# Burden of Proof

Professor of Mathematics Dennis Gaitsgory sketches a chalkboard full of esoteric symbols in Math 55, which he is teaching this semester for the first time.

At 10:02 a.m., the door to Science Center room 109 creaks opens, and 11 young men shuffle in. Some wear worn baseball caps and faded sweatshirts, others jeans and scuffed loafers. Whispering and rubbing sleep out of their eyes, they slowly settle into their seats, filling only two rows of their long and narrow classroom.

Class begins immediately as Professor of Mathematics Dennis Gaitsgory dashes in, dressed haphazardly in a button-down over a gray undershirt, most likely plucked from the same pile as his loose, wrinkled slacks.

Carelessly, he tosses his sweatshirt across his wooden desk, adjusts his round glasses, and rubs his facial hair. He makes eye contact with no one. Brand new stick of chalk in one hand, giant eraser in the other, he turns his back to the class and with dramatic, Van Gogh chalk strokes, he divides the wide blackboard into four squares.

He furiously covers the first portion of his canvas with a deluge of symbols. His studio—to stretch the artist metaphor a bit farther—is now ready. Thoughts pour from his mouth, obscured by his low voice and thick accent, a product of his Russian and Israeli background.

The students lean forward in their chairs, not wanting to miss a word of their lecture on Cauchy sequences. As if apprentices in the presence of a brilliant impressionist painter, they focus their attention on Gaitsgory’s canvas, their eyes darting to follow the flurry of bold, erratic strokes across the blackboard. Each nod marks a step closer to comprehension, but as the lecture progresses, brows begin to furrow.

Brett A. Harrison ’10 asks the first question: “Is the subspace closed under this topology? If so, we can conclude the lemma.” Harrison’s the trendiest of the group, wearing a cream Hollister sweatshirt and fashionably faded denim. On his feet are Velcro Pumas, a constant reminder that although he’s memorized pi out to a few dozen digits, he cannot tie his own shoes.

Patiently, Gaitsgory tries again to describe his steps. Anxious to help Harrison, the other students give their ideas. “Isn’t the space compact?” says Daniel A. “Choco” Litt ’10. His playful eyes belie his now-serious demeanor, making him appear as if he’s always in on a private joke. He’s wearing a knit sweater—blue, of course, the color he says he wears everyday to minimize fashion faux pas.

Harrison offers another question and answer, but trails off. “Never mind....”

“Well,” Gaitsgory says diplomatically, “the beginning is right.” The students laugh, as if they’re all in on one big joke. And maybe they are. After all, this course is far more cult than class. Welcome to Math 55.

AT WARP SPEED

A year ago, as high school seniors, math superstars across the country checked off the tiny “yes” box on Harvard’s acceptance reply postcard, partly motivated by this course.

Soft-spoken math enthusiast Elizabeth A Cook ’10, who started the year in Math 55, learned about the class one summer at a prestigious MIT research program from a Math 55 alum. “He brought up the subject, and my eyes bulged out of my head,” she says. “Everyone kept saying it was the hardest math course ever offered, and I said, ‘Cool! I want to take it!’”

Math 55, officially known as “Honors Advanced Calculus and Linear Algebra,” is essentially a nine-month mathematical boot camp. The course teaches four years of math in two semesters. “It’s an intense, warp-speed survey of the entire undergraduate math curriculum in one year,” Harrison says.

The class, in fact, includes subjects most math students will not encounter until graduate school. “This is probably the most difficult undergraduate math class in the country,” reads a page on the Mathematics Department Web site. While other universities like MIT offer similar, highly competitive courses, Math 55 is singular in its history and renown.

“It’s such a tradition in Harvard math culture,” says Raymond T. Pierrehumbert ’76, who took Math 55 as an undergrad and now teaches at the University of Chicago.

For mathematics professor and researcher at the University of Pennsylvania David Harbater ’74, Math 55 still holds sway. “If someone applies to graduate school at Penn, and I see that he or she was in Math 55, that would certainly get my attention. It definitely means something to me.”

Regardless of the course’s name brand value, Math 55 students face a single fact: It’s hard. Really hard.

Each week, their heads huddled together, these students dedicate 30 to 50 hours to problem sets—proving significant theorems with only definitions to guide them. Besides dark undereye circles and abandoned Expos assignments, this produces incredibly close friendships and camaraderie.

Most create their sets in LaTeX, a typesetting language, to produce 15- to 20-page problem sets. Add to that equation three hours of class and one hour of section a week, and these students essentially have full-time jobs.

Writing one’s own textbook, which is basically what the students do, is not for everyone. According to the Mathematics Department site, Math 55 is tailored for the dedicated: “You want math to be your most important class.”

SURVIVOR

On the first day of Math 55, it’s standing room only, a trail mix of serious mathematicians and the curious hoping for a quick glimpse of notoriety. This tremendous turnout is an annual phenomenon. “The first day each year, all the math kids who understand what’s going on are scared,” says Math 55 veteran Scott D. Kominers ’09, “and all the non-math kids who don’t laugh, because they think the class is so hard it’s overkill.”

