10,000 Problems in Analysis

Summary

In this video, the speaker reflects on the emotional and intellectual challenges of preparing for and failing a qualifying exam in real analysis. After months of rigorous study, the failure led to a period of self-doubt, but the speaker turned this setback into motivation. They set a goal to complete 10,000 problems in real analysis, focusing on practice as a means of mastery. This journey not only improved their understanding of the subject but also taught them the value of persistence, learning from failure, and the importance of challenging oneself to grow.

Takeaways

• 😀 The speaker shares their personal journey of preparing for a tough qualifying exam in real analysis, including the emotional challenges of failure.
• 😀 Despite extensive study (4-6 hours a day for 8 months), the speaker failed the test, which led to a period of emotional distress.
• 😀 The speaker's initial reaction to failure was sadness, but they eventually learned that failure was part of the learning process and accepted it.
• 😀 The speaker decided to tackle their discomfort with real analysis by setting a goal of solving 10,000 problems, inspired by the '10,000-hour rule' for mastering any skill.
• 😀 The 10,000 problem challenge aimed to help the speaker build expertise and confidence in real analysis, similar to the way athletes practice extensively to become professionals.
• 😀 The speaker solved 700 problems during one semester, using a variety of textbooks and online resources, but struggled to find time to complete the full 10,000 problems.
• 😀 Books used in the study process included an undergraduate analysis textbook, Bartle's book, and Komogorov's book, which the speaker worked through methodically.
• 😀 The speaker emphasizes the importance of practice and repetition in mastering complex subjects like real analysis, even if it means copying problems from books or the internet.
• 😀 The speaker discusses the value of learning from others, referencing conversations with peers who also struggled with the exam but approached it with different mindsets.
• 😀 While the speaker acknowledges the potential criticism of using online resources, they stress that active engagement and learning from others is crucial to solving complex problems.

Q & A

What motivated the speaker to begin taking extensive notes for the qualifying exam?

The speaker realized that without taking thorough notes, they would not be able to succeed on the qualifying exam. They needed to familiarize themselves with the material and feel comfortable with it, which led them to start extensive note-taking in preparation.

How did the speaker feel after failing the qualifying exam?

The speaker was deeply emotional after failing the exam, to the point where they cried for an entire week. They were disappointed, especially after dedicating months of hard work and daily study, but they later reflected on the failure with some humor.

What was the response of other students who also failed the exam?

The speaker mentioned two different responses from peers. One student had mentally prepared for failure, accepting it before even taking the exam. Another student remained hopeful up until the test results were announced but eventually accepted the failure as a learning experience.

How does the speaker describe their approach to algebra compared to real analysis?

The speaker feels much more comfortable with algebra, as it is intuitive for them and easier to solve. In contrast, they describe real analysis as more challenging and disorienting, often feeling lost and unsure of the right approach when solving problems.

What strategy did the speaker adopt to improve their skills in real analysis?

To improve in real analysis, the speaker set an ambitious goal to solve 10,000 problems in the subject. The idea was to use repetitive practice to become more comfortable with the material, similar to the 10,000-hour rule for mastering any skill.

How many real analysis problems has the speaker completed so far, and what is their current progress?

The speaker has completed around 700 problems so far but has not yet reached their goal of 1,000 problems due to time constraints. They plan to eventually finish the challenge and believe that the practice has significantly improved their understanding of real analysis.

What types of resources did the speaker use to solve problems in real analysis?

The speaker used a combination of textbooks (like those by Bartle and Komogorov) and online resources for solving problems. They also mentioned using undergraduate-level textbooks to practice foundational problems in real analysis.

What is the speaker's stance on copying problems from textbooks or the internet?

The speaker defends the practice of copying problems from textbooks or the internet, provided it is done with active engagement. They believe that copying exercises can help learners understand the material, as long as they don’t simply copy answers without attempting to solve them.

How did the speaker address the potential criticism of their method of practicing problems?

The speaker acknowledged that some might criticize the practice of copying problems, but they argued that the goal is not to mindlessly copy but to learn. They emphasized that time constraints often require practical solutions, and as long as the learner gains understanding from the process, it’s a valid approach.

What kind of emotional journey did the speaker go through after the exam, and how did they eventually cope with it?

After the exam, the speaker went through a period of emotional distress, feeling sad and down for about a week. They describe the experience as a difficult time, but eventually, they made peace with the failure and resolved to retake the exam in the future with better preparation and a more positive mindset.