Demystifying Math 55 – Harvard & The NSA

Mathematics 55A “Studies in Algebra and Group Theory” and Mathematics 55B “Studies in Real and Complex Analysis

DEMYSTIFYING MATH 55

Few undergraduate level classes have the distinction of nation-wide recognition that Harvard University’s Math 55 has. Officially comprised of Mathematics 55A “Studies in Algebra and Group Theory” and Mathematics 55B “Studies in Real and Complex Analysis,” it is technically an introductory level course. It is also a veritable legend among high schoolers and college students alike, renowned as — allegedly — the hardest undergraduate math class in the country. It has been mentioned in books and articles, has its own Wikipedia page, and has been the subject of countless social media posts and videos.

Most recently, Harvard junior Mahad Khan created a TikTok video dedicated to Math 55 that has received over 360,000 views to date. His is only one of many — his older brother created one, too — but it has the distinction of an insider’s perspective. “I thought it would be interesting if I cleared up the misconceptions about Math 55,” Khan said. While he hadn’t taken the course himself, he wanted to go beyond its reputation. “I wanted to get a real perspective by interviewing a former student and current course assistant.”

Over the years, perception of Math 55 has become based less on the reality of the course itself and more on a cumulative collection of lore and somewhat sensationalist rumors. It’s tempting to get swept up in the thrill of hearsay but while there might be kernels of truth to some of the stories, many of them are outdated and taken out of context. At the end of the day, however, Math 55 is a class like any other. Below, we take a stab at busting some of the more well known and persistent myths about the class. Or, at the very least, offering an extra layer of clarity.

MYTH #1: MATH 55 IS ONLY FOR HIGH SCHOOL MATH GENIUSES

Most articles or mentions of Math 55 refer to it as filled with math competition champions and genius-level wunderkinds. The class is supposedly legendary among high school math prodigies, who hear terrifying stories about it in their computer camps and at the International Math Olympiad. There are even rumors of a special test students have to take before they are even allowed into Math 55. But while familiarity with proof-based mathematics is considered a plus for those interested in the course, there is no prerequisite for competition or research experience.

In fact students whose only exposure to advanced math has been through olympiads and summer research programs can have a harder time adjusting. Their approach to the material tends to be understandably more solitary and that can be a disadvantage for the level of collaboration higher level mathematics require. “It has become a lot more open to people with different backgrounds,” said Professor Denis Auroux, who teaches Math 55,. “Our slogan is, if you’re reasonably good at math, you love it, and you have lots of time to devote to it, then Math 55 is completely fine for you.”

Also, there is no extra test to get into the class.

MYTH #2: JUST TAKE A GRADUATE CLASS, INSTEAD

Math 55 is hard. Whether you’re just 55-curious, or a past or present student in the class, this is something everyone agrees on. The course condenses four years of math into two semesters, after all. “For the first semester, you work on linear and abstract algebra with a bit of representation theory,” said sophomore math concentrator Dora Woodruff. “The second semester is real and complex analysis, and a little bit of algebraic topology. That’s almost the whole undergraduate curriculum.” Woodruff — incidentally, the student Khan interviewed — took Math 55 as a freshman and returned her second year as a course assistant. She is intimately familiar with the course’s difficulty level.

So why not just take an upper level undergraduate course to begin with or even one at a graduate level, if you’re really looking for a challenge? What justifies the existence of a class with the difficulty level of Math 55? One argument is that the course helps structure and systemize the knowledge with which many students come to Harvard. It gives them a firm background in preparation for the rest of their math education. Math 55 is difficult and it is purposefully structured that way as it’s meant to help students mature as mathematicians rather than as simple course takers.

But more importantly, “it’s just not true that Math 55 is at the level of a graduate class,” Auroux said. “It goes through several upper division undergraduate math classes with maybe a bit more advanced digressions into material here and there, but it sticks very close to what is taught in 100-level classes. The difference is we go through it at a faster pace, maybe with more challenging homework, and ideally as a community of people bringing our heads together.”

