Nash equilibria in human relationships

A Nash equilibrium is a state of equilibrium between two parties in which one only party changing their behavior will not make things better for that party. It will require both parties to change their behavior simultaneously to make things better for both.

It might surprise the reader to know that the author of this blog has had some experience with societal relationships; even some with the opposite gender. My college girlfriend and I started on a good note, and then due to mistakes made with time, things became more complex. Towards the end of our relationship, we settled upon an equilibrium in which we were both rude, uncompromising and unhelpful towards each other. Let me try and explain why this was a Nash equilibrium.

When one of us would feel bad about how things had turned out, and would try to be nicer so as to turn back time and return to how things were, the other would be unaware of such intentions, and only mock the other person for being fake and artificial. This would, of course, anger the person with good intentions, and we would both return to being terrible to each other. Note how one person trying to change their behavior for the better only made things worse for that person (invited accusations of insincerity). This is why this was a (bad) Nash equilibrium. A bad Nash equilibrium is caused by a lack of coordination. If both parties agreed that things had taken a turn for the worse, and that they had to change together for things to become better, there could perhaps be some hope.

I see relationships at bad Nash equilibria all around me. My extended family is often at bad Nash equilibria with one another. Many members of my extended family are not nice to each other during family functions, and one member unilaterally trying to make things better by being nicer or more helpful almost always invites only suspicion or derision from others (“Why is s/he suddenly being so nice? What is the point of this fake gesturing?”) This of course causes that one member to lose steam, and the state of affairs to return to the previous bad Nash equilibrium in which everyone is terrible to each other.

Any way out?

Is there any way out of bad Nash equilibria? One way of course is to tell everyone about your intentions. If you want things to become better between you and your family members, you should tell them that you want to try and become nicer to them so that things may become better with time, and that this bad Nash equilibrium was only hurting everyone. This may convince them to also be nicer to you, and soon things may turn for the better. But is it as easy as that?

The sad news is that two parties being good to each other is not a Nash equilibrium. If you and your friend are good and helpful to each other, and you decide to become inconsiderate and unhelpful while your friend remains good to you, you are of course at an advantage. You can take all the help you need from them, while being rude and dismissive towards them. This is why two people being nice to each other is not a stable state of affairs, as one is always tempted to defect (stop being nice). Similarly, if you and your romantic partner decide to be nicer to one another, as soon as you have an argument in which you try to prove your moral superiority, the person who brings up old issues to prove that they have historically been morally superior to the other will be at a temporary advantage in the argument. This of course causes the other person to bring up other old issues as well to prove their own moral superiority, and soon both parties return to the previous state of bad Nash equilibrium.

Money is always the answer

Is there any hope? I’d like to believe that there is some. If you and your partner are in a bad Nash equilibrium, you need to realize that being good to each other for the long term is a very difficult thing to achieve (it is not a Nash equilibrium), and that a single argument may send things hurtling down towards the bad Nash equilibrium (being terrible to one another). Hence, you need to resolve your old issues if there is to be any hope. But how does one do that? The answer to that is money (and my baniya-ness is only partly the reason why I came up with this explanation).

Suppose there was an incident in the past that causes you both to keep bringing it up and fighting about it. If you both blame each other (and both of you probably are to blame), you are only talking past each other without any hope for reconciliation. What you need to decide is how much both of you are to blame. You may come up with something like you were 7/10 to blame, and you partner was 3/10 to blame (this of course assumes that both of you are still cooperative enough to agree on these numbers). Both of you have concluded that you were more to blame. Now you can decide upon a monetary fine, depending upon your financial condition. Something in the range of ~$50 is reasonable if the issue involved is small, while larger amounts may be appropriate for more serious issues. Suppose you guys decide on a fine of $60. Transfer this amount to your partner. In my experience, this ends the issue then and there. If your partner ever comes up with this issue again in a future argument, you can remind them that you’ve paid your dues, and then they’ll have to switch to another issue in their search for moral superiority. Of course, if you’ve done this for every serious issues that you’ve had, there will be no other issues to go back to, and it will be easier to be nice to one another.

Can you do the same with relatives/co-workers who you are in a bad relationship with? I believe that you can. However, it will be harder to get them to the drawing board to sort these issues out. Your uncle who you’re in a bad Nash equilibrium with may not agree to go out for dinner with you to sort your old issues out. Even if he does agree to this meeting, he may be insulted if you ask him to assess his blame for any of your issues. If you have relatives like that, with whom there is a power differential, realizing that things are not going to get better is the only way out. However, there is hopefully no power differential between you and your partner, and you can mutually agree to sort out your issues using this method.

Perhaps the reason why people are (generally) bad to each other in this world is that being bad to one another is a stable Nash equilibrium, while being good is not. Thus ends my spiel on applying Nash’s rich and complex Game Theory in an overly simplistic way to our daily lives; with some buzzwords wrapped in a platitude.

Metarules are better than rules

In the last five years or so, I’ve defined certain rules in my life and tried to live by them. This has mostly turned out to be a not-so-good experience.

Why is it still good to have rules in one’s life? The world is infinitely complicated. Hence, given any particular situation, we will always possess incomplete information about it. Had we had perfect information, we would know exactly what to do in that situation to reach our goals. However, because we have incomplete information, a plethora of possible courses of action suggest themselves. Rules help us quickly decide between these possible courses of action, so that we don’t spend all our time ruminating and deciding.

How do rules come into being? Some rules are those that help form cohesive social groups (eg. Help those in need, be a dutiful spouse and parent, etc). Others are created from experience (eg. Go early to the airport, back up your data, etc), or are perhaps religious or constitutional. When you see a red traffic light, you have a plethora of options available to you. You may stop, refuse to stop, aim for other vehicles, etc. Traffic rules short circuit the process of deciding between these options, and force you to stop unless you’re ready to incur a heavy fine.

Because rules are artfifacts of a world that existed in the past, and often lack nuance and context, they are imperfect and ill-suited to a constantly changing world. Moreover, applying the same rule to different people and situations often leads to suboptimal results. If you have a rule to be nice to everyone, for instance, soon people who are prone to taking advantage of you will be even more empowered to do so. If you have a rule to wake up everyday at 6 and go to work, and you insist on doing so after a late night out, you’ll be sleep deprived and probably incapable of non-trivial work that day. Hence, I have found that metarules, which are rules about rules, are often more useful than rules. Given below are some useful metarules:

Metarules

All rules are imperfect, and one should be ready to change them with abandon, depending upon the context.

Rules are useful because they provide a way to quickly pick a course of action in response to external conditions. One should be aware of both their usefulness and their drawbacks.

Rules are ways to reach a desired goal. As conditions change, rules too must change.

Goals also change. Obviously, rules must consequently change.

These seem pretty generic and useless. However, just being aware of them can be useful. For example, if you create a certain rule for yourself in the belief that it will lead to a promotion of perhaps more research papers, and you’re unable to achieve your goal, being aware of the metarule that rules are imperfect and must constantly be changed will lead you to question your methods, and hopefully arrive upon a better rule.

Perhaps this game of ours has no rules; only metarules.

Just another day in Delhi

Morning

I got my visa renewed. It was not a miracle. Each and every student I know has successfully got their visa renewed. For my visa to have been rejected, I would have had to have committed some serious crimes in the United States. Maybe defrauded the government of millions; or not picked up a fake accent. But I knew that I wasn’t guilty of any of these crimes.

Despite knowing that my visa processing should have gone through smoothly, I was very nervous the whole morning before the interview. My hands were shaking, and my muscles (well, muscle) tensed. What if that Facebook status update I shared offended the government somehow? Should I not have written that article on Afghanistan? All these doubts made me sweat profusely, and also made my vision blurry….until my visa did get approved. Suddenly, colors became brighter, and I could breathe easy. For some reason, I strode out of the embassy like a triumphant general. I had convinced some officials to let me work for minimum wage in their country because I was sure I didn’t have it in me to make it in my own country. Alpha male in his prime.

