On Optimizing Time for my Research

 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 😈) 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. 

Organization

Now next, you need to organize and internalize. Which one should come first?

For organization, I remember my friend Kien mentioned how his Part III prof in Cambridge told them to summarize the subject three time. The first version can be as long as they want, second version should be three page long and the last version one page. This is what I usually do- do you see how I have so many writings on the same subject? My first notes on a subject are usually to flush out to details, to turn words into blobs. During that process, I realize a more efficient way to organize the knowledge. Usually books present the knowledge to me as a bunch of disjoint facts (may be they have a logic, but it is not “my” logic). Once I’ve flushed out all the details, turning them into blobs, I can perceive the links and the key ideas. (I should write more about this automatic “gistify” and “narrative-making” operation. Since noticing their power, a lot of my learning and thinking strategy has boil down to strategically exposing myself to the right thing at the right time. A way to solve a problem or to generate idea for example, is by collecting a bunch of “primers”, “triggers”. The juxtaposed exposure of them not only stimulates the brain but also forces it to take intersection, drawing connections and finding out the  gist, generalization or link them into a story. Notice how dreams are just an artificial narratives created to connect blobs, like a necklace for holding beads?).

(Drawing on the right side of the brain- perception and accent)

I then rewrite the theory based on my personal organization. That is a slice: i noticed the key points and relations (i.e. paths between them) in the blobs and thus cut through the blob with a sub variety including only those key points. All other points can be generated from those key points and relations (i.e. I was finding generators for my blob). The paths might not be the shortest distance, not geodesic. So I keep optimizing, reducing the number of points, changing the path and thus reducing the dimension of the “gist variety”- the generator of my variety. Each of this slice corresponds to a version of my summary. Note that the underlying “language” matter. Depending on my choice of language (i.e. ground field), I can perceive different things. So I can create a language in which things become simpler also. 

Blobification

They say that you can’t teach old dogs new tricks, but I have found that as I get older, learning has become easier. For one thing, I have a lot more things to link things to- a lot of “base blobs”, I guess. A lot of things are manifestations of ideas I have seen. Other things are rearrangements of blobs I have seen- thus being exposed to them allow me to see the gist- the core blobs- and also to strengthen the imprint/storage of the blobs. But more importantly, I have developed a much better understanding of the brain and know now how to learn optimally. 

For example, let’s say I want to learn Econ. What would be my plan? Well, let’s say I’m writing a plan for my future kid. That makes it easier to imagine and thus create an automatic response/momentum. It triggers a feeling because it’s another person. This was the reason why I used to adopt the idea of Jesus- to be able to externalize a part of me into a believable otherness that I could talk to and avoid the paralysis of not having a criteria to select an optimal explanation (a gradient to do gradient descent, ha ha).

OF course, the first step is to have an overview of the subject. You should use Wikipedia, table of contents, browse book- just window shop the subject for a bit to see what it is about. Then read and watch popular books/videos- layman stuffs for intuition. Stories are preferred, but the ideal materials are arguments. It’s best when stories are presented as evidence supporting some theory/arguments. The argument create an organic way of linking different things- unlike the artificial links created by the memory palace or the journey method. The argument, just like Jesus and my imagination of you, are also “auto trigger”, they will usually turn on a momentum. I guess we are wired to respond to conversations with others- dialectic anyone? 

To learn a concept, say “elasticity”, “Monroe doctrine”, or a fact about a person, look it up to read relevant facts, other manifestations, examples, usage, videos and arguments,  about it to turn it into blob and also to link it to other blob. Otherwise, it is just a word. The content of learning are the blobs, not he words. What I often call intuition is also a blob. What I want to get is the right blob. To find the right blobs, I also need to intersect the blobs- thus exposing myself to different manifestations of the same thing. Furthermore, I also need to convince my brain that the fact, the details, are worth remembering. If I cannot use emotions (evoking emotions or using an anchored sense, such as music, to store and evoke memory is an interesting topic I should explore more in future posts), then I should use repetition. BUt pure repetition does not do much, because the brain gets habituated and does not pay attention upon the second repetition- it just spaces out and does not perceive everything. So intersectional repetition/varied repetition is better. It is the same as learning natural language. You hear the same phrase spoken in many different contexts. Imagine the concept is an idol you like and stalk it. Read up on gossips about it. Don’t read books from beginning till end, but rather, follow ideas across books. Chase ideas instead of books. 

For math, this “blobification” is even more important. Concepts in math are often given without any motivation. Note that motivation and intuition are two different things though. So when you learn a concept, you need to get both motivation and intuition. You need to find some motivation: a place to start is by looking to see how the concept was develop historically. Another way is by seeing how it is used in proofs. A method of learning is find the key theorems in each book and learn backward from there. That is, imagine the goal is to prove these theorems, see how the concepts developed were useful for that purpose. Then everything is linked by these theorems. Everything is developed for the sole purpose of solving these theorems.  

