freshman
A freshman, fresher, first year, or frosh, is a person in the first year at an educational institution, usually a secondary school or at the college and university level, but also in other forms of post-secondary educational institutions.
A ...
undergraduate mathematics course at
Harvard University
Harvard University is a private Ivy League research university in Cambridge, Massachusetts. Founded in 1636 as Harvard College and named for its first benefactor, the Puritan clergyman John Harvard, it is the oldest institution of high ...
Shlomo Sternberg
Shlomo Zvi Sternberg (born 1936), is an American mathematician known for his work in geometry, particularly symplectic geometry and Lie theory.
Education and career
Sternberg earned his PhD in 1955 from Johns Hopkins University, with a thesis en ...
. 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.
Description
In the past, Harvard University's Department of Mathematics had described Math 55 as "probably the most difficult undergraduate math class in the country." But Math 55 lecturer for 2022 Professor Denis Auroux clarified that "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."
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
In mathematics, general topology is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differential topology, geomet ...
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.
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
Benedict Hyman Gross is an American mathematician who is a professor at the University of California San Diego, the George Vasmer Leverett Professor of Mathematics Emeritus at Harvard University, and former Dean of Harvard College.
In 2022, on average, students spend a total of 20 to 30 hours per week on this class, including homework. In addition to visiting their professor during office hours, students are encouraged to work together on homework assignments. Many have spent much time together working in the "war room" (a place in the Grays Basement) and have developed long-lasting friendships. Taking many other challenging courses and extracurricular activities in the same semester is ill-advised.
A
take-home final
An examination (exam or evaluation) or test is an educational assessment intended to measure a test-taker's knowledge, skill, aptitude, physical fitness, or classification in many other topics (e.g., beliefs). A test may be administered verba ...
ends the class.
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.David Harbater, a mathematics professor at the
University of Pennsylvania
The University of Pennsylvania (also known as Penn or UPenn) is a private research university in Philadelphia. It is the fourth-oldest institution of higher education in the United States and is ranked among the highest-regarded universit ...
and student of the 1974 Math 55 section at Harvard, recalled of his experience, "Seventy tudentsstarted 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).
Numbers of students dropping are due in part to the tendency of undergraduates to "shop around" for appropriate courses at the start of each semester.
Course content
Through 2006, the instructor had broad latitude in choosing the content of the course. Though Math 55 bore the official title "Honors Advanced Calculus and Linear Algebra," advanced topics in
complex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebra ...
, point-set topology,
group theory
In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups.
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ...
, and differential geometry could be covered in depth at the discretion of the instructor, in addition to single and multivariable
real analysis
In mathematics, the branch of real analysis studies the behavior of real numbers, sequences and series of real numbers, and real functions. Some particular properties of real-valued sequences and functions that real analysis studies include con ...
as well as abstract
linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as:
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as:
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces and through matric ...
. In 1970, for example, students studied the differential geometry of Banach manifolds in the second semester of Math 55. 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
In vector calculus and differential geometry the generalized Stokes theorem (sometimes with apostrophe as Stokes' theorem or Stokes's theorem), also called the Stokes–Cartan theorem, is a statement about the integration of differential forms ...
, though, time permitting, other relevant topics (e.g. category theory,
de Rham cohomology
In mathematics, de Rham cohomology (named after Georges de Rham) is a tool belonging both to algebraic topology and to differential topology, capable of expressing basic topological information about smooth manifolds in a form particularly adap ...
) might also be covered. 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'', an abstract treatment of calculus in the setting of normed vector spaces and on
differentiable manifold
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...
s, was tailored to the authors' Math 55 syllabus and served for many years as an assigned text. Instructors for Math 55 and Math 25 have also selected Rudin's ''Principles of Mathematical Analysis'',
Spivak
Spivak or Spivack is a surname of Ukrainian language origin, meaning ''singer''. It is also common among Ukrainian Jews, in which case it refers to cantor. The name may refer to:
* Charlie Spivak (1907–1982), American trumpeter and bandleader ...