I then decided to hang out with my friends around Delhi. For some reason, I was dismissive and condescending towards them. When they suggested that we hang out at a certain tourist spot, I scoffed at their suggestion and decided that we all go elsewhere. I felt supremely confident about myself, and decided that my opinions were more important than others’. Of course I checked myself soon enough, and with some effort returned to normal. It was only then that I realized how odd my behavior had been. I had only gotten my visa renewed; something that didn’t require any particular skills form my side. Despite this, I felt completely responsible for this “victory”, and decided that I was better than everybody else. I was taking credit for events that were only in a very limited sense my doing, and was convinced that all the good things that had happened to me were solely a result of my innate capabilities and drive.

This is probably what rich people think every day.

Evening

I was walking to a restaurant to have dinner with my friends, when an Audi with dark windows came screeching at us out of nowhere. It didn’t slow down when it saw us, and would probably have plowed right into us had we not rushed back. It was probably some smug rich kid trying to impress the plebs on foot….or maybe just a horrible driver who didn’t see us in time. I could have shaken it off, maybe made a couple of statements about “no traffic rules in India…” and “things are better in the US….” or something, and then just walked on. However, I was mostly just completely consumed with anger for the rest of the evening.

I began to imagine ways that I would avenge myself on that rich kid (definitely a rich kid!). I could empty my bank account, buy an even better car, and crash it right into the Audi. This way, I could signal that I was rich (mostly fantastical thinking), and also someone who couldn’t be messed with. I also wished that I knew some powerful people in the ruling party, who could ensure that that guy would get nicely beaten up in police custody. I wanted some police batons up peoples’ bottoms right then and there!

I was someone who had read up on status signaling a lot, and also explicitly wished to not ever be involved in it. However, it took just a fraction of a second with a badly driven car for me to want to devote all my money, time and influence to somehow signal my higher status to that driver. Perhaps this is just a demonstration of how deeply engrained our need for status signaling is. Reading up on status signaling doesn’t really take me out of the game. It just allows me to put a name to this phenomenon in retrospect when my primeval brain is done status signaling.

Experiences like these make me think that changing my brain into what I want is a much more difficult process than I give it credit for. Despite all my introspection and writing, I will mostly only be driven by my simian brain, and only later will be able to make sense of my irrational tendencies. This is something that the Rationalist community struggles with too. They spend a lot of time thinking about various biases and fallacies about the human brain, but often find themselves indulging in such biases despite the time spent trying to correct for them. Is there any hope?

Although I don’t have any concrete ideas about this, I do believe that writing things down helps in more ways than one. Maybe writing a small passage every morning would help? I don’t know. But I do hope to try and do this for a month, and hopefully report on whether this has changed things for me. But if you do hear about an Audi with dark glasses that was mysteriously trashed in the middle of the night….

The availability bias

The availability bias is commonly discussed bias in the Rationalist community. It says that examples that come to mind readily are thought to be more frequent in occurrence than they actually are. For example, we often come across headlines screaming “plane crash!!”. Hence, every once in a while when I get on a plane, I mutter a silent prayer to the gods above-er. However, I don’t do that when I get into my car, although the chances of me dying in a car crash are 2200 times greater than those of me dying in a plane crash. Because car crashes are relatively frequent, they’re discussed much less in the media, and are hence thought to be much less frequent than plane crashes.

Of avarice and men

What can the availability bias tell us about other aspects of our lives? All of us are constantly bombarded with stories of Elon Musk, Bill Gates, Mark Zuckerberg, etc. One possible reaction to these stories is getting inspired and trying to chart a similar entrepreneurial path to hundreds of billions, societal clout, and petty social media fights with other billionaires which make you look like an idiot despite the first two points. However, our knowledge of the availability bias tells us that the reason why these three individuals are discussed non-stop in the media is because it is very, very hard to rise to their position. If talent can be represented on a normal curve, we should have very strong evidence for assuming that we are anywhere but at the mean of this curve. And the further to the right (or left, but irrelevant here) of the mean we claim to be, the more evidence we need to support such claims. You went to Harvard? Fine, maybe you are 2-3 standard deviations above the mean. You got a 4.0 at Harvard? Well, maybe you are 4 standard deviations above the mean. But you still need a lot more evidence before you can claim you’re not the next Elon Musk! You need to take on Ford, NASA and all the banks at the same time, and beat them!

What does that mean for us? That we probably aren’t going to become billionaires. The jobs we are more likely to get are those that are not glorified as much in the media: tech workers in non-FAANG companies, or perhaps middle managers in failing companies.

Talent? What talent?

Whatever skill we want to “rank” ourselves in, we should always assume that we lie somewhere in the middle of the curve, unless we have strong evidence to the contrary. Because I did well in math in school and also managed to get into grad school for a PhD in math, I am probably at least one standard deviation above the mean in the mathematical talent normal curve. But because I haven’t written an Annals paper yet or won an IMO gold medal, I am probably much below the higher standard deviations. The same goes for my talents in music and other skills that correlate somewhat with Mathematics. Does that mean that I can never do amazing things in mathematics? No. It just means that it will be a statistical miracle if I do. Out of 10,000 people with my abilities, only one will do the amazing things that all 10,000 dream of. At Terence Tao’s talent level, those odds will probably be 1:1.5 or something. Hence, although Terence Tao is frequently talked about in mathematical circles, the high frequency of such discussions should tell us how frickin’ difficult it is to become a Terence Tao.

Is any of this useful?

Alright. Like everyone else, I am slightly above the mean in some skills, and (significantly) below the mean in others. On average, I am average. But despite this, I want to be successful in my chosen field, earn money, and have petty social media fights with other successful people at my level. How can I do that? A lot of people in the Rationalist community believe that people like me should play the numbers game. If I try enough things, it is statistically likely that I succeed in one. If I flirt with (flirt at?) enough girls, at least one should like me. If I apply to enough schools, at least one should take me. Experience tells us that this is a flawed strategy.

If I try a lot of things, I probably won’t get very good at any of them. My friend from another school (wink) hits on a lot of girls, and doesn’t succeed with anyone. You can apply to 50, or even 100 schools. However, if you don’t have the stats to get into them, you will be rejected at all. Moreover, the people who succeed at one thing, succeed in everything. Newton discovered his laws of motion, created calculus, split sunlight into seven colors, and also classified all cubic equations. Einstein discovered special relativity, discovered general relativity, discovered entanglement, and also won the Nobel for his discovery of the photoelectric effect. Elon Musk manufactures the best cars in the world, the best rockets in the world, and also the best solar panels in the world. He also helped create the best payment system in the world, and c’mon; we know he’s also gonna be the first person to Tweet from Mars through his Neuralink. Similar things can be said for Nikola Tesla, Gauss, Steve Jobs, etc (Jobs also won a frickin’ Oscar!). Hence, trying a bunch of different things hoping something hits is probably a failure-prone strategy if you’re not already absolutely brilliant at one thing.

So how does an average person succeed? There’s really no better answer apart from “get really good at something”, and “don’t assume you’re already good (3-4 standard deviations above the mean) unless you have strong evidence”. This seems like a particularly unhelpful answer. Wasn’t I already trying to get good at things? Is “try harder!” helpful advice? Well, one way to interpret this is the following: I can keep changing fields until I find something in which I can get good very fast, and can get evidence that shows that I am well above the mean. Another way is to keep changing my methods of learning and understanding until I can get evidence that I am getting beyond the appropriate number of standard deviations. I have tried the latter, but not the former as much. Regardless, I should try and get really good at something, and try and collect evidence for this. I am probably not lucky enough to become successful accidentally, and I should definitely not try to emulate people I hear about in the news. If I was good enough to be newsworthy, I’d probably already be on the news for my skills….or for my huge posterior (I never thought I’d end my article this way, but all roads do indeed lead to the Kardashians).

Learning to socialize

One of the biggest changes in my life in recent times is that I’ve gotten much better at socializing, making friends, etc. I can consistently make friends with new people that I meet, hang out with them without rubbing them the wrong way, and be invited in return to hang out with them, etc. Although this might sound easy for most people reading this article, it has been extremely difficult for me to learn, and has taken me almost three decades! Here are a few aspects of learning to socialize that I wish to talk about

Learning to socialize is a bit like learning English

I was often complimented for my ability to read and write English at an early age. I was lucky enough to be exposed to books at an early age, and even luckier to have developed reading as a hobby. Hence, I could read and write reasonably well, and this opened doors to prefectorial positions, scholarships, and even a seat at my undergrad institution, which bizarrely had a section on English in its entrance exam!