For intuition, one needs to compute and see examples, especially of usages. thus in learning a concept, you can skip even writing down definitions. All need you is to work out examples to identify the blobs. The best way to learn is to collect good examples. A few good key examples can be a great source in intuition to refer to when learning various different things. That is why often when one is goo at one domain/topic, that helps to excel at other domains, because one can transfer intuition across domains by drawing analogies. 

 

Catching the Initiation of an Impulse

 Yesterday, a wonderful thing happened: I finally was able to release my chronically tight muscles. It’s not just that- I was able to basically release the tension in all my body parts at will (except for the forehead). 

How did this happen? Well, I was doing a nighttime Yoga routine from Yoga dose. During the process, I started to develop a keen sense of different impulses coming and going in my body. I started to be able to “identify” the tensing impulse, before it could take over. By identifying, I mean two things: I notice the sensation associated with “tension is about to happen” and the “thought” (not quite a verbal thought, more like a mental activity) that triggers that sensation. In other words, I noticed both the stimuli and the reaction. Notice here means I being able to find the corresponding blob, establishing an isomorphism. 

This was like how Alexander fixes his voice. When reading about noticing the stimuli, I have always thought that meant the “external stimuli”. Now I realize that that should be the inner stimuli, the mental sensation/blob that triggers the impulse. And once I “find” the knob that turns on the impulse, I gain control over it. Not only can I turn it off, I can also adjust its intensity. This is the idea in TRIZ and Buddhism, as well as modelling curves, with enough data points at the extremes, we find the shape of the function (the knob) and thus can adjust the values at will. That is why it is better to experiment to see which input gives which output rather than aiming for one exact perfect output (cf. the story in Atomic habits about quantity over quality). Also, this is why, like Costa said, I should let myself fall. When I tense my body in order to keep balance, I block the sensory input to the vestibular system, so my body does not learn to balance (cf. Smart Yoga).

(I used to need to teach to generate ideas, but I guess now I can just write down my thoughts instead? The difference between this stream of consciousness approach and teaching is that without an audience to adjust to, I don’t have a way of eliminating and selecting optimum explanation. Without an audience, what would optimum mean- for what background? Thus I would skip steps in my writing, leaving holes undetected.)

Finally now I understood what that Russian book about stretch was saying. By understanding I mean now I could actually associate to the words in it a corresponding sensation, i.e. establishing an isomorphism between sensations and words. (Concretization) When I do I stretch, now I notice when the stretch impulse, i.e. the tension kicks in. At that moment, I will release it, and then continue to stretch deeper. This is similar to Amanda’s exercise with the feet crawl: crawl, notice tension then release, then repeat. The point is to break the process into many micro steps- micro in size to release tension before enough can accumulate. I think if I practice this enough to the level of conditioning and imprinting this in my memory, I will be able to release tension at will. Then I will be able to do split, yay! Would that help me with tension headache, singing and dancing too?

Why does my tension release so easily when I dance? Is that because it’s the music, not the conscious mind, that moves me? This is another kind of isomorphism, like the one in the treatment of the phantom limb. 

My dream (maybe fantasy) is that once I have establish enough trust with my body that it will allow me to do these stretches, I can transfer that feelings to ask other things from my mind and my body. What other things?

The feelings I get from teaching for example, are very useful. They become analogies that I can reuse and apply to my other situations, include my own studying. “Understanding” is always about establishing an analogy, and isomorphism with some feelings/blob you already have or you just created. So it is good to have as many blobs as possible for things to link to.

I was very intrigued by the study of French nuns with Alzheimer’s. Though they had Alzheimer’s, there’s were no symptoms. The brain was physically deteriorating, but the mind wasn’t. The reason was that thei knowledge and memories were linked in my ways, so there were many ways to retrieve the same memories. The number of links and things being links also strengthen the weight/strength/durability of the memory. Thickens the memory?

Charly tennis/open can/David Lynch

Can I apply this to thinking habits too? It’s much harder to notice the initiation of a thinking impulse. And what for? Can I redirect my thinking- is it like meditation? Does this leads to complete attention? Does complete attention leads to childlike spongelike plasticity and absorption (e.g. learning language by immersion) 

I probably should also write down thoughts about my thesis in the form of streams of consciousness like this. 

Attention, in the usual sense, only involve pushing the tip of the mind to the object of attention. Usually the background is still working on something else- thus creating a struggle. We need to full momentum- all cells to work on the same thing, creating a flow state, so that there is no resistance. This reminds me of what my weight lifting coach at Uchicago said about how lifting will become easy once the brain cells learn to fire together efficiently all at once. To apply this to thinking, there needs to be appropriate priming. I need to be mindful of what I exposes my brain to, since every such topic will start a momentum of thoughts in the background. Soon enough, so many apps will be running in the background, secretly leaking and exhausting my mental battery. 

