## 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?

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?

## 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.

I’m almost 30. And I think I’m finally learning how to talk to people

I lost my grandmother a couple of years back. I still vividly remember that I got to know this through a message that my sister sent me. My head began spinning, and I had to sit down for some time to process the information. My grandmother and I had drifted apart in the last few years, and it was disheartening to not have completely made up with her before her demise.

The last time that I had met her, she tried to talk to me about her impending death. Everybody knew that she was unlikely survive beyond a year or two. However, as soon as she brought up the topic of dying, I dutifully shushed her and told her that nothing would happen to her. She would get better very soon, and all would be well. She looked down with a sigh, and said “Let’s see what happens”. The conversation ended there, and I left after some time.

I’ve replayed that conversation in my head a lot of times. What should I have done differently? My grandmother probably wanted to reflect on the concept of death by talking to me about it. Had she laid out all her stories and feelings in front of me, she could have had an epiphany which would have helped her reconcile with the ending of human life. However, before she could do any of that, I just ended the conversation with a “don’t worry, you’ll soon get better”.

I was recently reading the book “Impro: Improvisation and the Theatre” by Keith Johnstone. In a small footnote, Johnstone talks about how nurses are trained to talk to patients. I will reproduce the notes that I took on this section.

Therapeutic techniques are the techniques that nurses should use to talk to patients, while non-therapeutic techniques are those that they should be careful to avoid. Of course, these techniques can (and should) be used with people in general, and not just patients in mental asylums.

On reading this section, I realized that I was basically doing almost everything wrong while talking to people. Let us explore some of those mistakes below:

• Reassuring- I chose to reassure my grandmother that nothing would happen to her, instead of just listening to her attentively
• Giving approval- If I wanted to get along with someone, I would just signal my approval with whatever they were saying. This didn’t always get me close to them, as they would probably suspect that I was being insincere.
• Rejecting/Challenging- I would commonly reject or challenge a political opinion during a conversation, instead of finding out why the person held that opinion
• Advising- A lot of times, I would offer unsolicited advice to friends who would tell me about their problems, without listening to them fully.

Instead of doing all of the things that I’ve mentioned above, I could have just offered the person I was talking to my complete, undivided attention. Let us explore some hypothetical scenarios below:

When people tell you their story or their opinion, they are often searching for clarity in their own thoughts. Help them find it. Ask questions, find connections between different parts of their story, ask them to explain something in more detail. Gift people your complete, neutral (don’t agree or disagree with things they say), undivided attention. Be curious about what they’ve been through. Let them arrive at their own epiphanies and meaning, on their own terms. Facilitating someone’s clarity of mind is often the greatest gift you can give them.

## Status battles in The Brothers Karamazov

The Brothers Karamazov is one of the most famous books in the history of literature. It was also Einstein’s favorite book. However, my appreciation for the book proves that (much) lesser mortals also have much to gain from reading it.

The book swings primarily between the author’s third person narration of events, and the psychoanalysis of the characters responsible for those events. The narration of events can sometimes be boring and dated. A gentleman of high rank goes to visit a lady, who looks upon him with suspicion because he is not adequately respectful in her esteemed presence. A lady of pure virtue wants to sacrifice herself to a man she doesn’t love out of the goodness of her heart. There are some allusions to “feminine” jealousy, the differences between the artificially sophisticated Europeans and the living, breathing Russians, etc. We don’t live in a world with nobility and fainting ladies and such anymore. Hence, this book can be very dated at times, and the reader begins to question Einstein’s sanity in recommending this book (I mean, he was also wrong about Quantum Physics, wasn’t he?).

Interspersed between the narration of said events is the author’s psychoanalysis of the characters, which by extension provides for a wide ranging discussion on philosophy, religion, etc. These sections are mind numbingly brilliant. I am not a sophisticated reader. I don’t know my Nietzsche from my Deepak Chopra. Even for idiots like me, the insights that Dostoevsky communicates, sometimes almost as an afterthought, make me stop reading and highlight furiously, amazed that someone could have this level of insight into the human mind. In my opinion, Dostoevsky anticipates both Freud and Carl Jung, and their philosophies lie embedded in this novel. Towards the end of this novel, I was basically highlighting whole pages on my Kindle. However, my intention in writing this post is not just to praise the book. I feel that Dostoevsky needed a little Robin Hanson to make some of his points about human nature even more transparent.

