Veblen a "Towering Figure" in Mathematics

It's gratifying to see Oswald Veblen being more widely recognized on the internet for his contributions to American mathematics. There are still many tellings of history in which Veblen remains hidden, however. Meeting a retired Princeton-based physicist/violinist recently, I naturally thought of Einstein and told him I was researching Oswald Veblen's influence in bringing Einstein to Princeton. He said emphatically that it was the Bambergers who brought Einstein instead, through their funding of the Institute for Advanced Study. He then mentioned Richard Courant, and gave credit to New York University for bringing this great jewish mathematician to America after he was displaced from Gottingen by the Nazis. 

But behind both of these stories of brilliant and impactful immigration is Oswald Veblen, who was quietly instrumental in bringing many displaced mathematicians and physicists to America. A succinct, attractively rendered telling of the story, called "Collaboration and Companionship," repeatedly mentions Veblen's involvement in bringing Einstein, Hermann Weyl, Emmy Noether, John von Neumann, Kurt Godel, and Richard Courant to the U.S.

That webpage links to another entitled "Towering Figures," in which brief stories are told of four "key individuals in pioneering and continuing the growth of the American Mathematical community": J.J. Sylvester, Felix Klein, E.H. Moore, and Oswald Veblen. 

Veblen had a connection to each of the three other "towering figures" who preceded him. His father, Andrew, would surely have studied with J.J. Sylvester at Johns Hopkins before moving to the University of Iowa in 1883 to teach physics and math--the same year Sylvester returned to Europe. And surely his father would have taken a 13 year old Oswald to hear Felix Klein speak in 1893 at the International Mathematical Congress held as part of the Chicago World's Fair. Oswald went on to study with E. H. Moore in Chicago, before moving to Princeton. 

A couple asides: Biographies of Sylvester and Klein mention their work to encourage women to pursue careers in mathematics, as did Veblen.

On a more autobiographical note, I expect the Veblens would also have witnessed the 40 inch refracting telescope on display at the 1893 Chicago World's Fair. When I was growing up 70 years later, my father, W. Albert Hiltner, was director of Yerkes Observatory, where that largest of refracting telescopes still functions beneath the big dome. The Collaboration and Companionship story also describes how the Manhattan Project convinced President Hutchins of the University of Chicago to "throw massive resources into the reorganization" of the physics and math departments near the end of World War II. That funding and resources may be why I grew up where I did, as my father was hired by the U. of Chicago astronomy department around that time. 

Math and Motion: When Our Limbs Become the Roots of Our Tree of Thought

I was told that Oswald Veblen did his best thinking while chopping wood. Similarly, a daughter of Joe Kohn, the great mathematician who died recently, told me that her father very much disliked reaching the point in aging when he could no longer walk, because walking was so important to his act of thinking. Though I'm not a mathematician, I've had insights and ideas come to me while washing dishes, chopping wood, or more generally when my hands and mind are working together. There is something in the connection between body and mind, as if our arms and legs were the roots of the tree of thought. Of course, the scientist side of my brain is immediately looking for evidence to support or contradict this thesis. Do those who have lost the use of their limbs also have eureka moments?

Recently I felt the urge to ask a mathematician if he did his best thinking while bicycling. The prompt for the question is something of a story. I was walking down the hallway of a medical building, fresh from my annual dermatology exam. There was an older couple walking just ahead of me, and they were disagreeing as to who would drive the car home. He wanted to drive, but she was insisting that she drive, given that he had just received some medication. He said he was just fine, however, and their back and forth seemed headed for an argument. In retrospect, it was as if I were a kid listening to his parents. Something came to mind, and I decided to go with it. Wishing to defuse the situation with some humor, I blurted out from a few feet behind them, "Okay, how about I drive you home?" 

They turned around, still in stride towards the door to the stairwell, and the man responded "Do you drive a stick shift?" This was music to my ears, as I love driving a stick shift. I told him about the '94 Ford Ranger I use for my work at Herrontown Woods, and as we headed down the stairs, we talked stick shifts we'd known, like the "3 on a tree" in my '63 Chevy station wagon--a car that was emphatically beige because my mother was big on beige. We continued to talk as we left the building. I said that when driving a stick shift, the driver must listen closely to the engine. That delicate interplay between clutch and accelerator--it's like a dance with the machine. (Though I didn't mention this, maybe that relationship is why my ear is also attuned to the sound of other machines, like when the furnace kicks on, and the implications for my personal contribution to climate change.)

