Category Archives: Reflections

Remembering Krishna Athreya


As I write this, he passed away about a week ago. While the end of a person’s life is a sad time, it also provides an opportunity to reflect on the past. The first memory that came to my mind was of sitting in Sid Resnick’s office in the Stanford Statistics department in 1974 (or 1975.) A group of us got together once a week to make our way through the new book on Branching Processes by Athreya and Peter Ney.  It was a welcome modernization of Harris’ 1963 book that started the subject. There were crisp proofs of the basic facts about Galton-Watson processes, Markov branching processes (with exponential lifetimes), the age-dependent case (general lifetimes), and multi-type branching processes. Over my career I have loaned this book to my graduate students many times to help them learn the subject. Remarkably I still have it but it is a little worse for wear.

#2 on my list of Athreya’s greatest hits is A new approach to the limit theory of recurrent Markov chains which appeared in the Transactions of the American Math Society, a paper that was written with Peter Ney in 1978. Again this is a contribution to an area founded by Ted Harris. While Markov chains on discrete state space are well understood, on a general state space numerous pathologies arise. Harris’ genius was to identify a class of these chains that have a tractable theory and cover a number of examples.

There is an elegant analytical theory described in the book by Revuz. However in 1978 several researchers, including Esa Nummelin who later developed a book based on this approach, had the same idea at the same time. I remember attending a session of talks at the 1978 meeting on Stochastic Processes and their Applications and hearing three talks on the topic. This was devasting for a Ph.D. student in the audience who was working on this for his thesis.

The idea is simple but brilliant: a Harris chain can be modified to have one state that is hit with positive probability starting from any state and having one such state is enough to carry out all the usual theory for the discrete case. Given this hint I am sure you can work out the details for yourself. I was so excited by the idea that I put it in the Markov chains chapter of my graduate text book.

Returning to a more traditional narrative: Krishna Athreya received his Ph.D. in 1967 from Stanford where he worked with Sam Karlin, a legendary probabilist with an impressive pedigree: son of Bochner, who was a grandson of Hilbert, and the mentor for 44 students including Tom Liggett and Charles Stone among many others. Athreya’s thesis topic was Multitype Continuous Time Markov Branching Processes and Some Classical Urn Schemes. Soon after he got his degree Athreya and Karin worked on Branching processes in random environments. Two papers were published in Annals of Mathematical Statistics in 1971 since the Annals of Probability which began in 1973 did not yet exist.

These two papers like many in Athreya’s top 20 most cited on MathSciNet contain a number of ideas that have not been fully explored. An example is the work with his Ph.D. student Jack Dai on random logistic maps. Last but not least, I would like to mention his 1994 paper on large deviations for branching processes which contains material that working probabilists should know. Athreya has left an impressive mathematical legacy that will enrich your life and research if you have the time to read it. It is sad that there will be no more work coming from him, but I hope others who read this will be inspired to continue his work. .

Tom Liggett 1944-2020

Tom Liggett passed away May 12, 2020 in Los Angeles at the age of 76. According to Wikipedia, Tom Liggett moved at the age of two with his missionary parents to Latin America, where he was educated in Buenos Aires (Argentina) and San Juan (Puerto Rico). He graduated from Oberlin College in 1965, where he was influenced towards probability by Samuel Goldberg, an ex-student of William Feller. He went to graduate school at Stanford, taking classes with Kai Lai Chung, and writing a PhD thesis in 1969 with advisor Samuel Karlin. Karlin had 44 students including my adviser Don Iglehart, which means I should call him Uncle Tom.

Tom’s first really impressive result was a 1971 paper with Mike Crandall on nonlinear semigroups.  At about the time this paper was written, Chuck Stone showed Tom a copy of Frank Spitzer’s 1970 paper on Interacting Particle Systems saying “I think you’ll find something interesting in this.” The rest, as they say, is history. In 1972 Tom wrote a paper proving the existence of interacting particle systems using the Hille-Yoshida theorem for linear semigroups. His St. Flour lecture notes published in 1977 helped spread the word about the field to a broad audience of probabilists.

His 1985 book, which has been cited more than 5000 times, helped grow interacting particle systems into a lively and vibrant area.  By the time he wrote his 1999 book, the field had grown so large that he concentrated only three examples: the contact process, voter model, and simple exclusion process. His books and papers are known for their clear and elegant proofs, though for those of us who are not as smart as he was they can take some effort to digest. A fuller account of Tom’s research can be found in the July 2008 article in the IMS Bulletin on his induction into the National Academy of Science.

