assemblyman

As a curiosity, the contrast between Grothendieck and Ramanujan is very striking. One famous story about Ramanujan from Wikipedia (https://en.wikipedia.org/wiki/1729_(number)):

"Hardy stated that the number 1729 from a taxicab he rode was a "dull" number and "hopefully it is not unfavourable omen", but Ramanujan remarked that "it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways"."

They, of course, were very different personalities, doing very different mathematics with very different impacts on the field. I always found it interesting that Ramanujan seemed to be very comfortable with numbers, their properties, patterns (continued fractions) and Grothendieck was very comfortable with structures and their rhythms without paying attention to concrete examples.

Peter Woit (https://www.math.columbia.edu/~woit/wordpress/) has occasionally posted about Grothendieck's life and work. E.g.

Articles on his life: https://www.math.columbia.edu/~woit/wordpress/?p=7335

Two Titans (Grothendieck and Witten) - https://www.math.columbia.edu/~woit/wordpress/?p=12868

AMS Math articles on Grothendieck - https://www.math.columbia.edu/~woit/wordpress/?p=78

My point is that it's really the storytelling/gregarious nature character. There is no shortage of people who were at the Manhattan project and won Nobel prizes or were prominent. A partial list: Oppenheimer, Bethe, Rabi, Teller, von Neumann, Compton, Fermi, Segre, Ramsey, Alvarez. There are easily many more. Schwinger was at Rad lab.

Schwinger was considered a tremendous educator. I think he had ~90 PhD students and four won Nobel prizes. His lectures were often described as Mozart symphonies. I have studied parts of his books and the experience was always eye-opening. But, his education was focused on students, mostly graduate students. He was also a shy character.

In any case, I still love reading Feynman and Schwinger's works. I would also include Sommerfeld, Pauli, Landau, Weinberg in that list.

It is very interesting to see completely different impressions of Feynman from comments here. As a physicist, I first got introduced to Feynman from his popular QED book as a freshman in college. Over the years, I came to admire his contributions and way of thinking but could also see the cult of personality that has formed.

Feynman is definitely not like Neil deGrasse Tyson or Kaku. He was a very creative and technically sophisticated physicist. All his popular books are based on lectures he wrote and gave but they were mostly "side projects" e.g. computation, lectures on gravitation, six easy pieces etc.

To get a better sense of his work, I would highly recommend:

- The Beat of a Different Drum by Jagdish Mehra

- Genius: The Life and Science of Richard Feynman by James Gleick

- Selected Papers - https://www.worldscientific.com/worldscibooks/10.1142/4270 (expensive and technical)

- QED and the men who made it by Schweber

There are also many historical physicists who are surprisingly unknown outside the field - Schwinger, Tomonaga, Landau, Sidney Coleman, Murray Gell-mann, Nambu, Steven Weinberg, Ken Wilson, Curtis Callan etc. I just randomly picked a few names before the 90s but these are all scientific giants. For example, Schwinger's papers are notoriously hard to read but his books are great after a first course. Sidney Coleman gave beautiful lectures on QFT. Landau is extremely famous for his 9 books with Lifshitz. It definitely is very surprising that Feynman has such an outsized share of interest. Maybe because he was a gregarious outgoing character?

Another interesting aspect is how a person is often viewed as an authority or even a genius because their work introduces an audience to the subject. Feynman with his lectures. To a far lesser extent (in my opinion), one sees this with Andrej Karpathy and Jeremy Howard. This is not to take away from their wonderful teaching work. I know how hard it is to distill material and convey it. But, there's a whole web of contributions that leads to a subject maturing enough to be taught clearly.

As I get older, I find it less useful to assign labels (names) to discoveries and contributions. As Feynman himself said in a lecture after drawing a Feynman diagram, "this is THE diagram" (and not the Feynman diagram).

I think you are writing your comments conditioned on not just what you are responding to but also a lot of internal assumptions about their intentions. The person you are responding to said or implied nothing about surveillance or Western assumptions about China. They are making the claim (apologies to them if I am misrepresenting) that societies or governments achieve extraordinary goals (i.e. goals that they were not expected to achieve within a certain time-frame) because of the physical, economic and social conditions and not because of cultural elements. Cultural explanations are post-hoc i.e. they are used after the fact to boost morale or give a sense of unity. More concretely, if China, the US, the EU, Japan, India, Russia can launch spacecrafts to the moon, so can Nigeria and Kenya given enough time, resources and the right incentive structure even if they are culturally very different from the countries above.