As we reached my bicycle (I clearly had not thought through the logistics of driving them home), he recounted having taken a bad spill on his bike. He had been biking on Canal Road and gotten so caught up in mathematical thoughts that he didn't notice a pothole. 

My ears perked up when he mentioned math, given of course that I am working to restore Oswald Veblen's house. "Are you a mathematician?," I asked. He responded affirmatively and told me his name. His story fit well with a video a botanist friend of mine had posted a week before on facebook, in which my friend is riding his bike along a broad trail, one hand on the handlebar, the other holding the camera, pointing at different plants and calling out their latin names as he passed by. The video ends abruptly when he falls off of his bike, breaking a couple bones in the process. He's on the mend, but high-speed botany, like high speed math, has its risks.

Biking home, I wished I had asked the mathematician whether he does his best thinking while riding a bike. I looked him up, found his home page, and sent him an email. His response? "I haven't done much good Math thinking while biking. The accident was more than 20 years ago and it taught me to concentrate on biking when on the bike."

Not the answer I was looking for, but all for the best, and a lot more fun than an argument over who is going to drive.

The Humanity Behind Science -- Mathematician Ingrid Daubechies

One role the Veblen House could play in a town so filled with vaunted institutions of higher learning is to make scientists better known as people. They tend to be presented in the media as remote figures stripped of personality, speaking about weighty subjects from their domain of rationality. As civilization has increasingly veered towards self-destruction, first with the nuclear arms race and now with a headlong radicalizing of the climate, scientists are primarily seen in the role of sounding the alarm. Caring people in a careless world, they are cast in the predictable, sober role of expressing concern while the rest of humanity goes its merry, scary way. Having grown up among scientists, I was able to see them as people who brought incredible perseverance, creativity, joy and humor to their study of the nature of things. 

One piece of recent writing that captures that side of science was Siobhan Roberts' portrait of Belgian mathematician Ingrid Daubechies, who in 2004 became Princeton University's first female full professor of mathematics. In her career, first in Belgium, then in the U.S., she has developed mathematical tools that have helped make possible the visual feast of the digital age, and she has even shown how artificial intelligence could be used to conserve and restore iconic artworks.

Entitled "The Godmother of the Digital Image," the article gives a portrait of Daubechies as optimistic, generous, creative, and fun-loving. 

The optimism perhaps comes from being able to solve problems others cannot:

"She revels in finding meaningful and practical problems — and solutions — where other mathematicians assume there are none."

And there's generosity in the way she uses this capacity to help her students:  

“I called her the deus ex machina adviser,” says Cynthia Rudin, a Duke computer scientist who is one of her former Ph.D. students. “When you’re in the depths of despair, your project has crashed and burned and you have almost proven that what you’re trying to do is impossible, Ingrid comes along and pulls you out of the pit of doom, and you can keep going.”

A mathematician can more easily think, and act, outside the box. How many people do you know who would throw a big shindig on their 64th birthday rather than their 65th, because 64 is a more compelling number, being a power of 2? 

"Daubechies booked a venue, a caterer, a troupe of majorette dancers known for farce — and then at the party made a surprise appearance in the baton-twirling cancan line, disguised in makeup and a tutu."

Along with her periodic "cathartic weeding in her garden," one of Daubechies' mottos, “Math can help! As always!”, reminds me of a letter about Oswald Veblen from 1992 in Princeton's Town Topics newspaper:

He and his friends spent Sunday afternoons clearing the poison ivy from the bank of the canal. He advocated washing vigorously with yellow soap after this. "You bet! "as he was prone to say. - ELIZABETH G MENZIES 926 Kingston Road

The letter writer, Elizabeth G. Menzies, turns out to have been the first female official photographer for Princeton University, who made a name for herself with a photo of Albert Einstein that appeared in a 1939 issue of Scientific American. 

Roberts' article also delves into Daubechies' efforts to help other women overcome institutional bias and gain deserved prominence in the field of mathematics. Veblen took on this role in the 1930s, helping the great mathematician Emmy Noether get positions at Bryn Mawr College and the Institute for Advanced Study. 

While it's easy to see mathematics as intimidatingly complex, it is not uncommon among mathematicians of Daubechies' caliber to find in their work an underlying elegance and beauty. Describing how she found a practical application for wavelets without sacrificing the beauty of the original concept, she said,

“It is something that mathematicians often take for granted, that a mathematical framework can be really elegant and beautiful, but that in order to use it in a true application, you have to mutilate it: Well, they shrug, That’s life — applied mathematics is always a bit dirty. I didn’t agree with this point of view.”

