My Journey Through Math 231: An Undergraduate Exploration of Graduate-Level Algebraic Topology at Harvard

As a sophomore at Harvard, I started my sophomore year taking what I believe is one of the most intellectually stimulating courses, Math 231: Graduate-Level Algebraic Topology, under Professor Hana Kong, a BP Fellow. This course has been a true revelation and I will provide my insights and review of the first term, 231A, here. My journey through this course has been an enriching experience, and I am eager to delve deeper into the material as we proceed with 231BR in the upcoming term.

The Course and Its Content

Math 231 is an advanced undergraduate course that delves into the realm of graduate-level algebraic topology. It serves as a sequel to the fundamental group and covering spaces discussed in freshman-level course 55. The course explores deeper topics such as homotopy and cohomology, which are truly beautiful and intricate fields of study.

The course is designed to provide a comprehensive understanding of these advanced concepts, building a strong foundation for students interested in pursuing further studies in topology and related fields. The material is challenging, but the rewards for tackling the complexities are immense. Professor Kong's lectures are well-delivered and filled with insightful observations that help learners navigate the abstract landscape of algebraic topology with ease.

Professor Kong's Insights

Professor Hana Kong is not just a professor but a guide through the world of topology. Her lectures are a masterclass in teaching. She emphasizes the importance of geometric intuition in understanding complex concepts and how to approach them with a topological mindset. Through her lectures, I have learned how to imbue geometric thinking into my problem-solving strategies, making the abstract concepts more tangible and accessible.

Professor Kong also stresses the significance of using apt analogies to make sense of homological algebra and category theory. This approach has been incredibly useful, as it enables me to bridge the gap between abstract theory and practical application. These insights have been transformative in my understanding of the subject, making the coursework not just a series of calculations, but a journey of discovery and insight.

The Textbook

While the course is challenging, the textbook, authored by J.P. May, is notoriously terse and dry. However, this challenge is mitigated by the quality of the lectures and the supportive teaching assistant, Lev Kruglyak, who has been instrumental in making the course accessible. The textbook is dense but provides a wealth of problems, which are both fun and intellectually demanding.

The problems in the textbook are designed to reinforce and expand upon the concepts discussed in the lectures. They are well-suited for developing a deeper understanding of the material, as they require not just rote memorization but rigorous thinking and application. This has been an excellent way to apply the techniques and ideas I have learned in class, turning abstract concepts into concrete problem-solving skills.

Teaching Assistant Support

Lev Kruglyak has been a standout teaching assistant for Math 231. His contributions to the course have been invaluable, providing additional support and clarification during office hours. Kruglyak's expertise and enthusiasm for the subject have made the course even more enjoyable and enriching. His insights and problem-solving techniques have been a significant boost in understanding and tackling the more complex problems.

Personal Reflection and Conclusion

Personally, I have found taking Math 231 to be one of the most rewarding experiences of my undergraduate journey. The material is beautiful and nuanced, and I have discovered that algebraic topology is a field that genuinely interests me. The course has provided an excellent mathematical training, requiring me to develop and apply a variety of reasoning strategies and work with high levels of abstraction.

In summary, I highly recommend Math 231 to anyone looking for a direct follow-up to 55 or a combination of courses 122 and 131. It is a fascinating and enriching course that will challenge you intellectually while equipping you with the tools to succeed. Whether you are a mathematics major or simply have an interest in topology, Math 231 is a course you should consider.

Stay tuned for updates as I continue my journey through Math 231BR and beyond.