's '' Calculus on Manifolds'', Axler's ''Linear Algebra Done Right'', and Halmos's ''Finite-Dimensional Vector Spaces'' 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 real analysis) and Math 122 (group theory and vector spaces) in Math 55a; and Math 25b (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 proof
A mathematical proof is an Inference, inferential Argument-deduction-proof distinctions, argument for a Proposition, mathematical statement, showing that the stated assumptions logically guarantee the conclusion. The argument may use other previo ...
s is listed as a course prerequisite for Math 55, while such experience is considered "helpful" but not required for Math 25. 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
IMO or Imo may refer to:
Biology and medicine
* Irish Medical Organisation, the main organization for doctors in the Republic of Ireland
* Intelligent Medical Objects, a privately held company specializing in medical vocabularies
* Isomaltooligos ...
). Typical students of Math 25 have also had previous exposure to proof writing through mathematical contests or university-level mathematics courses.
Notable alumni
Problem set A problem set, sometimes shortened as pset, is a teaching tool used by many universities. Most courses in physics, math, engineering, chemistry, and computer science will give problem sets on a regular basis. They can also appear in other subjects, ...
s are expected to take from 24 to 60 hours per week to complete, although some claim that it is closer to 20 hours. Many of those who are able to handle the workload and complete the course become professors in quantitative fields; alumni of Math 55 include Harvard mathematicians
Benedict Gross
Benedict Hyman Gross is an American mathematician who is a professor at the University of California San Diego, the George Vasmer Leverett Professor of Mathematics Emeritus at Harvard University, and former Dean of Harvard College. and Joe Harris, Columbia mathematical physicist Peter Woit, Harvard physicist Lisa Randall, Harvard economists
Andrei Shleifer
Andrei Shleifer ( ; born February 20, 1961) is a Russian-American economist and Professor of Economics at Harvard University, where he has taught since 1991. Shleifer was awarded the biennial John Bates Clark Medal in 1999 for his seminal works ...
and
Eric Maskin
Eric Stark Maskin (born December 12, 1950) is an American economist and mathematician. He was jointly awarded the 2007 Nobel Memorial Prize in Economic Sciences with Leonid Hurwicz and Roger Myerson "for having laid the foundations of mechanism ...
, and UC Berkeley economist Brad DeLong..
Past students of Math 55 also include business magnate and computer programmer
Bill Gates
William Henry Gates III (born October 28, 1955) is an American business magnate and philanthropist. He is a co-founder of Microsoft, along with his late childhood friend Paul Allen. During his career at Microsoft, Gates held the positions ...
, computer programmer and free-software promoter
Richard Stallman
Richard Matthew Stallman (; born March 16, 1953), also known by his initials, rms, is an American free software movement activist and programmer. He campaigns for software to be distributed in such a manner that its users have the freedom to u ...
, and television writer and executive producer
Al Jean
Alfred Ernest Jean III (born January 9, 1961) is an American screenwriter and producer. Jean is well known for his work on ''The Simpsons''. He was raised near Detroit, Michigan, and graduated from Harvard University in 1981. Jean began his wri ...
.
Demographics
Contrary to a 2006 article in ''
The Harvard Crimson
''The Harvard Crimson'' is the student newspaper of Harvard University and was founded in 1873. Run entirely by Harvard College undergraduates, it served for many years as the only daily newspaper in Cambridge, Massachusetts. Beginning in the f ...
'' which alleged that only 17 women completed the class between 1990 and 2006, 39 women completed 55a and 26 completed 55b. 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. In 2006, the class was 45 percent Jewish (5 students), 18 percent Asian (2 students), 100 percent male (11 students).Burden of Proof", Logan R. Ury, Dec 6, The Crimson , 2006. "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."
Demographics of students taking this course over the years have been used to study the causes of Achievement gaps in the United States#Math and science">gender and race differences in STEM.
Curtis McMullen
Curtis Tracy McMullen (born May 21, 1958) is an American mathematician who is the Cabot Professor of Mathematics at Harvard University. He was awarded the Fields Medal in 1998 for his work in complex dynamics, hyperbolic geometry and Teichmüller ...
Noam Elkies
Noam David Elkies (born August 25, 1966) is a professor of mathematics at Harvard University. At the age of 26, he became the youngest professor to receive tenure at Harvard. He is also a pianist, chess national master and a chess composer.
Ea ...