Thanks to a gifted writer like Siobhan Roberts, people can begin to see how much humanity and passion a scientist can bring to her work.

Other posts about great women in math and science:

2017: A Memorable Year for Women Mathematicians

Vera Rubin: The Courage of Her Curiosity

Math Writ Large in Hidden Figures

2017: A Memorable Year for Women Mathematicians

Before the cascading events of this fall put the national focus on women in the workplace, it was already a significant year for appreciating women and their contributions to mathematics.

Katherine Johnson, Dorothy Vaughan, and Mary Jackson
The movie Hidden Figures, released just days before 2017 began, tells of the critical role three black female mathematicians played in the U.S. space program. A hunch and some research uncovered a Veblen connection to that wonderful story.

This hidden role of women has played out repeatedly in the history of mathematics and computers. It's commonly said that though early computing machines were largely designed by men, it was the women who figured out how to actually run them. Women with keen mathematical ability had already proved themselves as the first computers, whether for NASA or for Veblen's ballistics work for the U.S. military during the World Wars. A little research yielded a story from my little home town of Williams Bay, Wisconsin, home of Yerkes Observatory. According to Wikipedia, Chandreskhar, a colleague of my father's and a future Nobel Prize winner, in the 1940s

"had used top performing female high school students from Williams Bay, Lake Geneva, Elkhorn and Burlington, Wisconsin to calculate immensely difficult mathematical equations entirely by long hand, and found that their abilities and vigilance were unparalleled."

Cathleen Synge Morawetz
There's also a Veblen connection to another great female mathematician, Cathleen Synge Morawetz, who died this past August. A writeup on the American Mathematical Society website tells of her distinguished career, including many firsts for a female mathematician. At age 90, she and her husband made a donation to significantly increase funding for the Veblen Prize in geometry, out of gratitude for a kindness Oswald showed to her father a long time ago. The touching story is taken from an AMS article.

"This long association with the AMS played a part in her decision, in celebration of her 90th birthday last year, to make a major donation to the Society. The gift from her and her husband, Herbert Morawetz, significantly increased the size of the long-underfunded Oswald Veblen Prize Fund, bringing it on a par with other AMS prize funds. Veblen was a good friend of Morawetz’s father, John Lighton Synge. How this friendship began is recounted in Synge’s article “For the 100th birthday of the American Mathematical Society”, which appeared in A Century of Mathematics in America: Part 1, edited by Peter Duren (AMS, 1988). The article is a written version of a talk Synge gave at the AMS Centennial Celebration in 1988. 

In the article, he recalls an AMS meeting he attended in December 1921 in Toronto. He had come to Toronto from Dublin the year before and found few colleagues with mathematical interests similar to his own. His encounter with Veblen at that AMS meeting and the kindness and consideration Veblen showed were important to Synge as he made his way in mathematics in a new land. At the time Synge wrote the article, he was 91 years old, the same age his daughter is now. One hears in his article an echo of the lively intellect and warmth of Cathleen Synge Morawetz. For those qualities and for her many contributions to mathematics and to the profession, she has evoked great fondness in the mathematical community."

After getting contact info from friend and distinguished mathematician Joe Kohn, I tried to reach out to her this past March to let her know about our initiative to save the Veblen House, but was unable to reach her. A video interview of her, late in life, tells of how her mentor at NYU, Richard Courant, had also been a supporter of Emmy Noether (see below) in Goettingen, how becoming pregnant actually hastened her getting a PhD, and how she was for a time a trustee at Princeton University. It appears that Veblen assisted not only Moravetz's father, but also played a quiet role in laying the groundwork for her own career. A book named "Hilbert-Courant", by Constance Reid, describes Veblen's role in encouraging Courant to take a position at New York University in 1933.

Maryam Mirzakhani

2017 also saw the loss at age 40 of mathematician Maryam Mirzakhani. She was the first and thus far only female mathematician to win the Fields Medal. Born in Iran, her brief but brilliant career as a professor began at Princeton before moving to Stanford.

Marina Ratner
She died this past July, after defying conventional expectations of a mathematician's trajectory by doing her best mathematical work in her later years, after turning 40.