Why don’t we have the ability to learn as fast as babies? Well, there is the plasticity problem and the basal ganglia problem. But more importantly, for the mind, there are two problems: attention and rounding. An example of rounding is accent. Another thing is how we interpret things in the way we think, or we only perceive, pay attention to what we care about. The stimuli aren’t fully perceived to begin with, how can they be store? Let’s talk a bit about learning next. 

    

First Law of Mental Motion: Law of Inertia

Recently I have been decorating my Algebraic Geometry notes- and I was surprised to find how “pretty” my “art” turns out. Why was that surprising? Usually if I am assigned to draw a picture, I would get quite stuck, paralyzed by the infinite possibilities. This is often the cause of my procrastination by the way- I always want to wait until I have time to examine all the possibilities to choose the optimum. 

What happened recently was I was short for time, and since the art was of no importance, I just decided to select a starting image at random and then go with my intuition. Once the first image was laid down, my brain immediately sees what else is needed- it picks up a momentum. This reminds me of the story of a sculpting competition. A competitor just stares at the clay without doing anything, then suddenly throws the piece of clay on the floor. Looking at the deformed clay, the competitor explains that he/she now can see what the clay “wants to be”. 

In a sense, my conscious brain has been able to delegate the task to my subconscious/automatic system. It is the same system/process/program that generates my hypnagogic images. I’m always so fascinated by my hypnogogic images. The incredible thing about that process is that seemingly unrelated concepts get linked. Sometimes I even had dreams that anthropomorphised math concepts. Thus at some levels, these “blobs of feelings” can be deconstruct and rearrange to create images, in a Lynchian way.   I want to understand “blob arithmetic”.

We will talk more about blob arithmetic, especially the intersection operation, in other posts. Right now, let’s just focus on one law of blob arithmetic: “inertia/momentum.” Conscious energy is only needed to start/“prime” the process- then momentum makes it continue. In practice, when anticipating energy needed for a task, we often make the mistake of assuming conscious effort is required through the whole process. In reality, effort is only needed in the initiation of the task. We are only the managers- we decide which tasks are to be done, then we delegate the task to the subconscious and get out of the way. Our usual over-estimation of effort leads to problems such as too much unnecessary physical (and psychological) tensions. I know this too well from my chronically painful thighs. 

This is the case with teaching. It’s very hard for me to just think of a lecture. But once the lecture starts (with an audience), my brain just naturally knows what to do next and it is basically effortless. 

This is the idea behind improv, isn’t it? One of the most memorable example in Keith Johnston’s improv book for me is the one where he guides someone to invent a story. He  asks them to close their eyes and then starts asking them questions about what they see. Magically, as if there is a hidden director in their head, images are projected to their conscious. Thus creativity is not so much conjuring up something out of nothing, but rather, to remove the obstruction. This is a bit like Milton Erickson’s philosophy of therapy: use the stream that is already there- simply redirect it. 

So I decided that instead of worrying about getting all my ideas organized before writing them down, I will just start the writing to actually discover the ideas. In other words, I’m giving my subconscious a chance and an outlet to talk to me, as well as a way to organize my thoughts.. 

I remember in 6th and 7th grade, every afternoon, the Literature teacher would make us write a short essay on the spot. The essays must satisfy some conditions, e.g. use certain words and certain sentence structure. I have always found that quite easy. I want to remember, recall and store that feeling to apply to any tasks these days. If I have kept up that habit, I could have written many mini articles and essays now. 

I have always found it mysterious how writing things down seem to create new ideas. I remember my first philosophy essay: nothing I wrote down was new to me. I have thought about those issues for years before. Yet somehow, after I wrote it, I felt like I learned something new. How could that be? Maybe it is the concretization of the ideas? Instead of letting them float out like blobs, I tied them down to words and thus making them easier to be retrieve. I do notice that when talking to people, I could explain some ideas much better if I have written or spoken about it before. Other ideas, even if I understand them well and have thought about them extensively, often fail to materialize into words. It’s not just that: another problem is that the blobs aren’t linked into a structure or a narrative. Why does that matter for communication with others (including with myself?) Why does that matter for thinking? Is it because then their coming and going to the front stage (the conscious) is rather random, instead of systematic? But why would that matter? It seems that this restricts and slows down the process of obtaining gist and depth, but why?

Dancing is another great example. If I tried to imitate the dancers in the video, I get really strained, confused and discouraged. What I have been doing these days is to look at those dancers only “primers/inspirations”. At each instant, their image caused my brain to fire an impulse, and then I just follow and improvise base on that impulse. So again, they only take part in the initiation of the movement. Since my conscious does not try to intervene during the movement, allowing the subconscious to fully express itself, the movement is more fluid.