I have not had trouble backing up claims or producing details for things I have proven. However, there is sometimes a huge expansion factor in translating from the encoding in my own thinking to something that can be conveyed to someone else.”

Thurston goes on to explain that his result went against the trends in topology for the preceding 30 years and it took people by surprise. He gave many presentations to groups of mathematicians but “at the beginning, the subject was foreign to almost everyone … the infrastructure was in my head, not in the mathematical community.” At the same time he began writing notes on the geometry and topology of 3-manifolds. The mailing list for these notes grew to about 1200 people. People ran seminars based on his notes and gave him lots of feedback. Much of it ran something like “Your notes are inspiring and beautiful, but I have to tell you that in our seminar we spent 3 weeks working out the details of …”

Thurston’s description of the impact his work had on other fields, is I sharp contrast to Jaffe and Quinn’s assessment. To see who was right I turned to Wikipedia which says “The geometrization theorem has been called Thurston’s Monster Theorem, due to the length and difficulty of the proof. Complete proofs were not written up until almost 20 years later. (which would be 2002, almost 10 years after jaffe-Quinn). The proof involves a number of deep and original insights which have linked many apparently disparate fields to 3-manifolds.”

Thurston was an incredible genius. He wrote only 73 papers but they have been cited 4424 times by 3062 different people. His career took him from Princeton to Berkeley, where he was director of MSRI for several years, then to Davis, and ended his career at Cornell 2003-2012. I never really met him but I could sense the impact he had on the department. Sadly he died at the age of 65 as a result of metastatic melanoma. A biography and reminiscences’ can be found in the

A Rainy Monday in Austin


After two great days of hiking, sightseeing, eating and drinking with my younger son (a CS assistant professor at UT Austin) and his girl friend who works for My Fitness Pal, the 98 degree heat was replaced by a steady rain. Trapped inside our hotel room, my wife read the New York Times and did a crossword puzzle, while I wrote a couple of referee’s reports on papers that were worse than the weather.

While it is not fun to be forced inside by the rain it is a good time to reflect on what I’ve seen while visiting Austin. Saturday afternoon we went to the LBJ museum on the UT campus. He served as president for five years after JFK was assassinated in 1963. Before that he was elected to the House or Representatives in 1937 and to the Senate in 1948.

His War on Poverty helped millions of Americans rise above the poverty line during his administration. Civil rights bills that he signed into law banned racial discrimination in public facilities, interstate commerce, the workplace, and housing. The Voting Rights Act prohibited certain laws southern states used to disenfranchise African Americans. With the passage of the Immigration and Nationality Act of 1965, the country’s immigration system was reformed, encouraging greater immigration from regions other than Europe. In short, the Republican agenda times -1.

On Sunday afternoon, we went to the Bullock Texas State History Museum. The most interesting part for me was the story of Texas in the early 1800s. In 1821 Texas won its independence from Spain and became part of Mexico. Between 1821 and 1836 an estimated 38,000 settlers, on promise of 4,000 acres per family for a small fee, trekked from the United States into the territory. The Mexican government grew alarmed at the immigration threatening to engulf the province. Military troops were moved to the border to enforce the policy but illegal immigrants crossed the border easily. Hopefully the parallel with the current situation ends there, since there were revolts in Texas 1832, leading to war with Mexico in 1834, and to the independence of Texas in 1836.

My third fun fact is a short one: Austin City Limits was a TV show for 40 before it became a music festival. Haven’t seen either one but Austin is a great place to visit.