Emmy Noether
Last but not least, and more appropriately first and most, Emmy Noether gained much deserved recognition in 2017 as part of a historical initiative at the Institute for Advanced Study entitled A Refuge for Scholars. Like Einstein, who called her one of the greatest mathematicians of all time, she was displaced by the Nazi takeover of Germany. It was Veblen who took the lead in finding her a position at Bryn Mawr and inviting her to be a Visitor at the IAS. Despite her talents being commensurate with male mathematicians of the time, or of all time, her career options were limited, as was her salary. She died in 1935 after complications from surgery. This poster about her has been on display at the IAS Fuld Hall.

Mathematics and Memory Meet in Tucson, AZ

This post is prompted by a texted photo from a friend of mine, Carl Hildebrandt. It wasn't a casual selfie. He took it only after riding his bike all the way up a mountain outside Tucson, AZ to reach Kitt Peak Observatory, where there's a telescope named after my father.

Carl, too, has math and science flowing through his veins. His grandfather Theophil Henry Hildebrandt's career as a mathematician paralleled Veblen's in many ways. Hildebrandt arrived at University of Chicago for graduate work just as Veblen was leaving for Princeton, had the same advisor, E.H. Moore, then went on to chair the math department at University of Michigan from 1934-57, and serve as president of the American Mathematical Society from 1945-6. Carl's uncle Theodore spent 1947-8 working at Princeton with von Neumann, Goldstine, and Julian Bigelow on the IAS computer project.

This snippet from Century of Mathematics in America portrays the University of Chicago as the academic center of gravity for American mathematics in 1900, spawning the PhD's who would then go forth to lead the growth of mathematics at Princeton, Harvard, Michigan and other institutions in the 20th century.

Concurrent with this fertile production of many of mathematic's future leaders, Chicago also built Yerkes Observatory in Wisconsin, which in 1897 was far from city lights. Yerkes, where I lived while my father was an astronomer there, had many qualities similar to Princeton's Institute for Advanced Study--an academic enclave surrounded by fields and woods on the outskirts of a small town. The observatory remained a center of cutting edge research until the second half of the 20th century, when more advanced telescopes sprouted in drier climes, such as Kitt Peak, AZ.

Tucson drew not only astronomers like my father, but also two of the Whiton-Stuarts--the family that built what would later be called Veblen House. They had already lived in Prescott, AZ, decades earlier, where Jesse spent his days on horseback herding cattle--a change of pace from running his high-end real estate firm in Manhattan. His wife, Mary (Marshall Ogden) Whiton-Stuart later moved back to Arizona to live in Tucson for the last 13 years of her life, as did her daughter, Silvia, for portions of that time.

Astronomy and the Whiton-Stuarts came together in the Nov. 16, 1964 issue of the Tucson Daily Citizen, which included Mary's obituary and, elsewhere on the same page, an announcement:

"To Speak At Dinner--Meet Dr. Gerard Kuiper, director of the Lunar and Planetary Laboratory at the University of Arizona, will speak Sunday at the annual Compact Day dinner meeting of the Society of Mayflower Descendants in the State of Arizona." 

Gerard Kuiper was a colleague of my father's at Yerkes, and Mary may well have been a member of the Society of Mayflower Descendants. They might have met had she lived a little longer.

Thanks to Carl for prompting me to weave all these threads together.

Weyl Fermion Discovery--The Veblen Connection

When I saw the headline "After 85-year search, massless particle with promise for next-generation electronics discovered" among the Princeton University website's top stories, my response was a bit cynical. So many headlines tease us with important discoveries that could transform society, only to reveal deeper into the text that the discovery won't find practical uses for decades, if ever.

The text reporting this discovery, however, got more interesting as it went along. Turns out the massless particle was first proposed "by the mathematician and physicist Hermann Weyl in 1929." If you look at the Robert Nolan biography of Oswald Veblen, you find that Veblen "was largely responsible for selecting the other members of the original faculty (of the Institute for Advanced Study): James W. Alexander II, Albert Einstein, John von Neumann, and Hermann Weyl."

The Princeton University article goes on to say:

"Weyl fermions have been long sought by scientists because they have been regarded as possible building blocks of other subatomic particles, and are even more basic than the ubiquitous, negative-charge carrying electron (when electrons are moving inside a crystal). Their basic nature means that Weyl fermions could provide a much more stable and efficient transport of particles than electrons, which are the principle particle behind modern electronics. Unlike electrons, Weyl fermions are massless and possess a high degree of mobility; the particle's spin is both in the same direction as its motion — which is known as being right-handed — and in the opposite direction in which it moves, or left-handed."

