Professor Richard Laver left his mark on mathematics and his friendsâ hearts
While descending one of the Cathedral Spires in Yosemite Valley, Richard Laver lost his route. But after a night stranded on a ledge in darkness, he found an answer that had eluded mathematicians for two decades.
Laver faced two challenges on that day in 1968. One was climbing down to safety. The other was the Fraïssé Conjecture, a problem had remained unsolved since 1948.
Cathedral Spires (center) in Yosemite Valley. Photo courtesy of the U.S. National Park Service.
Mike Olive, a friend and climbing partner was there. After a miserable night on a cliff ledge, he was exhausted and loopy. Laver was having an epiphany.
âHe suddenly said, âThatâs it. Thatâs it,ââ Olive recalls. âNow I see it.â
âItâ was the proof of the FraĂŻssĂ© Conjecture. Once back at school, Laver flushed out his proof and pounded out his Ph.D. dissertation. The Annals of Mathematics published an expanded version of his dissertation in 1971.
Within the field of set theoryâa branch of mathematics that deals with the nature and relations of setsâLaver had attained early fame. Laver, who passed away in September, spent most of his academic career at the University of Colorado Boulder as a mathematics professor who continued to advance and enlarge his field.
Laver was born on Oct. 20, 1942, died on Sept. 19, 2012, and spent the last four years of his life battling Parkinsonâs Disease, a degenerative central-nervous-system disorder.
In eulogy, the European Set Theory Society described Laver as a âcelebrated set theoristâ whose work was âbrilliant,â âinfluentialâ and âelegant.â Friends and colleagues say he was also funny, playful, generous and kind.
Keith Kearnes, a mathematics professor at CU-Boulder, notes that set theory is a foundational system for mathematics. The axioms of set theory, and hence of mathematics, are known to be incomplete, he said, adding that much of modern research in set theory involves investigating the consequences of different strengthenings of axioms.
âProfessor Laverâs significant contributions involve the identification, study and application of new âlarge cardinalâ axioms, and the development of refinements of the method of âforcing,â which is used to establish the relative consistency of new axioms,â Kearnes said.
Professor Laverâs work has enriched the mathematical literature with the notions of âLaver forcing,â âLaver reals,â the âLaver diamond,â âLaver treesâ and âLaver tables,â Kearnes said. But Laver himself taught set theory without using any of the phrases that bore his name.
The gangly thinker
Ralph McKenzie, now a distinguished professor of mathematics at Vanderbilt University, first noticed Laver in the fall of 1966. McKenzie had just earned his Ph.D. from CU-Boulder and had joined Berkeleyâs faculty. He observed that Laver was a gangly young man with a âsevere abstracted look ⊠usually hanging around the lounge thinking (apparently) or playing chess.â
Laver appeared in McKenzieâs office in spring 1968 and asked McKenzie to be his thesis adviser. By that time, Laver had spent years working on a âdifficult and famous open problem in mathematics,â FraĂŻssĂ©âs Conjecture.
In summer 1968âjust after the Eureka-experience climbâLaver telephoned McKenzie and informed him that Laver had finished his proof of the FraĂŻssĂ© Conjecture.
McKenzie does not take credit for that. âI probably had almost nothing to do with Richâs formation as a mathematician who would become a leading international figure in research,â McKenzie said.
Richard Laver, seen here during his time as a doctoral student in mathematics at the University of California at Berkeley. Photo courtesy of the Archives of the Mathematisches Forschungsinstitut Oberwolfach in Germany.
Olive, Laverâs friend and climbing partner, is still amazed that Laver solved an intractable problem after spending a night, waiting for dawn, clinging to the side of a cliff. âWe were wasted on the way down.â
Theyâd reached that point because theyâd gotten off route on way up, arrived at the summit late, and descended as darkness fell. On the way down, Laver dropped their flashlight. âWithout the light, it gets desperate,â Olive noted.
âBut weâre not stupid. We stayed on the mountain.â
Jane Twigg, who was married to Laver at the time, was waiting at the bottom of the spire. She knew the climbers were safe, because they yelled down to her. Twigg recalls that mathematics were always on his mind. âIf there were a square piece of paper, heâd just start writing whatever mathematical construct that was in his mind.â
âIâm not surprised that he was up on a ledge thinking about math.â
When he wasnât thinking about math, he reached out to friends or sought adventure, his friends and family say.
Olive recalls a young woman who spurned him at a Berkeley dance. Laver, sensing his friend was in pain, leapt on Oliveâs back, drawing Olive into a minor scuffle that distracted Olive from the rejection.
Laver suggested that the duo try rock climbing. Olive surmises that Laver was interested in climbing, but also eager to help his friend. It worked.
