As a high school student in Romania — when he won three gold medals with perfect scores in international math competitions — Ciprian Manolescu already knew about the William Lowell Putnam Mathematical Competition for university undergraduates.
"I found old problems from the Putnam competition in a University of Bucharest library, and I used them to prepare for the high school Olympiad," recalled Manolescu, who is now the coach of UCLA's Putnam competition team.
The preparation certainly paid off: As a Harvard undergraduate, he finished in the top five for each of three years, competing against thousands of undergraduates who took the Putnam exam. (Putnam does not reveal the order of the top five finishers, so Manolescu still doesn't know what his scores were or where in the top five he placed.)
Established in 1938, the annual Putnam exam consists of 12 mathematical problems to be completed in six hours, with a break midway between two sessions. If that sounds like a long time for a math test, in truth it's not nearly long enough.
The test is fiendishly difficult, and the grading is strict. Students can receive up to 10 points for each problem, but if they make even a small mistake, they get almost no points.
"It's important to solve the problem completely," said Manolescu, who is currently an associate professor of mathematics but has been promoted to full professor, effective July 1, 2012.
In the most recent Putnam competition, held last December, the median individual score — out of a possible 120 points — was 1. Many areas of mathematics are covered, including advanced calculus, differential equations, number theory and probability. One problem in number theory was solved by none of the students; everyone received a score of 0.
You don't have to be Terence Tao to figure out that with Manolescu as coach, UCLA's three-student Putnam team is likely to be very successful. Still, UCLA's mathematics department had fairly modest expectations for the team. For while UCLA's mathematics faculty is world-class and its graduate program is ranked nationally in the top 10, the undergraduate program has lagged behind.
"We were trying to get in the top 25 teams in the Putnam competition for now," said Sorin Popa, chair of UCLA's mathematics department and a math professor.
"Ciprian [Manolescu] is brilliant, extremely talented," Popa added. "We are very fortunate to have such an outstanding coach for our undergraduate students."
When the results of December's competition were announced recently, the UCLA team's performance likely raised some eyebrows. Of the 460 university teams from the U.S. and Canada that competed, UCLA finished 12th — the university's best result since 1970, nearly a decade before the 33-year-old Manolescu was born.
"The Putnam result is much more than we expected," Popa said.
The UCLA team consists of freshman Tudor Padurariu, sophomore Francisc Bozgan and junior Cheng Mao.
Manolescu and his students seemed pleased, but not thrilled.
"It's a very good result for the department, but we hope we can do better," Manolescu said. "It would be nice to be in the top 10. I think it can be done."
"I'm glad for the result, but I hope we can do better next year," said Padurariu, who is from Romania, as is Bozgan.
One reason UCLA is doing better, Manolescu said, is that Popa started an initiative in 2010 called the UCLA Math Undergraduate Merit Scholarships to attract outstanding high school students and raise the caliber of UCLA's undergraduate mathematics program; Padurariu and Bozgan both came to UCLA on this merit scholarship.
"The scholarship provides a great opportunity to come to UCLA and be taught by the best mathematicians in the world," Bozgan said.
"Like in sports, you can have the best coach, but you cannot succeed without superbly gifted students," Popa said. "Talent in mathematics is easily detectable. At UCLA, the talent is increasing at every level, including among our undergraduates. We hope to attract a donor to endow the UCLA Math Undergraduate Merit Scholarships so that we will elevate our undergraduate program to national prominence. We have already achieved that level with our graduate program."
Harvard, the Massachusetts Institute of Technology and Princeton offer many undergraduate scholarships, he noted.
Manolescu gives the credit for the team's success to the students.
"Tudor [Padurariu], Francisc [Bozgan] and [Cheng] Mao are outstanding young mathematicians, and they have worked very hard," he said. "They have devoted much more work in preparing for the exam than I did in coaching them, and they deserve to be proud of their hard-earned success."
Preparing for the annual Putnam competition is an important part of the students' education.
"It's very useful," Bozgan said. "Training for the Putnam exam, we develop our problem-solving skills, which are very important for a mathematician, and we learn how to manage our time during the exam."
"This is very helpful, and I can use what I learn here in other math classes," Padurariu said.
This year, 4,440 students took the Putnam exam, and Padurariu's score (43 points) was higher than 4,400 of them, but not as high as he had hoped.
"I start with the easier problems and try to do as many as possible," he said. "My goal for Putnam was to solve three or four in each session. At the time, I thought I had done it, but looking at the results, I probably did not."
"Working on difficult problems helps us improve our technique, and we learn not to fear such problems; we learn we can solve them," said Cheng, who is from Nanjing, China. "I came to UCLA as an engineering major, but I prefer math now."
Cheng hopes to become a mathematics professor, as do Padurariu and Bozgan.
"This year, the Putnam exam was particularly hard," said Manolescu, who added that coaching is quite different from teaching.
"It took me a long time to figure out what is the best way to coach the Putnam team," he said. "I'm still not sure if I've found the right way of coaching. I'm experimenting with various methods. It's about problem-solving rather than explaining a theory. You cannot just be at the blackboard and explain the theory to the students; you have to make them work the problem. I ask the students to look over books during the week and bring problems to our meetings. I try to get the students much more involved than in a usual class and ask them to present their own solutions."
Manolescu, who has won a distinguished teaching award in UCLA's mathematics department, teaches a problem-solving course for about 16 of UCLA's best undergraduates and a second course with six top undergraduates, who meet weekly to solve mathematical problems and review practice tests.
The kinds of mathematics problems Manolescu works on now in his research are "quite different" from when he was winning student competitions, but the way of thinking he developed helps. He works in two extremely difficult branches of mathematics: topology and symplectic geometry.
In mathematical research, problems often take months, not hours, to solve.
"In more advanced research, you have a bigger problem that you have to solve over many months, and you break it into dozens of small problems," Manolescu said. "The little problems can sometimes be attacked by methods similar to the ones we use on Putnam problems."
Like Popa, Manolescu wants to continue improving undergraduate mathematics education.
"We hope we will attract more excellent undergraduates to come to study math at UCLA, including more American students, especially from California," Manolescu said.
UCLA mathematics "is a great department and is getting better, in terms of research and the graduate students and undergraduate students that we are attracting," he added.
In a recent meeting in Manolescu's office, his students said they are very grateful for the opportunity to study with him.
"I don't think it is possible to have a better coach," Bozgan said.
A bit embarrassed by the compliment, Manolescu tried to change the subject, but Padurariu indicated the topic had been fully covered.
"I think," Padurariu said, "Francisc just said everything that we all feel."