Maybe someone's imagination, massless and possessing a high degree of mobility, will find useful applications for the Wehl fermion much sooner than later, in which case we will need to invent a dance in which we all spin both directions at the same time.

Veblen's Mathematical Genealogy

I was talking to a classical pianist the other day, and told her I'm three generations removed from Bela Bartok. She replied that she's 6th generation Beethoven. My link to Bela Bartok stems from having studied piano with Michele Cooker, who studied with Gyorgy Sandor, who was a student of Bartok's. Though Michele taught me piano rather than composition, maybe some quality of Bartok rubbed off, because I went on to write many short piano pieces for my students, in the spirit of Bartok's Mikrokosmos.

Similar lineages are traced in the world of mathematics. Jon Johnson, who's on the board of Friends of Herrontown Woods, sent me this link documenting Oswald Veblen's mathematical descendants. By this count, Veblen had 16 students and 10,126 descendants. The second number will steadily increase as descendants advise each new generation of mathematicians. The majority of Veblen's mathematical descendants can be attributed to three of his students, Alonzo Church, R.L. Moore, and John Whitehead. That Veblen played the advisor role for just 16 students must owe in part to his having left Princeton University to join the Institute for Advanced Study, which did not have the traditional advisory relationship between faculty and graduate students.

Turing, Veblen, Bicycles, Movies and Theater

I'm posting with permission this email from Steve Kruse, who among many other interests is a strong advocate for better bicycle routes in Princeton. He offers all sorts of interesting links and connections related to Alan Turing and the movie about him that's showing in town. Other recent events related to Turing in the Princeton area were a talk by author Andrew Hodges ("Alan Turing: The Enigma") this past December at the public library, and a Passage Theatre performance I happened to catch by the extraordinary actor June Ballinger of a solo piece she is developing about her mother, who worked for Turing at the Bletchley Park codebreaking center.

Various themes associated with Veblen House can be found in Kruse's email. First of course is Veblen's role in building the Princeton math department that attracted Turing to town for two years. There's also the reference to prefab housing, which the Veblen House is a unique example of, the house-in-the-woods theme of Snow White, and the bicycles we hope will one day have good routes out to Herrontown Woods from town.

Steve Kruse's email:

I recently watched "The Imitation Game" at the Garden Theater,

and can recommend it. As a computer engineer, and having 

seen "Breaking the Code" on Broadway many years ago, I already

knew quite a bit about Alan Turing's triumphal/tragic story. Plus

my parents both grew up during WW2 in coastal towns where

convoys, escort warships, and long-range patrol planes were part

of the everyday scenery; our cottage had been a prefab naval hut.

Turing spent 1936-38 in Princeton, obtaining his doctorate in

mathematics. He resided at 183 Graduate College and likely

spent a substantial amount of time in Fine Hall, which has

since been renamed Jones Hall. His supervisor was Alonzo

Church, who was recruited by Oswald Veblen (no relation),

lived at 30 Jefferson Rd and is buried in Princeton Cemetery.

Turing defended his Ph.D. thesis in May 1938, at a time when 

his favorite movie, Snow White and the Seven Dwarfs, had

already been playing in the cinema for several months. Since

the Garden Theater opened in September 1920, there's a

chance I watched the Alan Turing biopic in the same space

where Turing watched and was inspired by the Disney movie.

However, since Turing actually wrote most of his thesis while

back in Cambridge, this coincidence seems almost as unlikely

as the odds that a machine-assisted human would crack Enigma.

There are several anecdotes with regard to Alan Turing and

the bicycle. One concerns cryptography, as it relates to the

links of a bicycle chain, as told in Understanding the Enigma:

   "Cryptonomicon by Neal Stephenson is an interesting and 

    informative novel tying together different generations of 

    cryptography. A passage I found most interesting was in 

    the chapter ‘Cycles’. In this chapter, Stephenson expands  

    on the fundamental mathematics behind the Enigma machine: 

    modular arithmetic. Stephenson compares modular arithmetic

    to Turing’s bike. For some reason, Turing’s bike has a rear wheel

    with one bent spoke and a weak chain link. When the spoke comes

    into contact with the chain at a certain position the chain will fall apart.