After tourist season ends, another Math 55 ritual begins. Students may have more room to sit down, but the next few weeks will be anything but comfortable. It’s a game of “Survivor”: Outwit, Outplay, Outmath. Before the fifth Monday of the term, students who can’t seem to stay in the game start dropping like flies.

“I thought it was completely unbelievable,” Harbater says. “Seventy started it, 20 finished it, and only 10 understood it.”

Knowing the reputation, Kominers kept an unofficial tally of last year’s drop-off: “Last year, we had 51 students the first day, 31 students the second day, 24 for the next four days, 23 for two more weeks, and then 21 for the rest of the first semester after the fifth Monday.”

This year’s class is no exception. “I guess you can say it’s an episode of Survivor with people voting themselves off,” Litt says.

Later that first night, the first problem set is released online: 13 questions, each consisting of multiple sections to make a total of 47 parts. While nearly everyone is alarmed by the amount of work, Litt says he’s not too concerned. The class can’t stay this hard for this long, right?

“I figure he’s just trying to get people to drop the class,” Litt says.

He figured wrong. As class attendance steadily thins, the workload does not. The first few problem sets each take about 40 hours to complete. The work burden is reason enough for many extraordinarily gifted students to drop.

Case in point: Ameya A. Velingker ’10 took Advanced Placement calculus his freshman year and ranked in the top 12 for the USA Math Olympiad the year after that. “It was a tough decision to drop,” Velingker says. “You’re around all these people who are beasts at math. But I realized it was not going to work out.”

THE TEST

Midway through October, the “Survivor”-like competition intensifies with the add/drop deadline looming frighteningly near, only five days away. Today, the Math 55 contestants are faced with their most severe challenge yet: one test of six proofs, an entire class on the line.

“We all know that this determines whether or not the professor thinks we are worthy or if we are going to be kicked off the island,” Harrison says the morning of the test.

Later on, Gaitsgory calls some students into his office to discuss their exam results. “He said he was unsure about my solutions,” Harrison says, careful not to sound resentful.

“He wanted a confirmation that I was comfortable with the material and understood all that I had written down. He had me reproduce all the answers to the quiz in front of him on the blackboard in his corner office in the Science Center.”

While Harrison ultimately chooses to remain in the class, such conferences motivate several more students to drop, including the only two females: Cook, who had looked forward to taking the class, and Laura P. Starkston ’10.

“The problem was that I wasn’t prepared to think that abstractly,” Cook says. “Gaitsgory pointed it out to me in our private conference. Eventually I just got the picture.”

The final course drop forms are dutifully submitted, finalizing the class roster: 45 percent Jewish, 18 percent Asian, 100 percent male. The tribe has spoken.

‘THE WAR ROOM’

On a recent night, four Math 55 students inhabit an all-white, fluorescent-lit workspace in the Thayer Hall common room. Dorm residents reverentially refer to it as the “War Room,” honoring the Math 55 soldiers.

Georges Bizet’s Carmen blares from the computer of Menyoung Lee ’10. The boys sit scattered around their gray worktable, their eyes telltale red and fingers sore from countless hours at their laptops, dutifully LaTeXing problem sets. They have been here since 2 p.m. and will work for almost 12 straight hours to complete the problem set due the following day.

As the hours pass, they discuss the problem set. They formalize and write the solutions on their own for academic integrity. Despite the class’s cutthroat ereotype, this session is about community, not competition.

“We’re a team trying to survive this class,” Harrison says. “There’s no pretension. No attitude of ‘We’re better than everyone else.’” After working through a particularly difficult problem, they take a short break. Someone suggests comparing how many numbers of pi they’ve each memorized.

“I know, or at least did a little while ago, over 200 digits of pi,” offers Zack R. Abel ’10.

“That’s impressive,” Harrison says. “I’m 50.”

“I’m 15,” says Litt.

Harrison looks surprised. “You’re clearly not one of us,” he says. Litt begins to count the numbers in his head. “Only 11. Wow.”

The conversation then quickly turns to one of their most popular topics of discussion: Professor Gaitsgory. This is his first year teaching Math 55 and his students admire him in a Grigory Perelman-meets-Chuck Norris kind of way.

“He can win a game of Connect Four in three moves,” Harrison jokes.

They each volunteer their own sycophantic Gaitsgory stories. “One day, Menyoung and I went to his office hours,” Harrison begins.

Menyoung interrupts. “Remember how you thought he would be like ‘Fight me. If you win you get three questions?’”

Harrison laughs. “He taught us a few things on a random scrap of paper. Later, he let us take the paper with us when we left. But the back of this piece of paper wasn’t blank. It was something he was drafting. I’ve never seen so many symbols in one equation in my entire life.”