A core goal of Math 55, according to Auroux, is to build a sense of community. Other schools might encourage advanced first-year students to take upper level undergraduate or even graduate classes, but Math 55 helps build a cohort of like-minded people who really like math, are good at it, and want to do a lot of it during their time at Harvard. That’s the experience Woodruff had, as well. “The community can be very strong,” she said. “You meet a lot of other people very interested in math and stay friends with them for the rest of college.”

MYTH #3: HOMEWORK TAKES BETWEEN 24 AND 60 HOURS

Horror stories of endless homework are synonymous with the class. You’ll read or hear about “24 to 60 hours per week on homework” in almost every reference to Math 55. But one, there is a world of difference between 24 and 60 hours that is never explained, and two, this timeframe is quite misaligned with reality.

Auroux frequently sends out surveys to his students asking how long homework takes them and the average for most is closer to 15 hours a week. Those with more extensive prior math backgrounds can take as little as five to ten hours. The key factor is collaboration. “This class doesn’t lend itself to self-study,” Auroux stressed. Once they have thought about each problem set on their own, students are welcome and encouraged to talk to their friends and collaborate. “As soon as I see that something took over 30 hours I ask the student, do you know you’re supposed to be working with people and come ask me questions when you’re stuck?”

It is true that between reviewing lectures, digesting the material, and solving the problem sets, students usually end up devoting between 20 and 30 hours a week to the class. However, that includes the time dedicated to homework. So while students are discouraged from taking too many difficult classes and extracurriculars in the same semester as Math 55, they are also not expected to spend the time equivalent to a full-time job on their problem sets every week.

MYTH #4: LESS THAN HALF OF THE CLASS MAKES IT TO THE SECOND SEMESTER

Math 55 is just as infamous for its attrition rate as it is for its difficulty. Most sources like to cite the 1970 class, which began with 75 students and — between the advanced nature of the material and the time-constraints under which students had to work — ended with barely 20. Since then, the rumor has been that the Math 55 class shrinks by half its original size or more before the first semester is over. The reality is much less shocking and a bit more complicated.

Enrollment in this past fall semester’s Math 55A peaked at (ironically) 55 students. Well into the spring semester’s Math 55B, 47 students were still enrolled in the course. “On average, a drop of about 10-15 percent is much closer to what I would expect,” Auroux said. And those numbers become even more flexible if one takes into consideration the weeks math students have at the beginning of each semester to try out different classes and “shop” around before they have to commit to anything. This means students find their way in and out of Math 55 in a variety of ways over the course of the academic year.

According to Auroux, some students shop Math 55 in the fall and switch to the less intense Math 25 for the remainder of the semester. Others start out in Math 25 and, if not sufficiently challenged, switch to Math 55. Even people who end up in academia are not exempt from this. During his time as a student, our own Department of Mathematics’ Professor Emeritus Benedict Gross switched to the lower level Math 21 after two weeks in Math 55. In fact, those two weeks almost made him reconsider his desire to pursue mathematics. “By the beginning of sophomore year, I had decided to major in physics,” he recalled. “But during shopping period that fall, I walked past a math class taught by Andrew Gleason and stopped in to listen. It turned out to be Math 55.” He enrolled and by the end of the semester had found his vocation in mathematics.

All this means that Auroux sees student numbers vacillate up and down throughout the academic year. “There are about four or five students in this spring semester’s Math 55 that took Math 25 or even Math 22 in the fall, and they’re doing mostly fine,” he said. “It’s a lot of work, but I think they’re having a great time.”

MYTH #5: 55-ER CULTURE IS CULT-Y AND EXCLUSIONARY

Even though her experience with Math 55 was a positive one, Woodruff is very aware of the unhealthy culture the class has been rumored to cultivate. It’s easy for students to form exclusionary cliques that consist only of other Math 55 students, and some look down on anyone taking lower level math classes. But Woodruff also stressed that the instructors are very aware of this and actively take steps to curb that kind of toxic behavior. She said Auroux frequently brings up the importance of keeping the Math 55 community inclusive through Slack messages and lecture references.