English, famously, is not a very logical language, in that it cannot be completely developed from a small set of rules. However, it is self-consistent. Hence, the best way to pick it up is not to buy a book of English rules, memorize them all, and then use those rules everywhere. There are a bazillion exceptions to those rules, and ultimately those rules may hinder more than help. Perhaps the best way to learn English is to read and write frequently, until one has had access to a critical mass of the literature. The best analogy that I can draw here is training a neural network. After grappling with a big and varied training data, our brain becomes pretty good at recognizing where the phrasing “sounds wrong”, and how one should convey one’s thoughts in words.

In contrast with my access to English resources, I never really had any friends growing up. One major reason for that was that I was horrible at sports, and playing sports was how most of the guys in my apartment complex bonded. Hence, I was not exposed to enough social interactions to be able to pick up on a lot of the delicate and nuanced facets of socializing. It may be clear to anyone reading this that one cannot become good at socializing by merely reading a book on it. It is an acquired skill, and only comes after a lot of practice, after which it begins to seem easy and almost second nature.

I finally began to acquire social skills after noticing how very social people that I knew interacted (reading some psychology blogs also helped). I could pick up on cues that I hadn’t read in any books or come across anywhere at all. It’s funny that I picked up almost all of my social skills in a pseudo-simian fashion: blatantly copying the more successful members of the tribe.

There’s a pattern to the madness

I am not the only one.

Many grad students in Mathematics are socially inept in ways similar to me! I don’t mean to push a stereotype of mathematicians being introverted loners or something. Most of us want to socialize and have friends like other people. However, we are unable to, and for very similar reasons!

Some of us try too hard to sound nice and humble, but it comes across as fake. Others are too obsessed with coming across as smart, and would rather be thought of as intelligent rather than nice (inevitably, they’re thought of as neither). In other words, we lack the ability to tune our social selves to the optimal setting. Although it might sound like I’m pushing a bad stereotype, this correlation is easy as day to see, and it would be dishonest of me to not talk about it. There is something about being a social idiot that makes you want to study Mathematics.

So what really are some aspects of socializing that I picked up on?

I’m going to present some things that I picked up on as bullet points:

  • It is a mistake to look too eager to please someone, smile too much, etc. One should mostly adopt an attitude of helpful-but-uninterested. It converts the impression that you’re willing to help out if needed, but are not overly interested in getting close to them. Through this, you portray yourself as high status, and people mostly want to befriend high status people.
  • It is important to look like you’re willing to walk the extra mile for the community. It is also important to seem like you’re doing it for society at large than doing it for specific persons. Essentially, the more you make it clear that you’re helping out without expecting anything in return (including appreciation), the better the social returns might be.
  • Deeply religious people or people who demonstrate a willingness to follow any other arbitrary social code are often found to be more trustable than people who are not. If you can demonstrate that your behavior is tightly bound by known rules and is hence fairly predictable, you become easier to trust. This is stolen from Robin Hanson.
  • The people who try the hardest to sound humble are often the ones who are the most self-obsessed/arrogant. Perhaps the single most attractive quality in a person is not indulging in themselves at all. If someone tells you that they really enjoy playing music, don’t tell them that you suck at music and hence are impressed by them, or that you’re very good at it yourself. Don’t insert yourself into the conversation at all. Just ask them more about it.

    This is stolen from Freddie deBoer (in spirit).
  • A recent article on Putanumonit says that getting good at anything requires two steps: first learn the rules well enough so that they become instinctual, and then forget the rules so that you can approach any situation with a fresh mind. I would say that this works very well with learning how to socialize. One can indeed learn some rules first: be nice, don’t be overly eager, don’t put yourself in the middle of a conversation, etc. However, life can get complicated, and using these rules blindly may be sub-optimal. Hence, one should try and forget these rules after getting some practice with them, and play it by the ear.

    I have suffered on multiple occasions by directly copying something that a socially aware person did in a different situation, and then realizing to my detriment that a lot of what they did didn’t translate to my situation. Playing it by the ear is indeed a very important piece of advice.
  • Don’t be afraid to be rude with people when it is clear to both of you that they are trying to take advantage of you. People often have a deeply ingrained sense of fairness (even the evil ones), and if you hit back at someone trying to treat you unfairly, chances are that they will realize that they’re being unfair. Consequently, although your relationship will be strained, repair will be fairly easy. Remember to detail precisely how they’re being unfair to you, so that there is little room for misunderstanding, and you’re both on the same page.

It’s completely possible that this article is mostly unhelpful for you, as you’re already aware of all of the things that I’ve talked about. However, it has taken me a long time to figure these things out, and I am recording them here only to perhaps reflect on them years later when I’ve realized that everything that I’ve written here is wrong.

The Mundanity of Excellence

I had a great time reading The Mundanity of Excellence by Daniel Chambliss. I came across this article in an interview given by a famous researcher, who tried to read a new research paper everyday for more than two years. She said that this was the most influential paper that she’d read, and that it actually changed the way that she approached research.

Perhaps the main takeaway of this paper is that people who demonstrate excellence in a field aren’t necessarily working harder than you. They’re just doing different things. Small things like getting enough sleep, focusing on their work more, etc. And it is the cumulative effect of these “different” but obvious in retrospect, rather mundane things that lets them achieve excellence. There is nothing called “talent”.

It is on this “there is nothing called talent” sentiment where I’d like to disagree with the author.

Talent, the way that I’ve been brought up to believe about it, is the extrapolation of the rate of improvement at any skill. If you’re a trained tennis player who has trained for 10 years to get to your level, and your cousin trained for just a month to become almost as good as you, people will generally say that your cousin is more talented than you. They have the potential to become better than you, given enough time and opportunity. Even if that potential is never realized because, say, your cousin becomes a drunk and ends up homeless, they will still be said to be in possession of considerable (albeit wasted) talent.

When I was in college in India, the people who could do well in academics despite little effort, were glorified tremendously. On the flip side, the academic achievers who did well after considerable effort were almost vilified. It was not enough to do well in college. It was important to demonstrate considerable raw intelligence in the process. Of course this led to some academic achievers to hide all evidence of their hard work, so that they could put up the appearance of having done well despite putting in no effort.

Does this generalize to life in general? Not particularly. If you’re super talented but never amounted to much because you never put in the effort, people might have a kind word to say for you. However, you’ll mostly be displayed as an example for people **not** to emulate. If on the other hand you become successful after putting in a lot of effort, you will be glorified as someone who has earned their place in society. In other words, if you get too comfortable being called talented but never actually convert that into results, societal sympathy for you will decline very fast.

This relates to a recent blogpost I read by Robin Hanson, which says that people respect potential a lot more than actual performance. I am going to use potential and talent interchangeably here. He gives the example of Oscar-nominated movies: when a handful of movies are announced as Oscar-nominated, there is a lot of excitement about watching those movies, and predicting which of them will win. However, when one of them is declared as the winner, the excitement for watching those movies plummets. Now that we know the result already, our judgement and prediction powers are left unused. This is a demonstration of the fact that we are supremely enamored of “talented” people, betting on which ones will succeed.

Is being enamored of talent irrational? Should we only care about actual performance? Well, our understanding of the world rests on a predictive model. And prediction rests on extrapolation. Hence, although it is indeed rational to bet on talented people, we should be suspicious of such bets, because extrapolation is always a risky proposition. A very intelligent student might later, due to underlying mental causes, become a drug addict and drop out of society altogether. We are all aware of at least one such tragic story in our extended social circles. It is also possible that our extrapolation was invalid, and that the talented student would find it almost impossible to pick up the more difficult parts of the skill later. Hence, although it is rational to glorify and be enamored of talented individuals, we should hold off on too much premature extrapolation.

The work of Alice Chang

Sun-Yung Alice Chang is one of the pre-eminent scholars of modern geometry, and I wanted to understand the nature of her work. Although I’ve read papers by her before, this article written by her students Matthew Gursky and Yi Wang serves as a solid introduction to her work.