While the Math 55 students may marvel at their professor, they have attracted their own dedicated admirers in the freshman class. “Even though we have all this work,” Harrison says with pride, “when we’re in the War Room and all the Math 25 and 23 kids are down there, there has never been a time when people ask us for help and we say we’re too busy. We always help people.”

Elena D. Butler ’10, a self-described “Math 55 groupie,” agrees.

“These guys are my role models.” She remembers the first time she trekked to Harrison’s room for help. “He’s lives on the top floor of Thayer, so we had to walk up five flights of stairs from the common room. It was like the yellow brick road leading to the Wizard of Oz.”

Even 30 years ago, Harbater and his Math 55 classmates garnered the same awe. “The stereotype was that we were the math wizards.”

‘WOW, YOU'RE IN MATH 55?’

But that admiration comes at a price of an unflattering stereotype. Reactions vary when others find out Litt is in Math 55. Some immediately request homework help, he says. “Sometimes people just spontaneously combust, swoon, faint, turn bright red, start giggling.”

Litt resents being defined by his enrollment in this one course. “It pisses me off when my roommate introduces me as, ‘This is Daniel, he’s in Math 55.’ By the time he gets to ‘in,’ I’ve usually kicked him in the shin.”

Kominers also resents the labeling. “A couple times I was introduced to people as ‘He’s Scott. And he’s in Math 55.’ But no one in 55 would ever introduce a fellow 55er like that.”

Some feel the stereotypes are unavoidable. “Once you take 55, you get known as a 55er,” says Velingker, who dropped from 55 to 25 at the beginning of the term. “Non-math people might not know the people personally but they are aware of what the class is. There is definitely some kind of respect.”

While the hard-working Math 55 students warrant admiration, Dean of the College Benedict H. Gross ’71, who both took and taught the class, worries that the reputation may lead students to remain in the class for glory.

“You get people who are taking it to impress their friends, and that’s not why you should take it,” he says. “Then you get people who are in it for the wrong reasons. That’s why we’ve always sort of kept the membrane between Math 55 and Math 25 fairly permeable.”

While many of the Math 55 students scorn their labels, Harrison thinks the stereotype stems from respect.

“It is an honor to be respected by every other person in the math community,” Harrison says. “People say, ‘Wow, you’re in Math 55?’ and they really respect us….Yes, there’s a lot of work, and yes, we have to devote a lot of time to the class. But the rewards from the math community, the other people in the class, and the satisfaction of learning so much in such a short period of time so greatly outweigh having to put that time in.”

JUST LIKE A FRAT?

Monday night section in Science Center 411 is far more casual than class with Gaitsgory. The course assistant, Dustin T. Clausen ’08, took Math 55 as a freshman. He sits on a desk, his long legs stretched out in front of him. Suddenly, his feet find the floor as he jumps down to write on the board. He smiles widely and raises his eyebrows with excitement.

Harrison asks a question, and Clausen realizes that he skipped a key step not yet covered in the course. “Sorry,” Clausen says. “Sometimes I just forget you guys are freshmen.”

All in all, the atmosphere is free and relaxed. Students shout out answers and challenge one another. This is certainly not Gaitsgory’s lecture. During section, they use the word “fun” 17 times.

To reinforce the fun, frat-like image comes a not-so-fun fact. According to the Registrar’s Office, only 17 women have taken the class since 1990. Though past rosters have included female students, Math 55 is a fraternity. Students rush. Eleven become pledges, and they are initiated with problem sets.

“For better or for worse, it’s such a close-knit social club,” Kominers says. Litt attributes the camaraderie to the long hours they spend together.

“When we do Math 55, we do it together,” he says. “When we have wild drunken parties, when we paint the town, when we piss on John Harvard’s foot, we do it together. Even if we’re calculating the trajectory of the piss at the same time, we’re doing it together.”

Both Harbater and Pierrehumbert confirm they experienced the same Greek life 30 years ago.

“It is definitely a cult,” Pierrehumbert says. “I view it as more of an ordeal than a course. People who have been through it don’t really forget it….It’s like changing fraternity rings. It identifies you.”

GAME ON

It’s 10:58 a.m., and back in Science Center 109, the 11 survivors are still excitedly discussing Cauchy sequences. With each new concept, they are delighted. They’re in their fraternity house and they know it.

Suddenly, Gaitsgory scribbles an impossible problem on the board. “In the remaining two minutes, I want to cover this,” he announces, daring the boys to enter a mathematical duel. They are unsurprised by this last-minute challenge. Game on.

But outside, the clock strikes the hour. As students pour from nearby classrooms, they pause to peer, wide-eyed, into the windows of room 109. They take advantage of this rare chance to snatch a glimpse of the notorious Math 55. Fascinated, their eyes explore the encoded chalkboard, entrancing them with its odd arrangement of foreign symbols, formulas, and proofs.

None of the Math 55 students seem to notice. They’re not concerned about being on time to their next class. Math 55 is why they’re here.

But now it’s 11:06 a.m., and another professor is knocking vigorously on the window, pointing at his watch. Gaitsgory nods. And continues to write.