Additional details are given in That Strange Bicyclist, Alan Turing:

   While working at Bletchley Park, Alan would use the bicycle to 

   commute to work as well as to get around Cambridge. The bike, 

   however, was an old and defective machine. It also had an interesting 

   problem. As you pedaled it, every so often, the chain would pop off 

   and disengage from the chain ring. Every time this happened, he had 

   to hop off the bike and put the chain back on. [...] He loved his dying 

   bike and would not give it up for something better. In fact, he enjoyed 

   riding such a poorly functioning machine that no one else could. So how 

  did he ride it? Well, legend (from reading an article by Ian Stewart in 

  Nature) has it that he chose the most tortuous path to devising a 

  solution for the problem.     

  [...] That's not all. Turing had a bad case of hay fever allergy from an early 

  age. He rationalized that to filter pollen away from irritating and exacerbating

  the allergy, he would strap a gas mask on his face while riding his bicycle in

  town, even in the rain. He was indifferent to what others thought about this practice.

In cycling lingo, you eventually decide it's more cool to say "I shipped the chain"

than "my chain has fallen off". So whenever this happens, remember Alan Turing

and all those ships he kept from going to the bottom of the North Atlantic ocean.

Veblen, Reidemeister, Colin Adams, Knots, Sailing, Topology and Pi(e)

An interest in knots and Veblen lured me to a lecture on the mathematics of knots two weekends ago at the Princeton Plasma Physics Laboratory (PPPL). One of my ideas for an offbeat daycamp experience at the Veblen House is to reintroduce knot tying to the younger generation, whose exposure to knots may not have advanced beyond tying shoes.

Little did I know that I would witness a mathematics professor being very nearly devoured by a great white shark during his lecture entitled "Blown Away: What Knot To Do When Sailing." Part of the PPPL's Science on Saturday Lecture series, the lecture explored the mathematics of knots through the travails of two characters--a Texan and an Englishman. Colin Adams, of Williams College in MA, was the mathematician/storyteller.

The tall tale he told was of being in a sailing race off the coast of England. In the midst of the race, he discovers there's a knot in the sheet (a sheet, as everyone knows, though I had forgotten, is a sailing term for one of the ropes on a sailboat), preventing him from coming about. Adding to the dilemma, the "sheet hit the flan" that he had brought along for a snack, making it too slippery to handle. He dips the sheet in the water to clean it, a great white shark grabs the rope, and he is suddenly confronted with the question of whether the knot can be undone without having one end of the sheet free.

This segued into an inquiry into how to analyze knots visually and mathematically, beginning with Kurt Reidemeister, who in the 1930s developed what Adams calls three types of "Reidemeister Moves". Knot to be undone, Adams eventually wrestles the rope from the shark, unties the knot, and sails into the sunset.

Reidemeister, it turns out, was a German mathematician who was transferred to a less prestigious university from his post in Konigsberg in 1933, due to his opposition to the Nazis. A search of the internet reveals a Veblen connection: Oswald Veblen wrote a letter of support for his fellow geometer Reidemeister (p. 89 in this online book). The strong connection between mathematics and the arts that is a recurrent theme at the Institute for Advanced Study can also be found in Reidemeister, who "wrote about poets and was a poet himself. He translated Mallarmé."

In another parallel with Colin Adams' thinking: An idea of mine for the Veblen House was to name the farmstead either the Pi Farm or the Pie Farm. The symbol for pi, after all, is shaped like a house, and most everything a farm grows can be put in one sort of pie or another. Turns out that Colin Adams co-wrote a book entitled The Great Pi/e Debate, in which they answer the question, "Which is the better number, e or pi?" Princeton has an annual Pi Day on March 14, to celebrate Einstein's birthday.

Another book by Colin Adams that combines math and whimsy, "A Deprogrammer's Tale", appears in recommendations next to "The Dot and the Line", a childhood favorite of mine that is given a connection to the Veblen's Herrontown Wood in this post. Adams also, like Veblen's family, has a Wisconsin connection, having received his PhD at U. of Wisconsin in Madison.

Yet another book by the prolific Adams is an Introduction to Topology. Here's a description of Veblen's contribution to the field: "Veblen's first work on topology appeared just before he arrived in Princeton. He went on to establish Princeton as one of the leading centres in the world for topology research. Analysis Situs(1922) provided the first systematic coverage of the basic ideas of topology and contributed to the development of modern topology."

And sailing? Well, sailing was one of Einstein's favorite activities, and the Veblens had a cottage in a town in Maine known for its wooden boat building. It's not clear if Veblen ever took to sailing. He may have been more comfortable in the woods, clearing brush.