I feel the American media and general public has gotten psyched by the recent announcements coming from China. This might be new weapons or Deepseek or conveniences in the cities. I wish the Chinese people all the best and sincerely hope they prosper. At the same time, I really wish Americans would get out of this panic mode. You answered several challenges over the last 250 years. Some were existential threats. America generally lays out its problems for everyone to see. Americans tend to be extremely self-critical. This is often misconstrued by some to be signs of weakness. Anyone who believes this is, in my opinion, delusional.

As an aside, there are some comments about the "Chinese way of thinking" and the "American way of thinking". I generally think these discussions veer off into notions of cultural superiority. That, also in my opinion, is the mark of weak minds. The fact is once something is shown to be possible, it is exponentially easier to duplicate and improve it. America did this with German technology, China did it with American technology and I am sure countries like India are going to quickly get there too (I am not suggesting the Germans didn't learn from others themselves). This sets a firm base for iterative improvements.

To riff off another comment, China's progress wasn't done by God. America will learn and adopt what's valuable and discard what's not. If I have learned anything about Americans (of all backgrounds), they don't shy away from a challenge. For all its faults, I still personally will root for a society based on something like the American constitution.

Even with thick desires, I sometimes find myself day-dreaming about the state when I have mastered a skill or understood a topic deeply. At the same time, I know from experience that the process never ends. Even when one does master a skill, one is deeply aware of what one doesn't know or understand or what one is not good at within that domain.

What helps me is to focus on today. If I can spend even an hour on a topic and get lost in it or even get frustrated by it, it is time well-spent. I was going to say "it is progress" instead of "time well-spent" but even that's a trap. Progress implies moving forward in a preferred direction. While I can't say I don't want to make progress, I am training myself to care less about it. It is really the time spent engaging that's most valuable (at least to me).

As one example, I met quite a few graduate students from Indian Institutes of Technology (IITs) who ranked high enough in the entrance tests to study computer science/engineering or electrical engineering but really wanted to study physics. They all had significant pressure from their parents to choose the engineering branch and had to fit in physics electives where they could. My understanding is that the priority list was:

computer science/engineering > electrical engineering > mechanical engineering > ... > things like metallurgical engineering > ... > physics (and maybe other sciences)

Some of this is driven by job prospects while some of it is prestige driven because one's major lets one infer one's rough ranking in the entrance tests.

So it's very common to infer that if you weren't studying engineering, you didn't rank very high and barely made it past the cutoff ranks and had to study physics or metallurgical engineering.

When I was younger, I thought these rank-based systems (very common in Asian countries) are better than the fuzzier American system of grades + extracurriculars + reference letters. But my opinion is the opposite now. As soon as ranks are involved, a notion of prestige gets assigned. Once prestige is involved, people will climb over each other to get through the doors and suppress their instincts to earn social credits. I have seen enough people who are successful by traditional metrics but are miserable because they didn't spend time pursuing their interests (modulo concerns about jobs and money).

Edit: I'll add that my IIT friends were generally extremely bright, curious, creative and generally wonderful to work with. But they also had a competitive streak which could turn counter-productive. Against their own better instincts, they sometimes got locked into a path where outcomes could be measured vs exploring areas less traveled. If they saw a topic or area that attracted top minds (e.g. see AI at frontier labs today), they felt pulled in that direction because "that's where the smart people were going and they themselves were smart and therefore, should go into the arena". This is true of Asian Americans in general. After all, that's why there was an uproar that students with perfect SATs and GPAs of 4+ (5?! i.e. A++ grades) were sometimes getting rejected by Harvard. I agree with Harvard in this case. One doesn't want cookie cutter/prescriptive paths into top universities. Instead, there should be some randomness as long as students meet some decent baselines. I don't mean race-based or group-based selection. Just really random selection at least for a small fraction of students.

I came to the US for college from Asia to study physics (and mathematics). I actually came to study astronomy because I found it fascinating but didn't really like physics or math. My first physics encounter in college here transformed my life. There was no memorization. Instead, we had short quizzes in each class (first 5 min), weekly individual assignments, weekly group assignments (two students each), four "midterms" where one could get densely written "cheat-sheets" as well as weekly physics lab that often went on far beyond the time slot.