Some students come to Harvard just for the opportunity to take Math 55. Some view enrolling in the class as proof of their mathematical gumption and competence. A Harvard Independent article called Math 55 the “premiere mathematical challenge for overachieving and…ridiculously mathy freshmen” and a piece in The Harvard Crimson referred to it as “a bit of a status thing as far as math majors here are concerned.” Over the years, the Harvard Department of Mathematics has taken steps to correct these assumptions.

For one thing, neither the Math 55A nor the Math 55B official course descriptions boast the dubious honor of referring to it as “probably the most difficult undergraduate math class in the country” (don’t trust everything you read on Wikipedia). For another, “we’re trying to emphasize that there’s no magic to Math 55,” Auroux said. “It contains the same material as some of the other classes we have. People who take it are not intrinsically better or smarter than the ones who don’t.”

MYTH #6: YOU HAVE TO TAKE MATH 55 IF YOU’RE SERIOUS ABOUT GOING INTO ACADEMIA

One reason math concentrators could feel pressured to enroll in Math 55 is because they view it as a prerequisite for a career in academia. It’s a sort of badge of honor and proof of their commitment to the field of mathematics. It is true that quite a few graduates of the course have gone on to pursue a career in mathematics. Woodruff herself believes that will be the most likely path for her, and several faculty members in our own Department of Mathematics took Math 55 during their days as Harvard freshmen.

“Several times in my research career when I understood something fundamental, I would realize that this was what Math 55 was trying to teach us,” Gross said. “It was an amazing introduction to the whole of mathematics and it was transformative for me.” In fact, Gross met Higgins Professor of Mathematics Joe Harris when they took the class together, forging a lifelong friendship. When they returned to Harvard as faculty, they took turns teaching Math 25 and Math 55.

However, Auroux is quick to point out that while many graduates of the course do end up in academia, most professional mathematicians have likely never even heard of Math 55. “I would like to think that it’s a success story if people end up doing math, because the goal of Math 55 is to show students how beautiful math can be,” he said. “If they love it enough to go to grad school and become mathematicians, that’s wonderful. And if they want to take that math knowledge and do something else with their life, that’s just as wonderful.” BY ANASTASIA YEFREMOVA source

Mathematicians Urge Colleagues To Refuse To Work For The NSA

In January, the math community had its big event of the year — the Joint Mathematics Meeting — where 3,000 mathematicians and math students gathered to talk about new advances in the field and jostle for jobs. The National Security Agency is said to be the largest employer of mathematicians in the country and so it always has a sizeable presence at the event to recruit new candidates. This year, it was even easier for the agency as the four-day conference took place at the Baltimore Convention Center, just 22 minutes away from NSA headquarters in Fort Meade. Thomas Hales, a professor at the University of Pittsburgh, who describes himself as a “mathematician who’s upset about what’s going on,” is dismayed at the idea of the brightest minds in his field going to work for the agency. In reaction to the Snowden revelations — which started exactly a year ago —  about NSA’s mass surveillance and compromising of encryption standards, Hales gave a grant to the San Francisco-based civil liberties group Electronic Frontier Foundation to fly a representative to Baltimore to try to convince mathematicians young and old not to go help the agency with data-mining and encryption-breaking.

“Mathematicians aiding in national defense goes all the way back to Archimedes, defending against the Roman siege and designing the catapult. Mathematician Lewis Fry Richardson destroyed his work after realizing researchers in poison gas were looking at it. Mathematicians were involved in the Manhattan Project, developing nuclear weapons,” says Hales. “Many mathematicians work for the NSA or organizations with ties to it. They’re involved in facial recognition development and big data aspects of mass surveillance. If privacy disappears from the face of the Earth, mathematicians will be some of the primary culprits.”