Given below are my notes as I try to understand the article. I first attach a screenshot from the article, and then below that write my notes and questions and such. I often make speculations which are either verified or proved wrong in the best next screenshot. I learned a lot from writing these notes. These can hopefully be useful to anyone working on PDE theory or Riemannian geometry.

Note: This article is unlikely to be helpful to anyone trying to learn about the topic. It is perhaps more useful as a record for myself of how I try and understand Mathematics (or anything at all). One of the reasons that I’ve started blogging less frequently on research papers is that I now try and write detailed notes on every paper before I even think of blogging about it, and that often leaves me too little time to devote to my day job.

    • Why is it important to bound the norm of the function by that of its derivative? Why can’t we just measure both separately without having to compare both of them? I don’t know. It seems that we want to make the deduction that W^{1,p}\subset L^{p*}, where 1/p+1/p*=1/n. Why would we want to make such a deduction though? Imagine that you’re trying to find out about an object. You see what it looks like. You hear what it sounds like. You taste what it tastes like. You throw it around and see how it flies in the air. It is the accumulation of all of these infinite observations that tells you all that object is and can be. Seeing if a function belongs to a certain W^{k,p} or an L^q is like that. We want to study the membership (or non-membership) of functions in all of these infinite sets to get a sense of what that function is like.
    • Things like Sobolev’s inequality tell you that if a ball flies quickly through the air, it must be aerodynamically shaped, etc. Measuring one property tells you another. It gives you embedding theorems.
    • Why do we need to consider weak derivatives? What is wrong with regular derivatives? Well it turns out that the mathematics of proving that smooth or differentiable functions indeed solve PDEs does not exist. Smooth functions do not form a nice enough algebra for which functional analysis can prove the existence of solutions. However, if we enlarge the set a little bit, functional analysis can prove that an element within this set satisfies this PDE. But in order to be able to access all elements of this set, we will need to weaken the question itself; ie weaken the definition of the PDE. It should now incorporate weak derivatives. After we have found the solution, we can prove by some limiting methods that the solution is indeed smooth. Hence, we have found a solution for the original question.
    • Was this the only method possible? I am not sure. Maybe a new branch of mathematics would have rendered all of this obsolete. However, if we want to use functional analysis, we have to use this method.
    • What is happening here? It turns out that the extremal function is not just one function. It is a family of functions. Moreover, by changing the parameters a,b, we can get a function almost completely concentrated at a point. Hence, the Sobolev constant is independent of the domain.
    • We choose a cutoff function, and then let it remain fixed. Now we change a,b such that as soon as the domain is inside the domain of the cutoff function, the Sobolev constant is realized.
    • Why is the Sobolev constant not attained? I thought that it is attained for extremal functions. Let us walk back a little bit. In the second image, we are only considering compactly supported functions. We are able to reach the same upper bound with the much smaller set of compactly supported functions. However, the extremal function defined above is not compact supported, and it doesn’t become compactly supported for any values of a,b (even though there is pointwise convergence). Hence, the Sobolev constant is not attained.
    • Let us walk back a little bit. In the second image, we are only considering compactly supported functions. We are able to reach the same upper bound with the much smaller set of compactly supported functions. However, the extremal function defined above is not compact supported, and it doesn’t become compactly supported for any values of a,b (even though there is pointwise convergence). Hence, the Sobolev constant is not attained.
    • The assumption in the formula is that n\neq p. However, when n=p we see the inequality shown above. This looks like the \log formulae we generally see in extremal cases.
    • Why are we restricting ourselves only to compactly supported functions? Probably because we don’t have the concept of an extremal function here. Hence, we want to better understand this space.
    • What does the optimal value of \beta mean? I think they mean an explicit value, which depends only on the dimension and not the domain.
    • The surprising thing about this theorem is that they were able to find an extremal function. But I thought extremals don’t exist…? But that’s only for p\neq n.
    • Why should volume forms have anything to do with the metric? Why not just be completely coordinate dependent? I suppose we could have that. However, most manifolds don’t have a fixed coordinate system. Hence, we have a notion of integral that is invariant with respect to coordinate charts. That is why we include the metric in the definition of the volume form.
    • I think that in the theorems given above, we were assuming that the dimension of \Omega was the dimension of the space. Hence, when we have domains of lower dimensions, we need to modify the inequality, which is precisely what we have done on the sphere.
    • Why should we care about these inequalities at all? I think we are making an observation about these functions. We are establishing a property of these functions.
    • Why do we linearize differential operators? Probably because they are easier to solve, and help us figure out important properties of the actual operator. But the solutions of these linearized operators only give us a hint as to the nature of the actual solutions. A major fact in mathematics is that these linearized operators give us MUCH more information than we’d expect.
    • Why must questions about geometry lead to PDE’s? Because the very basic notions of geometry, like curvature, are differential operators of the metric. Hence, differentiation is built into the very fabric of geometry.
    • What is a variational approach? It is an approach in which you find the answer by minimizing some functional, because differentiation is easy! Hence, the hard part of this approach is finding the right functional.
    • A conformal map of the sphere also changes the metric conformally. Why do we care about this? Because it turns out that these conformal maps generate an infinite family of extremal points for every extremal point that you can construct by yourself.
    • What does it mean to say that the conformal group of the sphere is noncompact? Well it is a noncompact set in the space of all maps of the sphere to itself. But how does it matter? I don’t know. But it probably helps prove that some manifolds are not the sphere, or something.
    • What does bubbling mean? Is there a bubble being formed somewhere? We are constructing a family of conformal maps on the sphere such that the conformal factors progressively accumulate at the north pole. Well….so what? J_1 is still minimized for all of these factors right? Yes, they are. However, this just shows us that J_1 doesn’t satisfy the Palais-Smale condition. But how does that matter? First of all, the round metric minimizes J_1. Hence, all of these metrics also minimize J_1. Who cares about the Palais-Smale condition? I think the point is that extremal points other than these would be difficult to find.
    • Does bubbling happen only when we start with the round metric, and then construct the family described above? I think so. What is the point though? Why do we do it? Apparently it is the only thing that “messes up” the geometry of the sphere. We will probably have occasion to talk about it later.
    • What does this theorem intend to say? I don’t think it is a generalization of the previous problem, because \Delta_0 K\neq 0. However, as long as there are at least two maximum points and all the saddle points are positive, there exists a metric that is conformal to the round metric such that its Gaussian curvature is given by this function.
    • Note that we are not just concerned with any metric here. This metric here has to be conformal to the round metric. Hence, this is not like the Yamabe problem, where we start with a round metric and then try to get a conformal metric whose scalar curvature is constant.
    • Is a saddle point also a critical point? Yes. Why do we want to find saddle points? Are those the points that satisfy the \text{Gauss curvature}=K condition? I don’t understand why this has to be the case. How does J_K come in? Well, we just insert the desired K somewhere into the formula. Critical points will have exactly that Gaussian curvature. Why a saddle point though? Well maybe all of these critical points have to be saddle points. Critical point does not mean minima
    • What is the conformal invariance of this problem? It just means that the pullback of the metric after a conformal change is also a critical point. Is this what conformal invariance means in general? Yes, conformal invariance means that a conformal change in the metric implies no change or very manageable change
    • What does conformal invariance have to do with this? I think that through conformal invariance, we construct a saddle point of J_K which is not even bounded in W^{1,2}. So what? Why is that a problem, as long as the J_K value is finite? So remember that we are trying to find solutions of the original PDE, and hence the solution must lie inside W^{1,2}. Therefore, even though we may find a saddle point, it might not be the solution of the PDE. Therefore, we have not solved the problem we set out to solve. The key step in the proof is showing that at the saddle points where this might happen, \Delta_0 K<0. Hence, we don’t care about those solutions
    • Why is the spectrum of the laplacian important? In some sense, it is the skeleton of the linear operator. If there is an eigenbasis, then we know exactly how the laplacian acts on the whole space. Note that this condition doesn’t come up naturally in Physics. We have equations like \Delta \text{potential}=\text{charge} and \Delta Q=\partial Q/\partial t. The need for eigenvalues of the laplacian probably comes up somewhere else. Does it give us information about the metric or the manifold? It gives us (partial) information about the metric. Note that if the spectra of two linear operators (counter with multiplicities) are the same, then they are the same (provided they are of the same rank). However, this is not the same with metrics with the same spectra of the laplacian. Why did we have to choose the laplacian though? Why not a different linear transformation? That’s a good question. Maybe we can construct a linear transformation which satisfies this property? I’m not sure.