In high school, physics was mostly based on memorization. There were a few problems but all based on some patterns. None made you think extremely hard.

I also found that many American students (who were extremely good in my experience) seemed to have a much better practical sense.

One of the key steps in the development of a physicist is the transition from solving textbook problems to creating your own problems. In essence, the skill one learns in graduate school is defining/crafting problems that are solvable and interesting. The primordial phase starts in college as one is solving many problems. Initially, the new problems are straightforward extensions of existing ones (e.g. add an air resistance term for parabolic motion). Eventually, one (hopefully) develops good taste and essentially is doing research.

Interestingly, I also find very different attitudes to physics in the west (at least in the US) and other parts of the world. In US universities, physics is still seen in glowing terms. In many other places, physics is what you study if you couldn't do engineering. Young people (well, all people) are impressionable and this subtle bias affects what kind of students end up studying the subject.

Renting can be much better financially than buying.

Edit: all % numbers are per year

Consider the case of condos in cities. If you were to buy outright, you effectively get a return by not paying rent (i.e. paying yourself rent). Rent is usually ~5% of the condo cost. HOA + property taxes is 2-3% so subtract that from the rent return i.e. net return 2-3% (5-2/3%). The rest of the return is appreciation from the underlying real estate prices. I am excluding maintenance costs because they are negligible in condos.

On the other hand, if you rent and put the entire amount (that you would have paid to buy the condo), you get ~10% per year. To break even between the two scenarios, you would need real estate prices to grow 7-8% (2-3% + 8-7% = 10%).

Beyond this, there are psychological reasons to buy vs rent. Buying - ability to customize the space, peace of mind because of perceived stability etc. Renting - flexibility, peace of mind because of no long-term obligations etc.

A mortgage is an interpolation of the two cases at the cost of the interest one pays. It is noteworthy, at least in the US, that for most people, this is the only time they can borrow several hundreds of thousands at relatively low costs.

I haven't thought about or learned a systematic way to add roman numerals. But, I would argue that the difference is not notation but a fundamental conceptual advance of representing quantities by b (base) objects where each position advances by a power of b and the base objects let one increment by 1. The notation itself doesn't really make a difference. We could call X=1, M=2, C=3, V=4 and so on.

I also don't know what historically motivated the development of this system (the Indian system). Why did the Romans not think of it? What problems were the Indians solving? What was the evolution of ideas that led to the final system that still endures today?

I don't mean to underplay the importance of notation. But good notation is backed by a meaningfully different way of looking at things.

I am not in Seattle. I do work in AI but have shifted more towards infrastructure.

I feel fatigued by AI. To be more precise, this fatigue includes several factors. The first one is that a lot of people around me get excited by events in the AI world that I find distracting. These might be new FOSS library releases, news announcements from the big players, new models, new papers. As one person, I can only work on 2-3 things at a given interval in time. Ideally I would like to focus and go deep in those things. Often, I need to learn something new and that takes time, energy and focus. This constant Brownian motion of ideas gives a sense of progress and "keeping up" but, for me at least, acts as a constantly tapped brake.

Secondly, there is a sentiment that every problem has an AI solution. Why sit and think, run experiments, try to build a theoretical framework when one can just present the problem to a model. I use LLMs too but it is more satisfying, productive, insightful when one actually thinks hard and understands a topic before using LLMs.

Thirdly, I keep hearing that the "space moves fast" and "one must keep up". The fundamentals actually haven't changed that much in the last 3 years and new developments are easy to pick up. Even if they did, trying to keep up results in very shallow and broad knowledge that one can't actually use. There are a million things going on and I am completely at peace with not knowing most of them.

Lastly, there is pressure to be strategic. To guess where the tech world is going, to predict and plan, to somehow get ahead. I have no interest in that. I am confident many of us will adapt and if I can't, I'll find something else to do.

I am actually impressed with and heavily use models. The tiresome part now are some of the humans around the technology who participate in the behaviors listed above.

