Reflections on the Last Month

So I managed to blog every day in November with varying degrees of quality. So what did I learn? First, while writing every day might be a good idea, blogging every does not seem to be great. I felt quite burnt out by the end as evidenced by some of my latter posts. Second, while blogging every day does force me to avoid procrastination through excess preparation, it also means that ideas that require some more time to develop don’t get that. I don’t have a solution to resolve this tension, but it’s something I should think about. Third, I think I want to transition to talking about mathematics strictly on this blog. This was difficult to attempt in the previous month, since writing every day was difficult even without restricting the subject matter. It seems that the readers agree, since more views came from mathematical posts.

So what’s in store for this blog? I can’t make any promises, but here’s some plans.

  1. Progress on research. I still haven’t published my approximate diagonalization paper. I have some hopes of doing some further research along these lines, and as I get some insight I hope to write about it.
  2. Math education (broadly defined). I hope to (at least privately) write some lecture notes for possible future use or for hypothetical classes, since I am currently unemployed. I may also discuss either actual math education research or some popular mathematical videos or other media.
  3. Math in Plain English. I might try revisiting my own attempts at writing about mathematics for people without a technical bent. My original motivation for this series is gone, but it might be worthwhile, nonetheless.
  4. Mathematical reading. I’d like to read some mathematical texts, including some classics. I have a large backlog, so hopefully I’ll develop some thoughts while reading and might make some insights.
  5. The use of categories in analysis. I made a small post about this, but I think there’s some untapped potential here that I want to understand. Of course, there’s a lot that already done which I need to get caught up on.

Thank you for reading this past month. If all goes well, I hope to see you again soon.

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The Future of this Blog

Casey Neistat ended his vlog today. I’m a fan of his work, and while I’ll miss the daily vlogs, I look forward to whatever new projects he puts out. But what interested me the most was his reasons. Contrary to the thought that vlogging every day would be difficult, it seems that rather than encouraging innovation, the vlogs became habitual.

I couldn’t help but make comparisons with my blog and a similar project. Of course, it’s presumptuous to compare my tiny blog to a YouTube channel with more than 8 million subscribers. But this too was a project to force myself to be more creative and to actually produce something (maybe not stunning videos, but something). And the small scope and audience of this blog make me wonder what I’m trying to accomplish with this blog. Of course, the size of my audience doesn’t matter if my purpose doesn’t require a large audience. But my purpose will reflect my hopes and goals for this blog.

I’m afraid I don’t have any answers to this question. While I won’t stop blogging now, I find myself needing to reflect on what the future holds for this blog, and if the lack of purpose is the reason for the previous lack of motivation to post.

A Story about the Past Three Posts

I consider the previous three posts I wrote as a trilogy of sorts, and it was my procrastination of posting the last one that motivated me to start this blogging project. I’ve been meaning to write up yesterday’s post since June.During a conference at Hebei Normal University in Shijiazhuang, I gave a talk where George Elliott questioned, among other things, the fact that integer-valued continuous functions form a free abelian group. I gave the non-constructive answer that I gave at the beginning of the yesterday’s post. Later, during the conference banquet, I spoke with him about this problem, where he proposed a more constructive solution for the Cantor set. It was a bit difficult to understand without any visual aids, and since I was not entirely sober at the time. After this banquet, I spoke to my friend and colleague Wei Sun about this problem and he said that he had considered a similar solution for the Cantor set.

During a conference at Hebei Normal University in Shijiazhuang, I gave a talk where George Elliott questioned, among other things, the fact that integer-valued continuous functions form a free abelian group. I gave the non-constructive answer that I gave at the beginning of the yesterday’s post. Later, during the conference banquet, I spoke with him about this problem, where he proposed a more constructive solution for the Cantor set. It was a bit difficult to understand without any visual aids, and since I was not entirely sober at the time. In retrospect, I suspect he was correct. After this banquet, I spoke to my friend and colleague Wei Sun about this problem and he said that he had considered a similar solution for the Cantor set. It was much clearer with visual aids.

At this point, I thought this proof would make a nice little blog post once I was able to generalize the result to all compact Hausdorff spaces. But the generalization wasn’t immediately clear since both people talked about the Cantor sets in terms of the standard middle-thirds construction. This reminded me of the proof in Davidson that I talked about two days ago, and I switched views from the middle-thirds construction to a picture of the infinite complete binary tree (given by the Bratteli diagram) for the Cantor set. With the picture of rooted trees, it became clear how to do this for totally disconnected spaces and by taking the appropriate quotient, how to do it for all (compact Hausdorff) spaces.

Being reminded of that Davidson proof led me down a rabbit hole of how I reconciled that proof with this picture, which brought me to the Stone-Weierstrass theorem: the post from three days ago. So the ideas came in reverse from how I posted them, which seems how a lot of math works in my experience.

I hope you’ve enjoyed that trilogy of posts. They were ideas that I enjoyed thinking about and enjoyed writing about.

Blogging every day in November

In the first term of my undergraduate schooling, I took a math course that was considered one of the most difficult courses in the university. Given that, ultimately, I would go on to obtain a doctorate in mathematics, one could say that that decision changed my life. It was that challenge that, among other things, appealed to me. The course was quite difficult for me, but I wanted to be challenged and pushed to work harder than I ever did before. Throughout my life, whenever I looked back to the past, I always regretted not having worked harder and accomplished more. I didn’t really give myself enough credit; I did work hard, at least at times. It’s easy to regret the past when the troubles and temptations of the times are easily forgotten or diminished. But what became clearer over time is that the difficulty in working hard is mostly in the motivation. It is, in a sense, easy to work hard when faced with an explicit, external challenge (at least for me). The work needed to pass a difficult course is more tangible and immediate than most work I have nowadays, which is often vague with unremarkable incremental progress.

Of course, one solution to this is to make explicit the vague challenges that I want to face. When I taught, I would have a difficult time writing lecture notes for myself despite being a useful tool to have. It became much easier to write them when I posted them online. Writing a set of lecture notes became a much clearer task when I had an actual audience to write for. Also, it was much easier to write lecture notes when I had students emailing me asking when the next set will be available.

Some years ago, I decided to try to write every day in the month of November. After some thought, I have decided to try that again this year. While there’s not likely to be a demand for my silly diary entries or half-baked mathematical ideas, making a public announcement is likely to push me to write every day. It might not be the best motivation, especially since the last time I did this, I stopped writing on Thanksgiving.

So on the off chance that someone is reading this, thank you. I hope at least a few entries in this project are of some value to you.

New Year Resolutions

With the new calendar comes promises to improve, and so here are my aspirations for this blog:

  1. Write at least one blog post every fortnight, but aim for closer to once a week.
  2. Complete the “Math in Plain English: Topology” series.
  3. Write a cited post about pedagogy each academic quarter.
  4. Write a post related to research or C*-algebras at least once a month.
  5. Write a post accessible to undergraduate mathematicians at least once a month.

I hope this also demonstrates the direction this blog is headed. I hope you continue reading!

MaBloWriMo

Qiaochu Yuan recently made a blog post saying that he will be writing a blog post every day in November. I, being uncreative and wanting to actually blog for once, have decided to join in this activity. Since I will be posting every day, this will hopefully help me overcome my instinct of not writing a post for fear of it being trivial.

A instance of which is this post announcing that I am blogging every day in November.

See you tomorrow.