Math 55 at Harvard University

Introduction

• Math 55: Allegedly the hardest undergraduate math course in the U.S.
• Condenses 4 years of math into 2 semesters.
• High drop-out rate: Allegedly half the students drop after the first semester.
• Notable alumni: Bill Gates, former members of international math Olympiad teams.

Course Structure

• Math 55A: Studies in Algebra and Group Theory.
• Math 55B: Studies in Real- and Complex Analysis.
• Curriculum: Linear and abstract algebra, representation theory, real and complex analysis, algebraic topology.

Professor Harris

• Joe Harris: Esteemed mathematician in algebraic geometry.
• Makes complex topics more digestible by using related concepts.

Key Topics Covered

• Seifert-van Kampen Theorem: Fundamental group of a topological space in terms of two open path-connected subspaces.
• Algebraic Topology: Studies shapes and spaces using algebra, focused on the number of holes in shapes.
• Example: Donut and mug both have one hole; objects can be classified by the number of holes.

Class Experience

• Small classroom: About 20-30 students, actual enrollment around 60.
• Collaborative environment: Students ask questions openly, engage in discussions.
• Professor's enthusiasm: Infectious and makes the subject engaging.

Student Insights

• Backgrounds: Varying levels of prior experience, from competitive math to discovering a love for math at Harvard.
• Example: Won Jae-seo struggled initially but adapted by spending extra time.

Problem Sets (Psets)

• Rumors: Psets take 24-60 hours. Reality: Generally 15-20 hours, up to 30 hours max.
• Collaborative nature: Students work together and the grading is lenient.

Myths vs. Reality

• Drop-out rate: From 1970, outdated. Currently, only 3.5% drop out rate.
• Historical vs. Present: Course evolved with standard curriculum and better-prepared students.

Community and Support

• Sense of community: Strong bonds among students, supportive environment.
• Math department's acceptance: Welcoming to students who show interest and dedication.

Conclusion

• Challenging but rewarding: One of the hardest but also most supportive and collaborative courses.
• Importance: Community-based learning, students forming lifelong bonds and contributions to future mathematical developments.