I find software engineers spend too much time focused on notation. Maybe they are right to do so and notation definitely can be helpful or a hindrance, but the goal of any mathematical field is understanding. It's not even to prove theorems. Proving theorems is useful (a) because it identifies what is true and under what circumstances, and (b) the act of proving forces one to build a deep understanding of the phenomenon under study. This requires looking at examples, making a hypothesis more specific or sometimes more general, using formal arguments, geometrical arguments, studying algebraic structures, basically anything that leads to better understanding. Ideally, one understands a subject so well that notation basically doesn't matter. In a sense, the really key ingredient are the definitions because the objects are chosen carefully to be interesting but workable.

If the idea is that the right notation will make getting insights easier, that's a futile path to go down on. What really helps is looking at objects and their relationships from multiple viewpoints. This is really what one does both in mathematics and physics.

Someone quoted von Neumann about getting used to mathematics. My interpretation always was that once is immersed in a topic, slowly it becomes natural enough that one can think about it without getting thrown off by relatively superficial strangeness. As a very simple example, someone might get thrown off the first time they learn about point-set topology. It might feel very abstract coming from analysis but after a standard semester course, almost everyone gets comfortable enough with the basic notions of topological spaces and homeomorphisms.

One thing mathematics education is really bad at is motivating the definitions. This is often done because progress is meandering and chaotic and exposing the full lineage of ideas would just take way too long. Physics education is generally far better at this. I don't know of a general solution except to pick up appropriate books that go over history (e.g. https://www.amazon.com/Genesis-Abstract-Group-Concept-Contri...)

In the same vein, I always wondered if

* the vast majority (including me) are not really very intelligent. We have a lot of "state" that's transferred from generation to generation. Once in a while, a very small percentage of people make advances and they filter through society and improves (or maybe just changes) the state. We collectively gives humans credit for these improvements but it's not the species but those specific people who created that jump in capabilities.

* this notion of inherited pride or inherited achievement is very common. This leads to being proud of membership in a group (country, religion, tribe, corporation, university etc.) and also of instinctively rejecting ideas put forth by others (e.g. see the amount of derision vegetarians and especially vegans attract).

* achievement/progress is also time-scale dependent. While we get smug about our progress, if it ends up destroying the one planet we have, it will be incredibly stupid. Humans fundamentally are not capable of thinking long-term.

Everything around me was not made by me. I don't even understand how I would potentially make most of these from scratch without using machines made by other people or knowledge acquired over time (see first bullet above). Within the framework provided to me, I can convince myself to reason and act but the framework itself is my operating system. Of course, I like to think I am intelligent and reasoning but it's all in a box. I feel this describes almost everyone I know except for a few outstanding scientists I have worked with.

>You can’t avoid the reality that’s your life depends on something else dying. Either plant insect or animal

There are more nuanced ways of thinking about this. A good example is Jainism's version of vegetarianism which requires paying attention to what one consumes.

https://en.wikipedia.org/wiki/Jain_vegetarianism

"Jains make considerable efforts not to injure plants in everyday life as far as possible. Jains accept such violence only in as much as it is indispensable for human survival, and there are special instructions for preventing unnecessary violence against plants."

I keep seeing the argument that non-returners are creating jobs. Does that mean they also throw their trash on the streets so cleaners have to be hired? Should one randomly break into houses so every people need to hire security guards? How about scratching cars on the street so mechanics and painters have some extra work?

I always wondered if there are people who don't have a clear distinction between their private space/home and public space. Whether it's not putting their carts back, or talking loudly on the phone/playing music in a gym or any public space, tossing trash our of car windows, there are some people who seem to inhabit their own inner world so fully that it doesn't register that there are other people around them and that they are using a shared resource.

It's very heartening to read this sentiment. Thank you!

100% agree. In south asia and south east asia (maybe other parts too), it's very common to have insects like spiders, lizards, local bugs, ants inside apartments and houses. No one bats an eyelid and, if we felt, a larger insect was trapped and couldn't make it outside, we would just put them on paper tissue and take them outside.

In the US though, I frequently find people freaking out if they see any insect. There's a zero-tolerance policy for any living creature indoors. They are almost always killed. It leads to a disconnect between human beings and any other life. Other creatures are always a distant presence in zoos or on TV although exceptions are made for pets.