The main plot of the book is that a lowlife landlord by the name of Fyodor Karamazov is killed, and his eldest son Dmitri Karamazov is blamed for the murder. Both father and son were after the same woman, Grushenka, who was playing them off each other for sport. The father Fyodor had cheated his son of a sum of 3000 roubles, and had told Grushenka that he would give that money to her if she decided to choose him and become his wife. Dmitri, suspicious that Grushenka would indeed choose his father, breaks into his father’s property to prevent this. When the gardener of the property confronts him, Dmitri hits him in a moment of madness, and then runs away.

Let us delve a little deeper into Dmitri’s past. He was born into a high rank, and was a decorated army officer. However, he was loose with his money and morals, and tried to seduce virginal teenagers wherever he could, often abandoning them later. On the flipside, when a poor but beautiful woman asks him for 4000 roubles to save her ailing father, and was ready to sleep with him and do his bidding, he gives her all his property without asking her for any favors. That poor girl later becomes rich due to an unexpected inheritance, and also his fiancé. She tolerates his unfaithfulness, and also gives him 3000 roubles when Dmitri is penniless, despite knowing that he would only use the money to seduce the prostitute Grushenka. He takes the money, lowering his status to the lowest dregs, and does exactly that- try to seduce Grushenka. However, he also finds it beneath him to not try and return his fiancé’s money. Hence, he asks his father Fyodor for his 3000 roubles that he is rightfully owed. He is denied this, and then beats up his own father, threatening to kill him later. And so on.

When Dmitri is later accused in court of killing his father, the prosecutor explains his behavior to be like that of a pendulum, capable of containing both the highest of virtue and the lowest of vice. He gave away his last penny to a lady he didn’t ask anything of. However, he later took money from the same lady to cheat on her. He was ready to stoop to any kind of manipulation to get money from people. However, he only wanted the money to pay back his debt to his fiancé, so that she would not think he was a thief. Hence, he was a man who could be swayed by wild passions of any color, whether right or wrong, and he would be completely consumed with them without moderation. A man perpetually in control of his instincts and devoid of rational thought. An animal.

This is a fantastic explanation. However, a simpler explanation would probably suffice. Dmitri always wanted to maintain a higher status than anybody else. When he fought his father for those 3000 roubles, he didn’t really do it out of greed. Dmitri was famously generous with his money, and had reportedly spent 3000 roubles in a single night while partying with the villagers and raining champagne and chocolates on them. Also remember that he had given away all his money to his now fiancé without expecting anything from her. He didn’t need money for any expensive purchases for himself. However, he felt slighted by his father’s manipulation and control, and felt that his status had been lowered relative to his father’s. He had been outmaneuvered, and proven stupid. Hence, he beats his father and threatens him for the money so that he could prove himself to be the alpha, thereby raising his status in the process.

When he gave his now fiancé all his money, he didn’t do it out of a sense of generosity or love. In fact, it was implicitly understood between them that she would have to sleep with him for the money. However, at the last moment, he gives her the money and turns away, mocking her and sneering at her. He had gained something far more precious than intimacy- a clear status superiority in relation to another human being. She was ready to do whatever he said. And he turned her down on a whim. This was as big a status victory for him as he’d ever experience.

The only person that Dmitri didn’t try to defraud or bestow generous gifts upon was Alyosha. This was mostly because Alyosha never challenged Dmitri to a status duel. Whenever Dmitri talked to him, Alyosha never passed judgement or him or ask him for any assistance. Hence, Dmitri was never in a position to lose or gain status. Alyosha only lent a patient ear. Dmitri could be himself in front of him, without engaging in status battles.

Another person who engaged in frequent status battles was Grushenka, the prostitute who was playing with both father Fyodor and son Dmitri. She was abandoned by her husband-to-be, thereby lowering her status to the dregs. To compensate for this, she would charm men (like Fyodor and his son Dmitri), and then laugh at them as they would kill and maim each other to gain her affections. In this way, she would elevate her status to be above theirs. Her current status grab was in compensation for her status loss in the past.

I think that a lot of the world and people’s actions become simplified when looked at through the length of status. We don’t really need to work 12 hours a day at jobs we hate to earn buckets of money that we’ll only stash in the bank, or perhaps buy houses that are too big for us or cars that are too fancy for us that we’ll mostly only drive at 60 to jobs that we hate. We need the status that comes with all of that. We want the people in our lives to think that we managed to amount to something. That we have something that no one else has. That we are special. And The Brothers Karamazov shows that the fainting ladies and chivalrous gentlemen of the past centuries also had the same needs. Perhaps Robin Hanson and Keith Johnstone are on to something here when they say that society is mostly about status signaling and status battles.

