Harvard MATH 55: Studies in Algebra and Group Theory / Real and Complex Analysis
Math 55 (55A in fall, 55B in spring) is Harvard's legendary honors freshman math sequence, covering proof-based abstract algebra, group theory, and real and complex analysis at extraordinary speed and depth. It's widely described as the hardest undergraduate math class in the country.
Fennie is independent and not affiliated with Harvard University. This is an unofficial study guide.
What makes it hard
Everything. The course assumes serious proof experience (often olympiad-level), problem sets historically run 20-60 hours a week, and the pace covers material most universities spread over several upper-level courses. Most students who start drop to Math 25 or 22; finishing Math 55 is a credential in itself.
What you'll cover
- • Group theory
- • Linear algebra done abstractly
- • Rings and fields
- • Real analysis
- • Complex analysis
- • Topology fundamentals
The MATH 55 study guide
How to study for Harvard MATH 55, step by step.
- 1
Make every definition load-bearing
In Math 55, a fuzzy grasp of a definition sinks the proof built on it. After each lecture, restate the day's definitions from memory and construct an example and a non-example for each.
- 2
Work the pset in daily blocks, not weekend heroics
Problem sets historically run 20-60 hours, and the hard problems need spaced attempts. Start the day the pset drops and protect a daily block, because insight compounds across sleep cycles.
- 3
Reprove lecture theorems from memory
Close your notes and reconstruct the main proofs. The technique inventory you build this way, and the gaps it exposes, is exactly what the pset problems draw on.
- 4
Use the community, and know the off-ramps
Office hours and peer discussion are survival equipment, not weakness; write-ups stay your own. And if the placement isn't right, dropping to Math 25 or 22 in the early weeks is common and carries no stigma.
Today
Today's MATH 55 plan
What a Fennie Daily Plan looks like for MATH 55. Yours is built from your own syllabus and adapts every day to your deadlines and progress.
First plan free, no card required. Fennie is independent and unaffiliated with your school.
FAQ
Is Math 55 really the hardest math class in the country?
That's its enduring reputation, and the workload folklore (20-60 hour psets) is real for many students. The honest answer: it's as hard as the strongest freshman math cohort in the world can make it.
Who should take Math 55?
Students with substantial proof-writing experience, typically olympiad backgrounds or prior university-level abstract math. Harvard's Math 25 covers rigorous material at a more humane pace.
Can you switch out of Math 55?
Yes. Dropping down to Math 25 or 22 in the early weeks is common and carries no stigma. The sequence is designed with off-ramps precisely because placement is hard to judge in advance.
More Harvard courses
MATH 1A: Introduction to Calculus
Math 1A is Harvard's introductory calculus course, covering limits, derivatives, integrals, and the Fundamental Theorem of Calculus, for students who didn't take or don't have credit for calculus before college. It leads into Math 1B and the rest of the math sequence.
MATH 21A: Multivariable Calculus
Math 21A covers multivariable calculus (vectors, partial derivatives, multiple integrals, and vector fields) and is the standard math course for Harvard students in sciences, economics, and pre-med tracks who arrive with single-variable calculus done.
MATH 21B: Linear Algebra and Differential Equations
Math 21B covers linear algebra (systems, matrices, eigenvalues, orthogonality) and applies it to differential equations and Fourier series. It's a workhorse requirement for applied math, economics, CS, and science concentrators.
MATH 25: Theoretical Linear Algebra and Real Analysis
Math 25 (25A in fall, 25B in spring) is Harvard's honors first-year sequence in proof-based linear algebra and real analysis, the rigorous track below the legendary Math 55. It's the standard landing spot for strong students who want real mathematics at a survivable pace.