Tom was an Associate Editor of the Annals of Probability from 1979-1984 and became its editor 1985-1987. He lectured at the International Congress in Math in 1986, gave the Wald Lectures in 1996, and was a Guggenheim fellow from 1997-1998. At UCLA he was administrative vice chair 1978-1981, chair 1991-1994, and undergraduate vice-chair 2004-2006. I departed from UCLA in 1985, but I remember Tom telling me once that you know you are doing a good job as chair if EVERYONE is mad at you.

Tom had only nine Ph.D. students: Norman Matloff (1975), Diane Schwartz (1975), Enrique Andjel (1980) , Dayue Chen (1989), Xijian Liu (1991),  Shirin Handjani (1993), Amber Puha (1998), Paul Jung (2003), and Alexander van den Berg-Rodes (2011). Despite the small number of children his family tree goes deep in Brazil. As they say in the bible Andjel begat Pablo Ferrari, who begat Fabio Machado who combined for a total of 34 descendants. The sociology of how students choose their advisers is mysterious, but as Amber Puha wrote in the report on Tom’s 75th birthday party,(June/July 2019 issue of the IMS Bulleting) she found him to be an ideal adviser.

Over Tom’s career he mentored numerous postdoctoral fellows and young researchers. A glance at his publications since 2000 shows he had too many collaborators for me to list here. While I was at UCLA we talked a lot about math but we did not do much joint work since we had much different styles. It takes me months to years to pursue an idea to solve a problem, but Tom seemed to go from idea to solution and completed paper in a few days. One paper on which I could work at his speed was On the shape of the limit set in Richardson’s model. I was on my way to the Friday afternoon probability seminar at USC when Tom popped out of his office and said “it has a flat edge.”

Two of our other joint papers come from his solving a problem that I was working on with someone else, which I believe is a common occurrence on his publication list. We no longer have access to his agile mind, but you can still see it at work in 106 papers and his two books on interacting particle systems.

Two faces of Hurricane Florence

The title is supposed to evoke images of a woman who is a soccer mom by day and serial killing hooker by night. The coastal part of the state of North Carolina saw the second side of Florence. In Durham, which is 150 miles from Wilmington, we mostly  her softer side.

In the days leading up to Sunday September 9, the storm was hyped by the weather channel and the local news stations. Florence started as category 1 in the eastern half of the Atlantic Ocean but it was predicted that as she reached the warmer weather in the Atlantic, she would grow to category 4 and smash into coast somewhere in the Carolinas (and she did). The message of a coming disaster was reinforced by our neighborhood listserv. People who lived through Hurricane Fran in 1996, saw many trees go down, and lived a week or more without power, and they were not anxious for a repeat performance. Emails were filled with long lists of things we should do in order to prepare.

Monday September 10. At my house we had our semi-annual inspection of the heating/cooling system. It is hard to get work done while the technician is in the house, so I googled “How do you prepare for a hurricane?” The answers were similar to what I had seen on the listserv, with one new funny one: remove all of the coconuts from your trees, since they can become cannonballs in a storm. When the inspection was finished about 11:30 (with no repairs needed!), I went to the grocery store with my little list: water and nonperishable foods. The store was a zoo and water was flying off the shelves, but I came home with 48 quart bottles of Dasanti, canned fruit, canned soup, bread, peanut butter, energy bars, etc.

Tuesday September 11  was the 17th anniversary of a day that changed the world. Donald and Melania visited the new memorial to the plane that went down in Pennsylvania. I came down with a cold and stayed home from school. By this time, there was more weather than news on news. I recall hearing one weatherman pontificate “at this point we are as good with predictions 72 hours out as we used to be at 24 hours.” Then he showed a track that had the eye of a category 1 storm over Durham about five days later. Tuesday afternoon Duke announced that all classes were cancelled as of 5PM Wednesday. , with no classes on Thursday and Friday. UNC and NCState also closed and in a sign of looming disaster cancelled their football games. For UNC this was a blessing. Having lost to Cal and to the East Carolina in the two previous weeks, they were happy to escape from another loss at the hands of U of Central Florida.