The EFF sent to the conference a newly-hired staff technologist, Yan Zhu, a woman whose dyed red hair surely helped her stand out. She had a tiny table — between a math educational software company and Mathcamp — while the NSA had a huge “awesome” booth with multiple people and “lots of swag,” including laundry bags with the NSA logo. She said it was tough competing. “Students are mainly just interested in getting a job after graduation not in activism. People were around the NSA booth all the time when I walked by. When I looked at the NSA sign-up sheet for people who wanted to interview on site for summer internships, it was always full,” says Zhu. I imagine those interviews may have started like this scene from Good Will Hunting but didn’t end like it:

Zhu was dismayed by the NSA’s popularity at the event and the relative lack of attention to civil liberties concerns. “The NSA is illegally mass-spying on people,” says Zhu, who admitted she snagged an NSA laundry bag for herself. “I realize that people who have done pure math probably don’t have a lot of other career options, but I encouraged those who wanted to talk to learn to code or program, and pointed out that EFF has hired a lot of mathematicians.We weren’t doing recruiting just trying to inform them that there are people who are very against the NSA.”

Hales said he got depressed “every day walking in and seeing all the mathematicians gathered at the NSA booth,” but he is far from the only mathematician who is on the outs with the NSA right now. There have been a series of editorials written by mathematicians in the New Scientist and  Slate urging fellow mathematicians to speak out about how their research is being used in unconstitutional ways by the agency. Charles Seife is a mathematician who worked for the NSA briefly two decades ago, and is now a journalism professor. He wrote in Slate, “The agency insisted, over and over, that the weapons we were building—and weapons they are, even if they’re weapons of information—would never be turned on our own people, but would only be used upon our enemies. What do we do now that we have to face the fact that the Agency broke its word? … I feel compelled to speak out to say that I’m horrified. If this is really what the agency stands for, I am sorry to have helped in whatever small way that I did.”

The American Mathematical Society, a membership organization for mathematicians, has been addressing the anxiety within the profession in its newsletters. Last year, it printed a letter from Professor Alexander “Sasha” Beilinson of the University of Chicago asking the Society to cut ties with the NSA and stop accepting their grants for mathematical research. “The NSA destroyed the security of the Internet and privacy of communications for the whole planet,” wrote Beilinson. “If any healing is possible, it would probably start with making the NSA and its ilk socially unacceptable — just as, in the days of my youth, working for the KGB was socially unacceptable for many in the Soviet Union. Any relationship with an organization whose activity is so harmful for the fabric of human society is unhealthy. For the sake of integrity, the AMS should shun all contacts with the NSA.”

Beilenson says he “got some letters of support, mostly from the young mathematicians, which was very nice” but otherwise no response from the AMS after writing the letter. The leadership of the American Mathematical Society says it is not planning to deter members from working for the spy agency nor will it stop accepting administering the NSA grant program, noting that those grants support innocuous research on algebra, number theory, discrete mathematics, probability, and statistics. “Cryptology and classified research are specifically excluded from the grants,” say AMS president David Vogan and executive director Donald McClure in a statement. “The work of [the grant program] is directly in line with the mission of the American Mathematical Society ‘to further the interests of mathematical research and scholarship.’ It is strongly supported by the leadership of the AMS and, we believe, by a majority of the members.”

But the AMS is devoting six pages of its next newsletter — due out next week — to a discussion of the Snowden revelations. Notices of the AMS editor, Allyn Jackson, says she “solicited articles from mathematicians who we thought would write thoughtful and informative pieces.” I read an advance copy in which Andrew Odlyzko, a professor at the University of Minnesota, decries society’s preoccupation with terrorism but seems more troubled by the government’s failure in allowing the leakage of documents and secrets than by what those leaks revealed.  “I do not see the NSA as a rogue organization engaging in amoral activities,” he writes. He says it fills an “important role both in spying on numerous hostile actors and setting security standards” and that he will not discourage students from applying there.