    • What is the heat kernel?
        • The heat kernel is not the solution to \text{Heat PDE}=\delta. It is the solution to \text{Heat PDE}=0, with \text{temperature}=\delta(y). Hence, it is an actual solution to the heat equation, and not an attempt to form a general solution to \text{Heat PDE}=f
        • Why do we need to form an equation over \Bbb{R}^{d}\times \Bbb{R}^d\times \Bbb{R}_+, instead of just \Bbb{R}^d\times \Bbb{R}_+? I don’t know. I would have just done the latter and said \text{temperature}=\delta(a) at some a\in \Bbb{R}^d. I think that we want y to be a variable because we want to be able to assess what happens when y is varied. Hence, we might be able to solve a more general formulation of the problem.
        • What does \delta(x-y) mean? I interpret y to be constant. However, if this were a function on \Bbb{R}^{2d}, this would be a function that is infinite on the diagonal. Is the integral of the \delta function still 1? I think so.
        • I think that we are trying to form a solution for a generalized boundary condition, and not a generalized heat PDE.
        • What does \delta_x(y) mean? Well \delta(x-y) is kind of symmetric. \delta_x(y) makes it clear that x is the independent variable and y is fixed for the purposes of this computation.
        • How is the spectrum of \Delta related to the heat PDE? It turns out that the kernel of the heat PDE can be written in terms of the spectra and eigenbasis of the \Delta operator. That is how the spectra is relevant to Physics. A lot of PDE theory is about reducing solving a differential equation problem to a Linear algebra problem. That is exactly what we’re trying to do here as well.
    • What is the trace of the heat kernel? Assuming an orthonormal eigenbasis, maybe the trace is \sum e^{-\lambda_n t}\phi^n(x)\phi_n(x)? Yes, that would be the only way that this would make sense, as traces of functions don’t really exist.
    • What did Brooks-Perry-Yang show? They showed that if all the isospectral metrics lay in the same conformal class, then this set would be compact. Alice showed that we don’t need to assume that all these metrics lie in the same conformal class. I think that the conformal class needs to contain a metric of negative curvature so that bubbling doesn’t occur (which destroys compactness)
    • What does modulo conformal group mean? Well the action of the conformal group creates sequences that don’t have a limit inside the space. Thereby compactness is lost. But if all conformally equivalent metrics are considered to be one equivalence class, then the set is compact.
    • Why do we define the laplacian to be -\Delta? Probably because \Delta u +\lambda u=0 implies \text{laplacian }u=\lambda u. We want those \lambda‘s to be our eigenvalues.
    • Why do we have the \zeta function for eigenvalues? Well we have them for \{1,2,3,\dots\}. Why not for eigenvalues? But that doesn’t really answer the question as to why this could be important. I think it’s possible that \zeta(g_1)=\zeta(g_2) tells us something about the similarities between the metrics involves, although it is weaker than being isospectral.
    • Why the n/2 part? A function is defined if we have finite values. An analytic function is defined if it’s equal to its Taylor series. We see that if Re(s)>n/2, we have convergence, as \sum 1/n^k<\infty for k>1. The complex exponents don’t matter for convergence, which is all we care about right now. Why analytic though? How do we get powers of s in the expansion? I don’t know. Maybe there is an extended proof for this.
    • How do meromorphic functions make sense? They’re infinite! Well, they give us information on the nature of the poles, and that can help us determine important properties of the function in a neighborhood of the poles. Meromorphic functions are also important in Physics. Consider the electrostatic potential at the position of a charge. We still want to be able to study this potential, although it clearly has poles.
    • The determinant of the laplacian is essentially the product of all eigenvalues. Why would this be an important property? Again, this is a weakening of the notion of isospectral matrices.
    • What does a global invariant mean in this context? It doesn’t need to be measured at a point. “Niceness” is a local property of humans, because it needs to be measured within each human. The population of humans is not a local property. It’s a global property. In what sense is it an invariant? I’m not sure
    • What does equation (11) say? It says that there’s an upper bound to the fraction on the right, if you start with the round metric on the circle, and that it is 0 if the conformal metric is a pullback of a conformal map! Hence, not all conformal metrics are pullbacks of conformal maps. Therefore, because we have a \log on the left, we have invariance under action of the conformal group of the determinant as long as we start with the round metric.
    • The round metric only maximizes the determinant as long as we’re in the same conformal class.
    • How does formula (10) tell you anything about compactness? Maybe the boundedness of the determinant tells us about the compactness of the metrics themselves? That’s possible. But wouldn’t isospectral metrics have the same determinant as well? Yes. I don’t understand how the proof works.
    • What are the changes we make to the laplacian to make it conformally invariant? Well, we change the functional space we act on completely, and we also add a scalar curvature term. The good thing about this is what it matches the good old laplacian on Euclidean space, and hence it may be thought of as a “correct generalization” of the laplacian. As long as things “are the same” on Euclidean space, we’re fine.
    • Why does the laplacian change the weights of the weight classes that it acts on? Doesn’t a laplacian generally preserve the rank of the differential forms it acts on? Well in this case, taking \tilde{g}^{ij} brings along with it the factor \frac{1}{t^2}. This causes the weight to change.
    • Aren’t we “cheating” when we act on weight classes instead of considering \hat{g}=e^{2w}g? Well, it’s like studying a line bundle on \Bbb{R} instead of studying the space on all smooth functions on \Bbb{R}. We are saying that we can study f:C^\infty (M)\to C^\infty (M) by just studying \text{line bundle}\to \text{line bundle} because of the extraordinary homogeneity restrictions on the former map. Is that what is happening here? I think it is, because we are only consider the “extraordinarily homogeneous maps”- the conformally invariant maps.
    • Why are Paneitz operators important? Because they are conformally invariant. But so what? There are several conformally invariant operators! Maybe this is a new operator that cannot be generated by previously known conformally invariant operators, that arose in a specific situation. Moreover, it seems to be the natural generalization of the laplacian to four dimensions. Wait. Why only four dimensions? Moreover, isn’t the correct generalization the conformal laplacian? Well the conformal laplacian is a generalization. We can include other terms as well that become 0 in the Euclidean case, and get a valid generalization (of course, we need to have conformal invariance as well). If X_{\hat{g}}=e^{nw}X_g, is X the correct generalization of the laplacian in n dimensions? Shouldn’t it act on weighted classes? Shouldn’t it change the weight by 2? Moreover, shouldn’t we look for a generalization that works in all dimensions (like the conformal laplacian)?
    • Well first we generalize the notion of Gaussian curvature to Q, and then say that the Paneitz operator is the correct generalization of the laplacian. So a generalization of the laplacian doesn’t need to be equal to the usual laplcian in Euclidean space. It just needs to satisfy the same kind of PDE as the laplacian. Of course, equation (15) is only true in four dimensions.
    • What is the Q curvature? It is a geometric quantity whose integral is a topological invariant in four dimensions. Is this the only geometric quantity with this property? It is possible that it is not. However, it comes up naturally in the generalization of Gauss-Bonnet, which is called Chern-Gauss-Bonnet. But was this really needed? What does this tell us about the manifold? I think it mostly tells us what is not possible, given the topology of the manifold, because we can have really wild metrics on the manifold. Of course all of this is independent of the embedding.
    • Both K and Q are completely determined by the metric.
    • What is the point of this generalization though? It essentially tells us what a 4-dimensional manifold given a certain topology can “look like”. It restricts the class of metrics, and hence the ways that this manifold can be embedded inside Euclidean space isometrically. Generally, topology tells us very little about what a manifold can look like. This takes us towards that goal, but eliminating what a manifold cannot look like.
    • Why is scale invariance important? Well it’s sort of like conformal invariance, but weaker. Hence, when trying to construct conformally invariant quantities, we may mod out by scaling stuff, and then tweak the right things to get conformal invariance. I think of it as a stepping stone.
    • What is happening here? It is interesting that for all kinds of operators A_g, F_A can be represented as a sum of three functionals with different coefficients. We sort of have a vector space with a basis thing going on here.
    • How do we associate a Lagrangian to a geometric problem? Well the functional that we try to find the extremal point of is generally the lagrangian.
    • They’re not saying that I is the Euler-Lagrange equation given on the side. They’re just discussing the nature of critical points and the corresponding Euler-Lagrange equation on the right. Hence, if you want to make |W_g|^2, Q_g or \Delta_g R_g constant, you’ll have to extremize the functionals I, II or III
    • What is the theorem that they proved? They proved that the round metric extremizes every normalized functional of the form F_A on S^4. This is a pretty strong result. I think that we don’t have the tools to study S^{2n} in general.
    • So we are specifically considering A_g=L_g
    • How is the dimension being 4 relevant here? I think we are using Q curvature and Paneitz operator here. Note that II has Q_g=c its its Euler-Lagrange equation. It is possible that we extremize all three functionals I, II and III, and we get the Paneitz operator from II.