Wednesday September 12. The morning weather forecast announced a dramatic change in the track of the storm. It was now predicted to turn left after land fall, hang out at the beach for a couple of days and then head off to Atlanta. The new storm track was great news for Durham but not for Myrtle Beach South Carolina which was given a short deadline to evacuate. Wednesday was more or less a normal day. I met with a graduate student at 11, had lunch, taught my class, and announced the shifted schedule for the homework and exams due to the cancellation of class on Friday.

Thursday September 13 was the calm before the storm.  My wife and I took a walk around the neighborhood in the morning, went out to a Mexcian Lunch at La Hacienda at the northern edge of Chapel Hill, and grilled some chicken for dinner. (In NC barbecuing refers to cooking a large piece of meat over a slow fire until you can pull it apart with your hands. Grilling gets the job done in 15-20 minutes.) Feeling more confident about the future we scaled back from cooking four chicken breasts to have leftovers that could be eaten over the next few days, to making only two.

Friday September 14. Florence made land fall at Wilmington as a category 2 hurricane, which is definitely more than half as strong as a category 4. It was amusing to see several weathermen competing to see who could be filmed in the eye of the hurricane, where the winds suddenly drop to 0. A woman on the neighborhood listserv described it as an eerie experience. Most of the other things that happened along the coast were not at all funny. New Bern was about 30 miles from the coast, but it was on the banks of the Neuse River. Rains plus hurricane winds resulted in flooding that left hundreds of people (who stupidly chose not to evacuate) in need of rescue. Up in Durham things were much more sedate. There was very little wind or rain. However since the weatherman had told us to shelter in place, we stayed in for most of the day, as did most of our neighbors.

Saturday September 15 was more or less a repeat of Friday. Twisting a line from Big Bang in which Penny is talking about here relationship with Leonard. “This is a new boring kind of hurricane.”   I don’t know what normal people do during a hurricane, but it is a great chance to get some work done. On Friday I read a couple of papers from the arXiv and thought up a new problem for one of my grad students to work on. Saturday I decided I would use the lull to finally finish up the 5th edition of Probability; Theory & Examples, so I sifted through emails I had saved from readers and corrected some typos. Not everything I do is math. I watched the third round of the Evian Championship, the fifth major of the LPGA season. In more keeping with more manly pursuits I watched parts of Duke’s 40-27 victory over Baylor in Waco, Texas. The win was remarkable because Duke’s starting quarterback, a junior who might make it to the pros, was out. In addition, I watched Texas beat the USC Trojans, 37-14. It’s not just that my son is a CS professor in Austin. Having been at UCLA for nine years, when Peter Carroll was there, I love to see USC lose.

Sunday September 16. Duke’s severe weather policy (which covers not only the university but also the hospitals) ended at 7AM, so we figured that we had reached the end of the hurricane. We got up and went to the Hope Valley Diner for breakfast at about 7:30. When we first started going there it was called Rick’s. However, the owner got tired of having a restaurant named after her ex-husband. In keeping with the intellectual climate of Durham, our usual waitress is a second year medical student at UNC Greensboro. Weather was light rain as it had been for the last couple of days, so we went to the Southpoint Mall to get out of the house. Being in a Christian region almost all the stores don’t open until noon. At the mall Susan bought some Crocs, and I bought a couple of books that I read before going to sleep. However, mostly we walked around like many of the families who came there with their small kids.

Monday September 17. Our hopes of nicer weather were quickly dashed. It was raining very hard when we woke up. Curiously the direction of the flow was now from SW to NE in contrast with the last few days of SE to NW. Almost immediately there was a tornado alert on TV accompanied with its loud obnoxious noise, and robophone call to tell us of the event, which came a few minutes after channel 14 told us that the warning had expired.  The tornado warning came from an area well north of us, so we weren’t really worried when soon there was a second one even further north. This was too much excitement for Duke, so they cancelled classes until noon, irritating people who traveled through awful weather to get to their 8:30AM classes. Soon after the despair of facing another day of rain set it, the sun came out and Susan and I took a walk.

Tuesday and Beyond. Unfortunately the end of the rain does not bring the end of the misery for people near the coast, as we learned with Hurricane Matthew. Many of the rivers there have large basins (of attraction). Many will only crest Wednesday or Thursday. The Cape Fear River will reach 60+ feet compared to its usual 20. But don’t worry. Trump will be coming soon to inspect the damage. When he came he clumsily read from a prepared statement, that soon we will be getting lots and lots of money. I guess his advisers didn’t tell him that the state is so heavily gerrymandered that he will probably see 10 Republican congressmen elected from 13 districts.

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.