## Lessons from trying to go vegan

Not everybody hates Osama bin Laden. Even Hitler has some fans, and Thanos many more.

But everybody hates vegans.

I decided to go vegan a few years back….and I told everybody. I had expected some backlash. But it became a funny free for all in which the only thing people knew about me was that I was vegan (the socially accepted statistical correlation between being vegan and being stupid is ~1). I was too old by then to really let it affect me, and I also made quite a lot of friends this way! But it was still surprising to see why my diet would be that big an issue. I was still a regular idiot in a non-animal friendly way in every other aspect of my life.

I was prompted to write this article because I came across this youtube channel. It’s called the Vegan Teacher, and features a woman who is around 55-60ish, and talks about the merits of going vegan. She is pleasant, engages with the comments, and doesn’t act all high and mighty. I will reproduce some of the comments she gets below:

Someone: *Breathes* That Vegan Teacher: Is that air VEGAN?

Alternative title: “Crazy Vegan Lady goes apeshit all over opinions.”

Every comment was something along these lines; calling her stupid, commenting on her lack of breasts, etc. I felt really bad on reading these comments, and left the only supportive comment on her video. I thought I’d be violently abused, and was bracing myself for a long evening of online fighting. However, I was essentially ignored. It’s the indifference that always hurts more.

## Why do people hate vegans?

I think the most common sentiment expressed against vegans is that they act holier than thou. Vegans start proselytizing about their inherent moral superiority on the very day that they stop consuming animal products. No one wants to be told that some random dude (or old lady on youtube) is better than them because they stopped drinking milk.

Fair enough. If you’re truly a “good person”, you will stop consuming animal products and not declare it to the world. You’ll buy soy milk and keep it to yourself. No one wants to hear about how you are changing the world while we are apparently destroying it.

I find that this argument, although fair in its place, doesn’t quite capture the issue at hand. Eating animal products is indeed, at this very moment, worsening the long term climate of the planet. It is proven beyond doubt that animals feel pain (for the interested, pain is correlated with the complexity of the neural circuitry inside the body, and animals have been proven beyond doubt to pass that threshold of neural complexity. Even chickens. Trees have not). We are enslaving billions and billions of animals every year, keeping them in horrible conditions, forcibly impregnating them for milk, or killing them for meat. Will my not discussing veganism help in making the situation better for them? Am I not shunning some kind of amorphous social responsibility towards them when I buy soy milk quietly and mind my business?

Well I suppose you could make the argument that if I shut up, people will like me more, and might be convinced that I’m not trying to prove some kind of moral superiority. With gradual exposure, they might slowly become convinced to maybe give veganism a shot…..But this sounds like a slow process, assuming that it happens at all. And animals are dying in millions every single day. Shouldn’t we all be panicking for causing immeasurable pain to all the species on the planet? Wouldn’t the correct time to become vegan be this very moment? Well people could always argue that you will only gain haters by being preachy, but might gain supporters if you play the long game. This feels horrible if we think about the animals losing out on their lives everyday, but I suppose it would be a more sane strategy.

Another reason that people have given to hate vegans is that they’re hypocritical. Celebrities who declare their new found love for veganism on Instagram, but are soon caught eating seafood, are derided on the internet. Better to be a cow eating troll than a hypocritical vegan. I have also often failed to be vegan, and have had dessert or pizza when no other option was available at restaurants. I’ve faced some backlash over this. Although these hypocritical celebrities and hypocritical non-celebrities like yours truly were arguably causing less harm to the environment and less pain to animals on average, we compromised on our lofty ideals, and were hence untrustworthy.

This dovetails perfectly with a paper that was recently published in the Journal of Experimental Psychology. Let me reproduce the abstract below:

A lot of people argue that animals eat other animals, and humans are animals. Hence it’s fine. We’ve been eating animals for millions of years, and we should continue to do so. This argument is slowly losing favor with time. Animals are also known to abuse and rape females, eat their offspring, steal into nests and eat others’ offspring, etc. Should we also do the same? Modern humans have the option of raising themselves above the moral clusterfuck that evolution has hardwired within us. We might do well to take this opportunity.

## What would a correct pro-vegan strategy look like?

First, we need vegans to get down from their moral pedestal. How can they do that? They could detach their identities from the message. They could write anonymously. Hence, their preaching wouldn’t boost their real-life images or wages.

I don’t think this will work. PETA is a faceless organization, and its massive ad campaigns don’t really benefit anyone’s image. There are lots of anonymous vegans who talk about veganism on Reddit as well. However, both PETA and these commenters are universally hated.