    • I think that R affects the definition of L
    • How might the proof go? The positive scalar curvature might make L “nice”, leading to a relatively simple fourth order PDE that we’re able to solve. The \chi\leq 0 probably makes the PDE even easier to solve. Hence, we’re affecting both the metric and L to make things “easier”. Why do we need to be four dimensions though? Well we have to prove that II can be extremized. It’s probably difficult to do in other dimensions, but gives us a solvable 4th order PDE in four dimensions.
    • I find it difficult to understand theorems like these. How are these conditions intuitive? How do you know how many to apply and when to stop? Are these the weakest conditions that can be imposed? Let’s take an analogy. We know that compact sets are closed. OK. But is this the weakest condition possible? No. There are lots of non-compact sets that are closed. However, this is still useful information. There are probably some important metrics that satisfy the conditions given above. And it is good to know that this large class of metrics satisfy this property. We don’t always want to construct the largest class of objects that satisfy a given property. We mostly want to understand individual objects, or classes of objects. And sometimes specifying this awkward class of objects takes awkward wording and seemingly arbitrary conditions, like those stated above. These conditions may in fact encode very easy to see geometric properties that we are missing. Much like the formal definition of compactness hides the intuitive notion of “closed and bounded”. But how can we be sure that we are including all “nice” metrics that we want to study? I think that we are definitely including a large subclass. R>0 and \chi\leq 0 aren’t that hard to visualize.
    • How do we have a parallel for surfaces with high genus and four dimensional manifolds with \chi\leq 0? Well a high genus implies \chi\leq 0. The only difference is that we can discard the R>0 assumption in surfaces. I think the article suggests that we may be able to remove that assumption in four dimensional manifolds as well.
    • When you minimize II, you get Q_g=c. How do we get c=0? It has something to do with P_g having a trivial kernel. Maybe c is the dimension about the kernel, assuming P_g\geq 0? I don’t know.
    • Note that all our minimization and extremization is occuring in the same confomal class.
    • Why couldn’t we do this for the round metric? Well we know that the round metric maximizes everything in its conformal class, and also has constant Q curvature. Alice has studied this for all other conformal classes.
    • We no longer start with a functional that is the \log of determinants. We play around with coefficients \gamma_i such that we get an “easy” Euler-Lagrange equation.
    • If we find critical points of F_A with positive scalar curvature, then the Ricci curvature will also be positive. So? Well we want to construct metrics with positive Ricci curvature, and this is hard in general. However, playing around with values of \gamma_2,\gamma_3 so that we get the “right” second order PDE is easy. And that is what we do here
    • What does it mean to say that the conformal change of the curvature tensor is determined by the Schouten tensor? Well if the Schouten tensor is conformally invariant (which it is not), the curvature tensor will also be conformally invariant. However, if the Schouten tensor has a different transformation law, that will determine the transformation law of the curvature tensor.
    • Keeping \text{volume}=1 is equivalent to normalizing the relevant functionals so as to have scale invariance.
    • Why are \sigma_k curvatures important? They are an attempt to generalize the \int R functional. We want to minimize this functional, so as to get \sigma_k=c manifolds.
    • Why don’t we want to attain manifolds with K=c or Q_g=c? Why \sigma_k=c? I think we want to solve both problems. Of course the former is dependent on the embedding, while the latter isn’t.
    • Note the following interesting fact: \int K when n=2 and \int Q when n=4 are topological invariants. However, we can solve PDE’s so as to find K=c or Q=c solutions. For \int \sigma_1 when n=2 and \int \sigma_2 when n=4 on the other hand, we have conformal invariance (at least for n=4) instead of topological invariance. There are no PDE’s here, and obviously no minimization. But we should still be able to somehow make \sigma_2=c for n=4 right, even though the integral doesn’t change? I’m not sure.
    • It seems that Q_g and \sigma_2 are related. Are K and R related in the same way? I’m not sure. Maybe they are in two dimensions
    • I don’t understand why the conformal invariance of \int \sigma_2 precludes the possibility that \sigma_2=c
    • Note that K,Q are found to be constant only in dimensions 2,4 respectively. On the other hand, \sigma_1,\sigma_2 are found to be constant in all dimensions except 2,4
    • We assume local conformal flatness. Do we have the same assumption in the Yamabe problem? No. I suppose it’s needed for higher k. Yes, that’s exactly what is stated.
    • What is happening with \sigma_1? I think that we can find it to be constant in all dimensions. Hence, the problem starts only with k\geq 2. Therefore, the analogy with K,Q isn’t exactly clear. \int R may not even be a conformal invariant. In fact it clearly isn’t. We minimize it to find the solution of R=c
    • So I was right. In 4 dimensions, we can still have \sigma_2=c. Of course, the assumption is that \sigma_1>0. I don’t think we had to assume that K>0 to prove Q=c.
    • What does it mean to say that the conditions are conformally invariant? Is R>0 a conformally invariant condition? I’m not sure.
    • What are quermassintegral inequalities? They’re easier to describe in convex geometry, and compare volumes of the form \int \sigma_k(L)d\mu, where L is the second fundamental form. These inequalities are not expected to hold in nonconvex domains. However, Alice and her collaborators could prove such an inequality for (k+1)-convex domains.
    • But (k+1)-convex domains are also convex right? Where does the non-convex part come in? I’m not sure of this point. I think the implication is that the non-convex domains that lie inside this (k+1)-convex domain satisfy the inequality. Again, I’m not sure.
    • Why is this space called conformally compact Einstein? Well it is conformally compact because it is conformally equivalent to an almost compact manifold, which is the disc. Note that the Poincare disc is not the same as the Euclidean disc. It is the Euclidean disc that is almost compact. The Poincare disc has to have its metric conformally changed to make it the Euclidean disc. It is conformally Einstein because \text{Ric}(g_+)=-ng_+. It is actually Einstein, and no conformal factors are needed to make it Einstein
    • I think this is where the Einstein part of Poincare-Einstein comes in. g_+ doesn’t even have to be in normal form for the Einstein condition to hold.
    • I think g_+ is always fixed. Moreover, the conformal class [g] on the boundary is also fixed.
    • Why do we need a conformal class on the boundary at all? We want to construct quantities that are conformally invariant, so that we may use them in Physical laws.
    • I think that the volume is in terms of the metric g_+, and not \rho^2 g_+
    • What does o(1) mean? It refers to a quantity a(n) which satisfies \lim\limits_{n\to\infty} a(n)=0. Why do we need to use this notation? Why should we compare functions to powers of n? Moreover, why should we care about behaviour as n\to\infty? Computational costs depend only on n\to\infty. But why? This is because n often does get very large. And we want a good feeling for the computational costs involved. O(n^3) might be manageable, but O(n^{3.1}) might become completely impossible. We could also have had things like O(2^n) right? Well those problems are completely intractable, with impossible computational costs. We want to restrain ourselves to polynomials and monomials. I still don’t understand why knowing something is O(n^3) instead of O(n^4) would help. Well, suppose we want a result for n=100 in finite time. The former will take 10^6 seconds, which is approximately 9 days, while the latter will take 900 days, which is about three years. Hence, if we are already aware of the value of n needed in our computation, and it is high, then knowing the exponent of n is pivotal.
    • The coefficients are not conformal invariants, and depend upon curvature terms of the boundary metric. Hence, a defining function has already been defined.
    • I’m always surprised by inequalities leading to homeomorphism or diffeomorphism theorems. How do they work? Can’t we make the volume arbitrarily small? Not when we have a blowup near the boundary.
    • How is this volume renormalized? We are not talking about the whole volume expansion. We are only talking about the conformally invariant constant term V
    • What was the Anderson formula? It relates V to \chi, although it need to say V>\frac{1}{2}\frac{4\pi^2}{3}\chi (which implies homeomorphism to B^4)
    • It turns that this is true only when n=3. There is a generalization of this to higher dimensions, which involves a conformally invariant integral.
    • Have I constructed a fractional power laplacian in my paper? I have proven its existence. However, I haven’t factorized a fractional power laplacian. I have constructed GJMS operators anyway whole symbols would be fractional power laplacians.
    • Is the Paneitz operator a GJMS operator for k=2? Yes. That is correct.
    • How can Fourier analysis be used to construct fractional power laplacians? Well differential operators are analogous to multiplication by a variable. Hence, fractional differential operators are probably multiplication with fractional powers of polynomials, which are not that bad. We can then take the inverse Fourier transform.
    • What is an elliptic extension problem? I think it is just an elliptic PDE with a certain boundary condition.
    • How can the Cafarelli-Silvestre construction be interpreted as acting on the hyperbolic half space? Maybe we modify the elliptic PDE on Euclidean space to become the scattering operator on hyperbolic PDE? Yes, that is possible.
    • How does making the space a hyperbolic half plane give us the extended interval \gamma\in (0,n/2)? This is because of the usual ambient restrictions. But what if n is odd, which I think is the assumption here? I’m not sure. However, what I learned is that w=-\frac{n}{2}+k is always negative. As f\in \xi[w]\implies \tilde{f}=t^{-w/2}f, this implies that t will be raised to positive powers.