âWhen I was on the rock, I didnât think about the young lady at all. I thought about saving my ass,â Olive recalled. âHe wasnât one of these guys who could talk about touchy feely things. He would just make a good choice, one that would help people.â
Lumberjack in England
Keith Devlin, now executive director of Human-Sciences and Technologies Advanced Research Institute at Stanford University and the âmath guyâ on National Public Radioâs Weekend Edition, recalls meeting Laver at the University of Bristol, UK, where Laver worked as a post-doctoral researcher.
Within a short time, Devlin, Laver and their wives began a lifelong friendship. Laver, who appeared in Devlinâs door âlooking like a lumberjackââin jeans, a checkered shirt and hiking bootsâinvited Devlin to go climbing.
They became climbing partners who spent much of their time on the rock discussing mathematics.
Though their academic paths diverged, Laver and Devlin remained close. For a time, Devlin taught calculus at CU-Boulder during summers. He and Laver enjoyed long discussions.
âWe had two kinds of conversations,â Devlin recalled. âIf we werenât talking about mathematics, we were having whimsical conversations, like impromptu Monty Python sketches.â
âWeâd follow the logical illogicality of it and see how far it would take us,â Devlin said. In both mathematics and Monty Python, âyouâre just following logical threads,â Devlin observed.
âThose two things alone capture a lot of things about Rich.â
Laverâs pursuit of adventure, a constant throughout life, appeared just after graduate school, when he hitchhiked through Africa. There, he âdrank out of mud puddles,â got desperately ill and was hospitalized for weeks, Twigg recalls.
Although Twigg and Laver divorced, they remained friendly, with Laver sometimes stopping by her house to chat as he rode by on his bicycle. âHe was stubborn and he liked to win, but there was a lot of kindness in him as well.â
On the night Laver passed away, Twigg was surprised by the number of people in the room, many of whom she didnât know. âI got to meet this whole other side of his life, people who came after me, people who took care of himâ after he became ill.
âThe compassionate person he must have been to gather all those people wasnât all that evident,â Twigg said.
A rockslide plié
Kristen Marshall was among those who helped Laver after his diagnosis. When she was a graduate student at CU-Boulder, Marshall answered a knock at the door to find Professor Laver extending her an invitation to ascend Royal Arch near Chautauqua.
They were casual friends. But after earning her Ph.D. in applied mathematics, Marshall took a faculty position in New Jersey. There, she was hit by a car and suffered a head injury.
Laver flew out and stayed a week as she convalesced. âAfter I learned about his Parkinsonâs, I reached out to him because he was there for me,â she said.
Laura Michaelis-Cummings, associate professor and director of graduate studies in the Department of Linguistics at CU-Boulder, was Laverâs friend for more than a decade.
She recalls scrambling up a rocky incline near Guanella Pass south of Georgetown in 2000. She accidentally knocked a boulder loose and watched, horrified, to see it hurtling toward Laver.
âI was convinced that it would grievously hurt him, but he did an elegant pliĂ©, and it sailed right through his legs,â she said. (A pliĂ© is a ballet move in which dancers bend their knees while keeping their backs straight.) âHe looked delighted.â
Laver, Michaelis-Cummings said, was a ânotoriously shabby dresser.â Once, she spotted him wearing a bright pink T-shirt under his usual checkered button-down shirt. She registered her surprise.
âHe said, âI thought Iâd add a splash of color,ââ she recalls. âSubtle irony was one of his gifts.â
On another occasion, Michaelis-Cummings gave Laver Clorox tablets to place in each of his toilet tanks. The bleach apparently dissolved some rubber fittings in the toilets, and they began flushing themselves.
âRather than attempting to fix the problem, Rich reported later, he lay in bed listening to the two toilets flushing alternatively and devising a formula that would predict the time at which they would flush simultaneously.â
Michaelis-Cummings said Laver âelevated many livesâ with his wit, brilliance and simplicity. âI will miss his explanations, the language play and the laughter but, perhaps most of all, his belief in me. He felt that I could understand the math he was doing, and so I did, if only to a point, and he felt I could be a rock climber, and so I, an acute acrophobe, became a rock climber.â
Devlin and his wife bought a home in Boulder and planned to retire here. The idea was that the four friends who met in Bristol would eventually spend their lives together in closer proximity.
After Laverâs death, Devlin and his wife sold their Boulder home. âThe fact that I would go to Chautauqua and Rich wasnât there made Boulder completely different for me.â
âIâve had three really good friends in my life, and he was one of them.â
Marshall recalled Laverâs love of family, friends, logic, puzzles, music and mountains. He fervently wanted to live, she emphasized:
âHe struggled to overcome Parkinsonâs with some of the same skills and tenacity that he had used to solve math problems and decipher climbing routes. This was one riddle that he couldnât solve. None of us could.â
Now that Laver is gone, Twigg observed, âItâs like the universe isnât quite right.â
Added Olive, âItâs like they turned the sun down a little bit.â
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