I honestly think that the only way to convince other people to become vegan is to make it easier for them to become vegan. When cheap imitation meats flood the market that taste identical to actual meat, people will slowly be tempted to give them a go. I am tempted to say that governments should subsidize imitation meats at the beginning so as to create a market. But seeing as government subsidies for education, housing etc mostly serve to ultimately jack up prices, I feel that a technological push is the only thing that can really propel this market into sustained growth. If my grocery store sells both beef and imitation beef at the same prices, and they both taste exactly the same, I might decide to decide to give imitation meat a shot because **waves hands** cows and feelings and such.

If you’re vegan and reading this, a better use of your time than proselytizing is learning some food technology to make this possible (and I would do well to remember this as well). I mean c’mon. How hard can it be to make soy taste like milk, cow, chicken and pig?

## Ancient games in modern times

I really can’t play any sports. I was never picked for games of cricket or football, and the only sporting event I have ever gotten a prize for was in grade 4, when I was the engine for my house team.

Therefore, you should take my unsolicited opinions about Olympics and sports very seriously.

## The Olympics

India’s Neeraj Chopra recently won a gold medal in Javelin throw at the 2020 Olympics. Some people thought that this was amazing that we should be proud of our Olympic contingent, whilst others were quick to point out the fact that India had one only 1 gold medal, whilst our neighbors to the North had won 38. I, on the other hand, was confused about why we have a Javelin throwing contest at all.

Olympics were first held in ancient Greece, when it mattered how fast you could run or how far you could throw a javelin. The strength of your army, and hence, the security of your very household depended on it. If your army couldn’t hurl heavy objects at the enemy or your messengers couldn’t run fast enough to keep communication lines open on the battlefield, the enemy would roll into your city and kill you and your family for game. Physical prowess was often the sole determinant of whether one would survive to see the next day.

The games also had a political motive for the Greek city-states that were constantly at war with one another. Let me quote from the wikipedia article:

During the celebration of the games, an Olympic Truce was enacted so that athletes could travel from their cities to the games in safety. The prizes for the victors were olive leaf wreaths or crowns. The games became a political tool used by city-states to assert dominance over their rivals. Politicians would announce political alliances at the games, and in times of war, priests would offer sacrifices to the gods for victory.

Greek city-states used the Olympics to demonstrate their physical capabilities, and also gauge that of others, so that they could decide whom they should ally with if a war broke out. The utility of these games spilled onto the real world, and had very real consequences.

The world has obviously changed a lot since 400 BC. Most educated persons now spend most of their time sitting in front of computers, working hard to make rich people even richer. Farming is becoming more and more mechanized, and even armies rely more on their weaponry and less on their physical power. Suffice it to say, being able to throw a javelin really far has no real world utility.

So what does that mean? Should we stop these games completely? For some reason, this seems wrong. I enjoy watching team sports for instance. There is a lot of strategy involved in those, and team sports generate a lot of revenue for individual owners and countries and such. However, individual athletic events have no such relevance. 230 million people are not going to tune in to watch the English Premier Long Jump. Training athletes for these events costs countries a lot of revenue, and almost none of this revenue is recovered. Long jump cannot be a spectator sport, even if you throw in some cheer leaders and Rihanna. It’s just someone jumping really long.

But wait. Countries still get to demonstrate their physical prowess right? Let’s explore this a little bit. It would seem clear to any viewer of international sports that physical prowess is not national. It is genetic. If you are 5′ 8” and want to become a sprinter, there is probably a black person in Jamaica who is faster than you, even if you receive better training than them and are ten times more patriotic than them. Although training does make a difference at the highest level, there are some necessary genetic pre-requisites. I mean, C’mon. That is why India has to resort to fielding Dutee Chand, a female athlete with abnormally high testosterone levels, to be able to have a stab at athletic medals, and \$(&^%^(*& has to resort to widespread doping. A peek at the 100 m medalists at the modern Olympics should further convince you of this point. Countries gain very little by putting so much money into training athletes. No one is going to form a military alliance with Jamaica because they have a lot of people who can run very fast.

I’m still proud of Neeraj Chopra though, and I wish to become friends with him…..in case I ever need someone impaled 87.58 metres away.

## IMO 1981, Question 3

I had a great time solving the following question a couple of nights back. More so because I’d failed to solve this question in the past.

This question is completely unapproachable if you try to use algebra. However, generating simple examples helps.

Let me try and write down some solutions to $(n^2-mn-m^2)^2=1$: they are $(1,2), (2,3), (3,5), (5,8)...$ Do you see a pattern? The answer becomes a two-liner as soon as you observe this.