Learning algebraic geometry was difficult at first because of the lack of intuition. I made a terrible decision of taking the part III course on schemes without first learning classical algebraic geometry. It was all formalism. And while back then I memorized every single proof and theorem for the exam just in case, nothing stays in my brain afterwards. I mean, the content is in the intuition which I didn’t have anyway- what I had was just words.
Later, slowly, by reading various different books on classical algebraic geometry, curves, manifolds and Riemann surfaces, I start to get the intuition and things start feeling simple. It was like how in BSM, number theory was easier for me when I was taking six (related) courses at once rather than just three like in MiM. (Hey, more is more!) The key thing is that I get exposed to the same idea in different way, thus getting gist, intuition, as well as solidification of it in memory. I learn to speak it like natural language by hearing it spoken often, in natural context.
However, I am still on the journey. Now that I know how I should have studied it, how do I use that knowledge to optimize my current and future study? I still notice how after reading/learning about some concepts, I might still completely forget about it the next time I see it. Of course, ideally, for each such concept, I should do research on it, play with it, think about it, compute examples, search across books for different manifestations and links and research history. But I don’t have that kind of time. So what should I do considering the limited time I have?
So let’s talk time optimization. This has happened to me before several times in my life. I usually over-estimate the time it takes to do things and deadlines + time shortage forces me to be more efficient, to trim out the non-essentials. I should list those examples to store those intuition into one package, to zip/ compress them, I guess so that I can retrieve them. Oh, this compression is a bit like my dream about a dolphin. Somehow in the dream, it seems that I have had years of memories with that dolphin. How could that be? How could those years of memories be compressed into some instant like that? Is that time compression? Is that physics or psychology? And can I use that to optimize learning somewhat? This packaging of intuition is also very useful. I should explore it more. To that end, I should remember not only my conclusions, but also the specific examples and life experiences leading to them. For that, I should try to collect as many life experiences and feelings as possible- exposing myself to different circumstances and ideas. That is also good for my brain- so that all parts of it are used and none will be fired for not working, by neuroplasticity. Once I have a label, it its like forming a limit/colimit of the set of blobs which help me retrieve it.
(Bijection, phantom limb, brain’s way of healing- pain map)
I remember how I wanted to take a gap year to prep for the GRE subject, or to prep for high school math entrance exam or thought I need extra time to prep for part III exam but when I learned that was not possible, I managed to trim my strategy to get a leaner cut anyway. I notice the same when there are word limits: turns out a lot of things I thought were necessary to say aren’t so important at all. Word limit always force me to find the gist, to be more concise. Is that the appeal of Twitter? Similarly, when I have lots of money, I would buy lots of tools for a project. But when my budget is limited, I magically finds way to get things done with the tools I already have. That applies to skincare, making YOuTube video, exercise, house decor etc.
Let’s look at how I trim my strategies. By the way, small savings in time add up, just like what the book Atomic Habit says. When I used clockify to track my time, I was amazed at how much time I wasted everyday on things that didn’t seem to take much time at all. My estimation of time is really off. For example, in the morning, if I don’t get up immediately but instead daydream in bed, that would usually last about 30 mins. In 30 mins, I could have done a workout or finish an algebraic geometry video, or grading etc. Eating usually take extremely long time too. (I’m so happy that these days I barely need to eat ha ha). Usually it is activity in which I do not notice the passing of time that is very dangerous.
That is one class of time-sucking activities- those in which my attention is so absorbed that I lost track of time. Another class are small perfectionist stuffs, like rewriting a page with ugly handwriting, or always having to read a book from beginning till end, or having to read through a lot before feeling ready to start research. Do I need all of those? After all, for the Madison REU, I barely needed anything at all, I started backward from the project and fill in what is necessary. That approach will eliminate the non essentials and also will give me a perspective from which to link and internalize everything. Notice how the stuffs I learned about elliptic curves in order to understand Duarte’s paper stuck with me more than those I learned in part III? I need a narrative, a goal. The problem gives a criterion to sort through info, a utility function which allows me to put a measure on materials and thus ordering them.
When learning though, I do not need to learn every concept and every details. I just need a few generators: key ideas and examples. Everything else will be auto fill by the brain. Especially for proofs- I should only look at details.
Instead of reviewing, I could internalize things by reading more on it by from different sources and perspectives instead. That is, I want to approach it horizontally, across books and articles, rather than vertically. For example, say if I just read about Keynes once, especially in some list like 30 famous economists, I won’t be able to remember his theory. However, reading about him across books, hearing him mentioned in podcast and videos help me naturally remember him. It is like how I could not remember anything about Games of Thrones by watching a quick recap. However, had I watched the series itself, I would have been able to memorize the characters’ names fine.
I notice one of my super-power (wahahahaha đŸ˜ˆ) is that though I can’t generate ideas out of the blue, I can see gist and link easily. So my approach to research should be stealing- in the sense of T.S.Elliot “Good artist borrows, great artists steal”. For example, to make my cover art for each of my AG notes, I randomly think of a word, google a picture, use that as an inspiration to draw the cover art. Similarly, for my dancing, I watch Bai Shi fu and other dancers, then use those to prime my own movements. For ideas, especially in math, it is much subtler. For example, when I read Duret, I try to find the gist of what he did. He didn’t fully explore the general phenomenon. So once I know the gist, I can exploit that to generates other things. Since I steal the gist instead of the specific, it becomes mine. It is the same process I use for teaching and that is why I enjoy teaching so much. When I am presented with any material, my brain automatically find a way to organize and extract the gist out of it. So my process research should be by utilizing this ability. The key point is to find the right thing to take the gist out of.
What is great about this process is that it is not dependent on luck or the whim of any muse. I don’t need to wait for any creative spark to strike. I know the process is always guarantee to work. So I am always moving forward.
By the way, that is one great thing about writing things down. It allows me to move forward. When ideas aren’t written down, they just go around in my head in circle. And they won’t become a coherent theory.
I notice how a lot of sophisticated mathematical results are built from very simple ideas and proofs. I often marveled at the extent to which such simple thoughts could be utilized. Let’s try to absorb that kind of meta intuition and aesthetics and replicate that.
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