The other mathematician who wrote a piece, Keith Devlin of Stanford University, worked on Defense Department projects after September 11th and takes a far more critical view of the NSA after the Snowden revelations. He writes that he felt “intense betrayal when I learned how [the intelligence community] took the work I and many others did over many years, with a genuine desire to prevent another 9/11 attack, and subverted it in ways that run totally counter to the founding principles of the United States, that cause huge harm to the US economy, and that almost certainly weaken our ability to defend ourselves.”

“I think mathematicians should refuse to work for the NSA until they both follow the US Constitution and demonstrate responsible use of mathematical tools,” says Devlin in an email to me. “The latter is something they clearly failed to do by engineering weaknesses into mathematical crypto systems, which mathematicians know to be a very dangerous thing to do. I think it is very regrettable that the current NSA leadership has broken the immense goodwill that most of us in the mathematical community once had toward them.”

Thomas Hales, who sponsored the EFF representative to try to dissuade mathematicians from going to work for the NSA, says he taught a graduate level course on mathematical cryptography this fall and that it was influenced by Snowden.  “I would not have taught the course if not for the Snowden documents,” he says. He sees the spread of cryptographic practices as a defense against the NSA. When Google, for example, released an “end-to-end” encryption tool for Gmail this week, it placed a smiley face message in its code, an inside joke that was a subtle dig at the NSA, and a celebration of the fact that it will be harder for spying types to get access to messages sent this way by Gmail users. Hales says he has had students go work for the NSA in the past, but that he will discourage them from doing so moving forward.

“I’m a mathematician, I’m not in politics,” says Hales. “As a citizen, I’m outraged by what’s happening, and find the small size of the public response to be very disturbing. It seems that the most influence I can have is within the mathematics community. I really hoped that things would change for mathematicians as a result of the Snowden documents, but it’s happening more slowly than I hoped it would.”

Kashmir Hill source

Math 55

Math 55 is a two-semester long first-year undergraduate mathematics course at Harvard University, founded by Lynn Loomis and Shlomo Sternberg. The official titles of the course are Honors Abstract Algebra (Math 55a) and Honors Real and Complex Analysis (Math 55b). Previously, the official title was Honors Advanced Calculus and Linear Algebra.

1. Description

The Harvard University Department of Mathematics describes Math 55 as “probably the most difficult undergraduate math class in the country.”^[1] Formerly, students would begin the year in Math 25 (which was created in 1983 as a lower-level Math 55) and, after three weeks of point-set topology and special topics (for instance, in 1994, p-adic analysis was taught by Wilfried Schmid), students would take a quiz. As of 2012, students may choose to enroll in either Math 25 or Math 55 but are advised to “shop” both courses and have five weeks to decide on one.^[2] Depending on the professor teaching the class, the diagnostic exam may still be given after three weeks to help students with their decision.

In 1994, 89 students took the diagnostic exam: students scoring more than 50% on the quiz could enroll in Schmid’s Math 55 (15 students), students scoring between 10 and 50% could enroll in Benedict Gross’s Math 25: Theoretical Linear Algebra and Real Analysis (55 students), and students scoring less than 10% were advised to enroll in a course such as Math 21: Multivariable Calculus (19 students).^[3]

A take-home final ends the class.^[4]

1.1. Historical Retention Rate

In 1970, Math 55 covered almost four years worth of department coursework in two semesters, and subsequently, it drew only the most diligent of undergraduates. Of the 75 students who enrolled in the 1970 offering, by course end, only 20 remained due to the advanced nature of the material and time-constraints under which students were given to work.^[5] David Harbater, a mathematics professor at the University of Pennsylvania and student of the 1974 Math 55 section at Harvard, recalled of his experience, “Seventy [students] started it, 20 finished it, and only 10 understood it.” Scott D. Kominers, familiar with the stated attrition rates for the course, decided to keep an informal log of his journey through the 2009 section: “…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” (the beginning of the fifth week being the drop deadline for students to decide whether to remain in Math 55 or transfer to Math 25).^[6] In 2006, the class was 45 percent Jewish (5 students), 18 percent Asian (2 students), 100 percent male (11 students).^[7]