This shows up in language too. Instead of saying, a bear was killed for straying into settled human land or breaking into a house, the phrase used is "the bear was euthanized" (still accurate) or "the bear was destroyed" as if it was a piece of furniture. To contrast this with, say India, even tigers and elephants that kill, are mostly tranquilized and moved deep into a forest. This is very alien in the US where the trigger-happy reaction is to kill the animal.

I find this obsession with building strange. It's a very SaaS Silicon Valley mindset. There are whole swathes of very talented engineers who spend most of their time debugging, characterizing systems, doing performance analysis and resolving bottlenecks. Some of it might require writing significant code but mostly it's writing small test cases. The key skill is to treat a computing system as the object of study and to be a good empirical scientist (which requires understanding theory pretty well). These are people with deep expertise in networking, GPUs, CPUs, memory etc. One only has to look at national labs that do large-scale HPC (high-performance computing) to see examples.

One can argue that a lot of "building with AI" is commoditized by fine-tuning and RAG libraries or even reduced to prompt engineering. A lot of it is also tricks that might work on one dataset but not others. Putting together libraries fueled by pizza and coke gives an illusion of skill and speed.

Are there grifters who are jumping onto the AI bandwagon? Of course! In spades. Are there also engineers who want to build up their skills and are failing to do so or in the process of doing so? Of course, this happens too! But there are also people who are trying to understand, debug and improve models who are not necessarily "building". After all, the scaling laws paper (the original one) was a result of pure analysis of empirical data.

I am a big fan of using LLMs although in my own limited way. I don't work at Meta and don't feel strongly about him leaving or staying there.

It's possible that he will turn out to be correct in the long run. From his viewpoint, the primary goal is research and any usefulness of intermediate advances is maybe (speculating) "beneath him". If this is the case, I completely understand why a corporation would want to eject him. LeCun probably sees the pretty amazing developments since ChatGPT first came out as incremental hacks. I am neutral about this aspect too. Maybe they are but the hacks have been useful to me.

Eventually this feels like the correction of a real misalignment between LeCun/FAIR and Meta. Hopefully now, they can both focus on what they are good at. I must admit that I have great sympathy for open-ended research but industry has always been fickle about it. That's where the government and universities are supposed to play a key role.

A few people mentioned Meta burning through people like LeCun, Carmack and Luckey. We give a lot of credit to individuals in our society. At the same time, there's a frequent pattern of very successful people changing fields, organizations or general environments and suddenly looking very ordinary. In a way, this is a very strong argument to let people settle into a place they can be happiest in. A few examples:

* This is seen very often in Formula One. Schumacher when he went from Ferrari to Mercedes (after a short sabbatical), Vettel from Red Bull to Ferrari, Raikkonen from Lotus to Ferrari (his second stint), Hamilton from Mercedes to Ferrari, Perez at McLaren. There's a lot of Ferrari here so maybe that's the confounding factor.

* This also happens when physicists changed fields. They generally can't replicate their earlier success. Dirac, Feynman, Schwinger, Einstein all went through a transition like this. One explanation is that their early success was precisely so unusual (for anyone) that it would be hard to replicate in general.

* In my experience, this happens at companies too. Whenever we hired a "rockstar" from another company, they would generally struggle (across multiple companies I have been at). This could partly be a result of sabotage from a few vested interests at the new company. But often, it's hard to adjust to a new environment in a short amount of time.

The converse also happens. Sometimes a person considered ordinary goes to a different environment and flourishes. Palmer Luckey has been very successful at Anduril. Stephen Smale was almost failing out of his math PhD program but suddenly started flourishing in his third year IIRC and eventually got a fields medal. Ed Witten experimented with economics, history, linguistics, applied math before switching to physics in his second year and suddenly started making rapid progress.

This is not a very rigorous observation and I am missing many confounding factors.

I see a lot of criticism of LeCun and his views on LLMs as well as his inability to "deliver" products. I don't think that's what he cares about at all. His prominence led to him being picked by Meta. It was a chance to get massive resources that he couldn't get at NYU and the chance to work with smart people outside academia. The pay probably didn't hurt either. In return, Meta became a magnet for smart ML researchers and engineers. If I permit myself to speculate about his thoughts when he took the job, he had no intention of committing to product timelines and generating revenue. Now that Zuckerberg has clearly committed to something he like i.e. building a new product line and expanding the business, it was only a matter of time before LeCun would feel left out and under-resourced.