How to actually make friends

Don’t be yourself.

The most common advice to make friends is “be yourself”. Don’t be artificial. Don’t pretend to be someone you’re not. But what if you are, very deep inside, an obnoxious troll? What if you’re insecure, always looking to status-battle, and very intrinsically unpleasant to be around? Is there any hope?

Now that we’ve done away with generic advice on how to make friends, let’s take a peek at evolutionary biology. I am mostly going to be re-hashing Robin Hanson‘s arguments, who I think is either the most important thinker alive today or the troll-est troll I’ve ever seen. Either way, I’m very impressed.

We still do it like the hunter-gatherers did

People often think that if they can impress others with my academic prowess or their sporting or musical skills, others would think highly of them and want to hang out with them. This turns out to mostly be misguided. The most obvious reason is that no one likes show offs. The less obvious reason is that we are all very deeply convinced that we are incomparable. We believe that we have achieved top status from our perspective, and anyone who thought deeply enough would arrive at the same conclusion. Hence, if some show off comes along and tries to, well, show off, we’d think that they were trying to claim a superior status to ours. And we are evolutionarily hard-wired to not let such a challenge go unanswered. People mostly respond to show-offs by making fun of them or avoiding them completely. At least the cool people you want to hang out with.

Another common technique in making friends is flattery. People often think that buttering up the other person is a sure shot way to getting into their good books. I would recommend against it for two reasons. The first is that when you butter someone up, you set up your status to be below theirs, which you’ll now have to maintain for as long as you know them. Hence, although they might have goodwill towards you (because you have recognized their genius that no one else could or something), you can’t really be friends with someone whose status you’ll have to maintain constantly above yours. For example, if you praise someone to the skies in order to be friends with them, you can’t ever disagree with them about important things. Otherwise they’ll realize that you’d only falsely raised their status, and hate you for manipulating and betraying them. Hence, although buttering up your boss for a promotion is probably kosher as they’ll probably always have higher status than you at the workplace, buttering up your way into a friendship has too many disadvantages.

The second reason is that unless you’re a seasoned flatterer (and there are some of those around, for sure), your flattery might sound insincere. And that would choke-slam any hope of friendship. I suppose I don’t need to explain this too much. If you’re already socially awkward, you’re unlikely to be convincing as a flatterer. Don’t even try. Just pretend to be shy and hope that someone takes pity.

So how can one make friends? In friendship, as in life, one is defined more by the things not done than those done. In pre-historic times, when society was mostly defined by small (<150 members) groups of hunter-gatherers, meeting a stranger (anyone outside of your hunter-gathering group) was a situation fraught with danger. Most of these encounters would end up in deadly battles. Hence, it was important to convince the other person that you were not a threat to them. Friendship, for clear evolutionary reasons, is something like that. When you meet a stranger, you don’t really have to be cool or smart or successful to become their friend. Most times, you just have to convince them that you will not try to harm them or their reputation as soon as they turn their back to you. That’s it. That’s all you need to do to become friends with someone. Convince them that you are not a threat to them.

So how do you do that? Well, you first have to convince them that you won’t badmouth them to others. One convincing way to do that is to praise your other friends, and not criticize people in general. That will tell them that you’re not prone to badmouthing, and this will help build mutual trust.

The second thing to do is to not assume higher status. If everyone is sitting on the carpet, don’t be the lone jerk who sits on the chair. Don’t take take the floor and regale everyone with stories about your fancy school or your high paying job or the time your quick thinking saved someone ten bucks. This will convince them that you’re not a status fiend who may challenge them for a status battle. It’s easy to join a group if you’re not straight away gunning for alpha. Of course, it will help even more if you claim a slightly lower status by helping them in the kitchen, etc without looking like a sucker for appreciation.

If people are convinced that you’re not a status challenger or a threat to their security and reputation, they are likely to invite you to parties and want to know you better, etc. And with enough of these out of the way, you’ll soon be friends! And note that you didn’t have to do that much! You just had to convey that you would not be a threat to their security, reputation or status.

It’s really about the mistakes you don’t make that often determine the quality of your social life. For evolutionary reasons. Because we really are, in all our social interactions, glorified primates.

Here’s to hoping that you soon get down to monkey-ing around and make a lot more friends!

Afghanistan and the Turing Machine

How do societies modernize? One day they realize that their ways have gotten older, and they transform overnight into liberal democracies with equal rights for all.

Not really.

Afghanistan

The US invaded Afghanistan to spread democracy, women’s rights and universal love (there was something about hegemony in the Middle East, but those are just rumors). It rained money and support (and bombs, and missiles, but again those might be rumors) on the Afghans, providing them with a red-carpeted ramp to democracy and prosperity. Music academies were encouraged and funded, women were encouraged to take leadership roles, and there was even a women’s football team!

So what happened?

https://www.prb.org/resources/most-women-in-afghanistan-justify-domestic-violence/

Why wouldn’t Afghanistan appreciate and adopt the awesomeness that America was gifting to them on a silver platter? Why do they stick to their “medieval” practices, and refuse to engage with the changing world?

The brain as a Turing machine

A Turing machine is a computer-like device that “manipulates symbols on a strip of tape”. How does it do that? Imagine that the Turing machine is trying to manipulate the symbol on a particular cell. The machine can exist in any of n states. Depending upon the state it currently is in and the current symbol on the cell, it decides how to manipulate the symbol and which cell to move on to next.

In other words, the state of the Turing machine keeps changing depending on its current state and what symbols it encounters on the tape.

I have recently begun to realize that this is a fantastic description of all kinds of complex networks, whether we’re talking about the brain or society. For example, how our perspective on a certain issue changes depends almost entirely on what our current perspective is, and what we encounter in real life. Imagine that I am an 18th century slave owner in provincial America who is brought up to believe that slaves are essentially animals and hence must be treated like beasts of burden. This is my current state. If I have an encounter which convinces me that slaves are treated unfairly (think of this as the symbol on the strip of tape), I won’t immediately start thinking that slaves are the same race as us and hence should be freed immediately. I will probably be led to this conclusion only after multiple encounters of a similar nature, through which my mental state would change several times before I got ready to fight for my suppressed brethren.

