I am a first-year graduate student, and I aim to shift my focus to geometric analysis because I have recently realized that it is likely the field I want to pursue. In the past, I was more focused on algebra, which I still deeply enjoy, and I find algebraic geometry particularly fascinating. However, I now need to solidify my analytical foundations. Previously, I studied analysis from Rudin’s Principles of Mathematical Analysis, but I disliked its style and ended up skipping many exercises. As a result, I now plan to use Pugh’s Real Mathematical Analysis (2nd ed.) due to its more intuitive and geometric approach.
I would greatly appreciate any suggestions or advice to help me on this path, particularly regarding how to effectively use Pugh’s book to build a strong foundation for geometric analysis. For example:
Which chapters or sections are most relevant to geometric analysis?
Are there specific exercises that are particularly valuable for developing intuition and skills in this area?
How can I balance working through Pugh’s text with learning other topics essential to geometric analysis?
While some have expressed doubts about my ability to succeed in this transition, I am deeply passionate about learning this material and am committed to putting in the effort required.
