羞羞视频

Professor Richard Laver left his mark on mathematics and his friends鈥� 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 Fra茂ss茅 Conjecture, a problem had remained unsolved since 1948.

The first problem vanished with the sunrise, and the ability to see the route. The second resolved in a flash of insight. At the time, Laver was a doctoral student in mathematics at the University of California at Berkeley.

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.

鈥淗e suddenly said, 鈥楾hat鈥檚 it. That鈥檚 it,鈥欌�� Olive recalls. 鈥淣ow I see it.鈥�

鈥淚t鈥� was the proof of the Fra茂ss茅 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 theory鈥攁 branch of mathematics that deals with the nature and relations of sets鈥擫aver had attained early fame. Laver, who passed away in September, spent most of his academic career at the 羞羞视频 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 Parkinson鈥檚 Disease, a degenerative central-nervous-system disorder.

In eulogy, the European Set Theory Society described Laver as a 鈥渃elebrated set theorist鈥� whose work was 鈥渂rilliant,鈥� 鈥渋nfluential鈥� and 鈥渆legant.鈥� Friends and colleagues say he was also funny, playful, generous and kind.

Keith Kearnes, a mathematics professor at CU-羞羞视频, 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.

鈥淧rofessor Laver鈥檚 significant contributions involve the identification, study and application of new 鈥榣arge cardinal鈥� axioms, and the development of refinements of the method of 鈥榝orcing,鈥� which is used to establish the relative consistency of new axioms,鈥� Kearnes said.

Professor Laver鈥檚 work has enriched the mathematical literature with the notions of 鈥淟aver forcing,鈥� 鈥淟aver reals,鈥� the 鈥淟aver diamond,鈥� 鈥淟aver trees鈥� and 鈥淟aver tables,鈥� 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-羞羞视频 and had joined Berkeley鈥檚 faculty. He observed that Laver was a gangly young man with a 鈥渟evere abstracted look 鈥� usually hanging around the lounge thinking (apparently) or playing chess.鈥�

Laver appeared in McKenzie鈥檚 office in spring 1968 and asked McKenzie to be his thesis adviser. By that time, Laver had spent years working on a 鈥渄ifficult and famous open problem in mathematics,鈥� Fra茂ss茅鈥檚 Conjecture.

In summer 1968鈥攋ust after the Eureka-experience climb鈥擫aver telephoned McKenzie and informed him that Laver had finished his proof of the Fra茂ss茅 Conjecture.

McKenzie does not take credit for that. 鈥淚 probably had almost nothing to do with Rich鈥檚 formation as a mathematician who would become a leading international figure in research,鈥� McKenzie said.

Laver was among the most 鈥渉ighly gifted鈥� of the 25 people who earned Ph.D.s under McKenzie鈥檚 guidance, he added.

Olive, Laver鈥檚 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. 鈥淲e were wasted on the way down.鈥�

They鈥檇 reached that point because they鈥檇 gotten off route on way up, arrived at the summit late, and descended as darkness fell. On the way down, Laver dropped their flashlight. 鈥淲ithout the light, it gets desperate,鈥� Olive noted.

鈥淏ut we鈥檙e not stupid. We stayed on the mountain.鈥�

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. 鈥淚f there were a square piece of paper, he鈥檇 just start writing whatever mathematical construct that was in his mind.鈥�

鈥淚鈥檓 not surprised that he was up on a ledge thinking about math.鈥�

When he wasn鈥檛 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 Olive鈥檚 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.

鈥淲hen I was on the rock, I didn鈥檛 think about the young lady at all. I thought about saving my ass,鈥� Olive recalled. 鈥淗e wasn鈥檛 one of these guys who could talk about touchy feely things. He would just make a good choice, one that would help people.鈥�

Lumberjack in England

Keith Devlin, now executive director of Human-Sciences and Technologies Advanced Research Institute at Stanford University and the 鈥渕ath guy鈥� on National Public Radio鈥檚 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 Devlin鈥檚 door 鈥渓ooking like a lumberjack鈥濃�攊n jeans, a checkered shirt and hiking boots鈥攊nvited 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-羞羞视频 during summers. He and Laver enjoyed long discussions.

鈥淲e had two kinds of conversations,鈥� Devlin recalled. 鈥淚f we weren鈥檛 talking about mathematics, we were having whimsical conversations, like impromptu Monty Python sketches.鈥�

鈥淲e鈥檇 follow the logical illogicality of it and see how far it would take us,鈥� Devlin said. In both mathematics and Monty Python, 鈥測ou鈥檙e just following logical threads,鈥� Devlin observed.

鈥淭hose two things alone capture a lot of things about Rich.鈥�

Laver鈥檚 pursuit of adventure, a constant throughout life, appeared just after graduate school, when he hitchhiked through Africa. There, he 鈥渄rank out of mud puddles,鈥� 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. 鈥淗e was stubborn and he liked to win, but there was a lot of kindness in him as well.鈥�

On the night Laver passed away, Twigg was surprised by the number of people in the room, many of whom she didn鈥檛 know. 鈥淚 got to meet this whole other side of his life, people who came after me, people who took care of him鈥� after he became ill.

鈥淭he compassionate person he must have been to gather all those people wasn鈥檛 all that evident,鈥� Twigg said.

A rockslide pli茅

Kristen Marshall was among those who helped Laver after his diagnosis. When she was a graduate student at CU-羞羞视频, 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. 鈥淎fter I learned about his Parkinson鈥檚, I reached out to him because he was there for me,鈥� she said.

Laura Michaelis-Cummings, associate professor and director of graduate studies in the Department of Linguistics at CU-羞羞视频, was Laver鈥檚 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.

鈥淚 was convinced that it would grievously hurt him, but he did an elegant pli茅, and it sailed right through his legs,鈥� she said. (A pli茅 is a ballet move in which dancers bend their knees while keeping their backs straight.) 鈥淗e looked delighted.鈥�

Laver, Michaelis-Cummings said, was a 鈥渘otoriously shabby dresser.鈥� Once, she spotted him wearing a bright pink T-shirt under his usual checkered button-down shirt. She registered her surprise.

鈥淗e said, 鈥業 thought I鈥檇 add a splash of color,鈥欌�� she recalls. 鈥淪ubtle irony was one of his gifts.鈥�

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.

鈥淩ather 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.鈥�

Michaelis-Cummings said Laver 鈥渆levated many lives鈥� with his wit, brilliance and simplicity. 鈥淚 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.鈥�

Devlin and his wife bought a home in 羞羞视频 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 Laver鈥檚 death, Devlin and his wife sold their 羞羞视频 home. 鈥淭he fact that I would go to Chautauqua and Rich wasn鈥檛 there made 羞羞视频 completely different for me.鈥�

鈥淚鈥檝e had three really good friends in my life, and he was one of them.鈥�

Marshall recalled Laver鈥檚 love of family, friends, logic, puzzles, music and mountains. He fervently wanted to live, she emphasized:

鈥淗e struggled to overcome Parkinson鈥檚 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 couldn鈥檛 solve. None of us could.鈥�

Now that Laver is gone, Twigg observed, 鈥淚t鈥檚 like the universe isn鈥檛 quite right.鈥�

Added Olive, 鈥淚t鈥檚 like they turned the sun down a little bit.鈥