Let us take a more modern example. Most people in the world eat some form of meat. We are brought up to believe that animals are inferior to us and hence we ought to eat them to survive, even Jesus ate meat hence we are justified in doing the same, meat is too delicious to let go off for flimsy reasons, worms and insects die in agriculture too, etc. This is our current state. If we have encounters in which we see sheep and cows getting killed for meat (think of this as a symbol the Turing machine encounters on the strip of tape), we might develop a slight aversion to meat. However, we may still keep on consuming meat. If we have multiple such encounters (we get equally delicious soy alternatives to meat, the environmental impact of meat becomes even more apparent, it is proved beyond reasonable doubt that a vegetarian diet reduces chances of cancer and cardiovascular disease, etc), we may slowly move towards a state of veganism.

Much like a Turing machine, our perspective cannot just jump from the initial state to its final state. It has to go through multiple intermediate states before it reaches its destination.

How is any of this relevant to Afghanistan?

America’s attempts to spread democracy and women’s rights in Afghanistan are misguided because they want Afghans to jump directly from a state of conservatism to a state of western liberalism. We can’t just expect women, who are brought up to believe that it is for their own good that they must be beaten by their husbands, to start discarding these beliefs. They will slowly and painstakingly have to realize that they are the societal and moral equal of their husbands before they can start believing that they should not be beaten. Neither can we expect Afghans to overnight want a secular society with freedom of speech. They have been brought up to believe that their true ruler has to be a divinely appointed Caliph. They have to read world history and realize how misguided this idea has been in the past, before they can accept democracy as a workable alternative.

Poking holes in this model

Let us try and poke some holes in this model to see if this holds up.

Afghanistan was not always a conservative society. Kabul was resplendent with sights like those below in the 60s

Would it be so hard to just hop back to the modernity that Kabul possessed?

Let us draw an analogy to present-day India. It has a quickly ballooning youth population that mostly lives by liberal western ideals, dates freely, and is steeped deep in women’s rights. Does that mean that India has a liberal society?

Societal beliefs can be well represented in the form of a normal curve.

If we imagine “liberal beliefs and lifestyle” to increase from left to right, in every society we have a small number of people who are either very conservative or very liberal. Both the Afghanistan of the 60s and the India of today have a small number of liberals in the cities, and a small number of ultra-conservatives outside of urban centers. However, most people lie in the middle. A typical man in India may believe that women should not work but take care of the household instead, but may refrain from regular beatings. Hence, it would be erroneous to say that the Afghanistan of the 60s or the India of today are liberal. They just had (have) liberal subpopulations.

What the Taliban has done is that is has quashed those liberal subpopulations in Afghanistan. Hence, although it may be unpopular amongst educated Afghans with liberal ideas, it is mostly aligned with current Afghan values and beliefs.

I think it may be even more complicated than that. Regardless of the state of liberalism in the country, Afghans may believe that the Taliban represents the “purest values of Islam”. Hence, Taliban may be representative of religious perfection. This is an aspect of societal acceptance that I got from reading a Pakistani’s take on Taliban’s recent takeover of Afghanistan. Taliban may be the embodiment of religious aspiration in Afghanistan.

So how does all of this fit with my model of society and perspective as a Turing machine? Well, although liberal pockets may exist in the Afghan society, most of the population is still in a conservative state. In order to create lasting change in Afghanistan, it will not do to just create some liberal pockets through American propaganda. The mean of the normal curve will have to be shifted. This can only be done like one changes the state of a Turing machine: slowly, and through lots of intermediate stages.

A liberal Afghanistan won’t be built in a day.

Discovering ourselves

Descartes said we think, therefore we exist. But do we really know what we think?

Note that whatever I am going to describe in this post is based completely on my own experience, and I am not aware of any research in this direction. Hence, make what you will of this n=1 experiment.

Introduction

I was recently listening to the audiobook of How to Change by Katy Milkman. It was unusually good when compared to other self-help books, and offered great advice on how to beat procrastination, etc. The emphasis of the book was on human psychology. One section that particularly struck me as insightful was that giving advice to people (about how to improve their lives, treat their kids, etc) is almost always misguided, because everyone is acutely aware of their flaws. They know that they are lazy, short-tempered, whatever, and can also think of insightful ways to overcome them! Despite knowing all of this, they are unable to change.

This struck me as remarkably true. I know that I procrastinate on work, take on ambitious projects that I don’t complete, devote all my time and energy to something else when I should be focusing on completing my assigned tasks, spend too much time on my phone etc. I also know that I can overcome these by getting accountability partners, paying money to someone when I don’t accomplish them, setting locks on my phone, etc. However, despite knowing both the problem and the solution, I fail to stop these and improve my life. Was it impossible for me to get some control over my own life?

I had recently started writing a couple of pages every morning to clear my head. I would often write about procrastination, getting work done on time, not spend time reading other books when I had my textbooks to read, etc. I would often give myself advice on these pages. “Keep your phone inside your room and don’t use it except at one hour intervals” or something of the sort. This is the sort of thing that the likes of Katy Milkman and Jordan Petersen ask us to do: to advise ourselves as if we were advising a friend. Despite standing on the shoulders of such behavioral therapy giants and writing very useful advice to myself, I would inevitably find myself lying on my couch and scrolling endlessly through my phone within the very hour.

Something had to change.

Experiment and Observation

Instead of giving myself advice, I started using my morning pages to try and write out my exact mental process that leads me to destructive behavior like procrastinating on projects that are important to me, using my phone obsessively, etc. After a couple of days, I was shocked by the results.

It turned out that I wasn’t even (completely) aware of my thought processes that led to such behavior. It sometimes took 5-6 iterations of writing my thoughts out before my “deeper thoughts” could come to the fore and bleed on to the page. Although what I discovered about myself is perhaps too personal to put on a public blogpost, I could trace a very clear line to my past errors in judgement, and how these underlying thought processes led to those. I will try to explain one of those thought processes below, that was completely hidden to me before this.

I discovered that like most other people, I wanted to be someone special (famous, important, etc), instead of doing something special. These are related concepts, but they are slightly different. I have never done homework assignments well, for instance. Whenever I would get a homework assignment, I would, without fail, keep it in my drawer, and look at it only on the day of submission, when I would scribble something half-arsed and get mediocre marks. But it wasn’t that I was lazy and spent all of that time just sleeping instead of working on my assignment. I would instead work hard and try to read up on the latest research and science, hoping against hope that I would land on a project that would make me a famous scientist. Would doing homework make me famous? No. Would doing my shitty school project make me famous? No. Then I’m better off trying to study other things that might give me more of a chance to become the person I envision myself becoming.

This sounds egotistical and misguided. And of course it is! It is grossly stupid, and has led to a lot of self-destructive behavior on my part in the past. I wanted to be someone great instead of doing something great. And the subtle difference between those two made all the difference. Of course this is not the full extent of what I discovered.

What is miraculous is that as soon as I discovered these hidden thought processes, I found it very easy to change my behavior. On thinking about it more, I found a useful analogy for this phenomenon: these thought processes are like missiles that are shot from your unconscious mind. Your conscious mind, on the other hand, is equipped with an anti-missile system that may save you from these destructive missiles. However, you have to be aware of the exact location of the missiles so as to be able to shoot them down and prevent damage to your life. Giving yourself bookish advice is like shooting those anti-missile systems in the dark, hoping that you’ve hit something. You have to know exactly how your subconscious mind works, so that you can prevent it from torpedoing your life.

One great way to figure out these hidden thought processes is writing. I don’t have a clear explanation of why writing helps with this while just sitting on the couch and thinking doesn’t, but it does work. Hence, if you’re looking to change things about yourself, you first have to find out who you really are; in other words, how your subconscious really works. And for that, you will have to write and re-write about yourself, until you’re convinced that you know who you are and what you really think.