2. Course Content

Through 2006, the instructor had broad latitude in choosing the content of the course.^[8] Though Math 55 bore the official title “Honors Advanced Calculus and Linear Algebra,” advanced topics in complex analysis, point set topology, group theory, and differential geometry could be covered in depth at the discretion of the instructor, in addition to single and multivariable real analysis as well as abstract linear algebra. In 1970, for example, students studied the differential geometry of Banach manifolds in the second semester of Math 55.^[5] In contrast, Math 25 was more narrowly focused, usually covering real analysis, together with the relevant theory of metric spaces and (multi)linear maps. These topics typically culminated in the proof of the generalized Stokes’ theorem, though, time permitting, other relevant topics (e.g., category theory, de Rham cohomology) might also be covered.^[9] Although both courses presented calculus from a rigorous point of view and emphasized theory and proof writing, Math 55 was generally faster paced, more abstract, and demanded a higher level of mathematical sophistication.

Loomis and Sternberg’s textbook Advanced Calculus,^[10] an abstract treatment of calculus in the setting of normed vector spaces and on differentiable manifolds, was tailored to the authors’ Math 55 syllabus and served for many years as an assigned text. Instructors for Math 55^[11] and Math 25^[9] have also selected Rudin’s Principles of Mathematical Analysis,^[12] Spivak’s Calculus on Manifolds,^[13] Axler’s Linear Algebra Done Right,^[14] and Halmos’s Finite-Dimensional Vector Spaces^[15] as textbooks or references.

From 2007 onwards, the scope of the course (along with that of Math 25) was changed to more strictly cover the contents of four semester-long courses in two semesters: Math 25a (linear algebra) and Math 122 (group theory) in Math 55a; and Math 25b (calculus, real analysis) and Math 113 (complex analysis) in Math 55b. The name was also changed to “Honors Abstract Algebra” (Math 55a) and “Honors Real and Complex Analysis” (Math 55b). Fluency in formulating and writing mathematical proofs is listed as a course prerequisite for Math 55, while such experience is considered “helpful” but not required for Math 25.^[1] In practice, students of Math 55 have usually had extensive experience in proof writing and abstract mathematics, with many being the past winners of prestigious national or international mathematical olympiads (such as USAMO or IMO). Typical students of Math 25 have also had previous exposure to proof writing through mathematical contests or university-level mathematics courses.

3. Notable Alumni

Problem sets are expected to take from 24 to 60 hours per week to complete,^[1] although some claim that it is closer to 20 hours.^[16] Many of those who are able to handle the workload and complete the course become professors in quantitative fields;^[5] alumni of Math 55 include Harvard mathematics professors Benedict Gross and Joe Harris as well as Harvard physics professor Lisa Randall,^[17] Harvard economics professors Andrei Shleifer and Eric Maskin, and Berkeley economics professor Brad DeLong.^[18] Contrary to a 2006 article in The Harvard Crimson which alleged that only 17 women completed the class between 1990 and 2006,^[6] 39 women completed 55a and 26 completed 55b.^[19] Math 25 has more women: in 1994–95, Math 55 had no women, while Math 25 had about 10 women in the 55-person course.^[3]

Past students of Math 55 also include Bill Gates,^[20] Richard Stallman,^[5] and Simpsons executive producer Al Jean.^[21]

Demographics of students taking this course over the years have been used to study the causes of gender and race differences in the fields of mathematics and technology.^[22]

4. Historical Instances of Math 55

5. Fictional References

Math 55, along with several other high-level mathematics courses, was brought up by Dr. Spencer Reid in a 2015 episode of Criminal Minds entitled “Mr. Scratch.” However, graduates of the class are not forced to join the NSA, as the show states.^[36]