Interestingly, Yoshua Bengio is the only one who hasn't given into industry even though he could easily raise a lot of money.

Ah no, I meant these ones by Graham:

https://www.amazon.com/Intelligent-Investor-3rd-Ed/dp/006335...

https://www.amazon.com/Security-Analysis-Seventh-Principles-...

I'll paste a couple more readable ones too :)

https://www.amazon.com/Concentrated-Investing-Strategies-Gre...

https://www.amazon.com/Value-Investing-Graham-Buffett-Financ...

Like I said to someone else above, I myself put almost all my savings in index funds and slowly read/study these just for curiosity. Eventually, I hope to actually spend some time looking at real data but even if I don't find any interesting stocks or lose my modest starter funds for this activity, I would be okay with it.

I think a lot of this depends on where you started. I don't disagree that the details of the methods don't apply especially to technology stocks. On the other hand, one learns a principled way of looking at companies. It's easy, if one hasn't paid attention to anything in the financial world, to think of stocks as abstract numbers that "will go up" or "go down" or "has good prospects because everyone is talking about it". Any of these books are a good antidote to that because they teach a framework to reason about companies.

I also do find it strange that this idea gets so much push back. For people who don't want to think about investing, just put your savings in index funds (and/or cash equivalents depending on your plans). For people who are curious about these aspects, read some of the books and invest with small amounts and learn. Maybe you won't consistently beat SP500 and maybe you will. After all, it's not like all active investors shut their hedge funds down and started investing in SP500. The usual argument is that the funds have much more resources but it's actually easier to invest with smaller amounts than millions or billions of dollars.

zipy214 below has given much more eloquent explanations about the non-applicability of the EMH than I can. For me, the vast majority of my personal savings are in index funds. The reading list above, even if the methods are outdated (the broad details still apply even though many ways of analyses no longer apply especially to tech stocks), are mainly to get the curious started on one principled path to investing with the caveat that, if they do experiment, they should do so with amounts they are willing to lose.

Just one more comment to what you said below:

"If you personally believe markets are not efficient, and prices are not accurate, then perhaps you should take up day trading."

Value investors famously oppose any kind of day trading or "in and out" trading. People like Phil Fisher used to advocate never selling unless the fundamentals change drastically (change of management, new technological developments that make a company's products obsolete etc.). Of course, one doesn't have to be that extreme but equating value investing with day trading is misleading.

I don't think feeding pigeons is objectively a bad thing. While I live in a highly urban area and don't doubt they get food, a couple of additional sessions of seeds per week is not going to make them dependent or obese. I generally confirm I am doing something reasonably appropriate by asking my local bird sanctuary.

I don't follow social media at all so not sure if there's a trend going on about feeding birds. A few times someone has shown me clearly fake (i.e. spliced together) videos of animals being "rescued" and living with families. They do make me angry because they are clearly made for views.

If one has time, a few books I would highly recommend:

The Intelligent Investor by Ben Graham

Security Analysis by Graham and Dodd

Common Stocks and Uncommon Profits and Other Writings by Philip Fisher

The Little Book That Still Beats the Market by Greenblatt

Warren Buffett's annual letters

Actually, anything by Ben Graham or Joel Greenblatt is worth reading if one is interested in the investing world. I don't know if I'll ever invest enough time doing fundamental analysis and actively (value) investing but I am making my way through these just to understand value investing properly.

As an aside, pigeons are deeply misunderstood. Here's a very nice article about it: https://www.bbc.com/future/article/20240620-why-do-people-pe...

From the article: "One study found that, between 1941 and 2004, there were just 207 reports of pathogens transmitted from pigeons to humans – anywhere in the world. In all, there were 13 recorded deaths. The true number may be higher, but it would have to be off by several orders of magnitude to compete with the scale of infections from other domesticated animals – particularly some of those with more favourable reputations."

I occasionally (2x per week per location) feed a group of pigeons both in my apartment building's garage as well as my grocery store's open parking lot. They love sunflower seed kernels (unsalted since salt can be really bad for them) which has led to me hoarding sunflower seeds every time I go food shopping. They now recognize my car and swoop in as soon as I drive up to a parking spot (which is not consistent at the grocery store). Hopefully, one day I'll convince myself to stop/slow working and dedicate my free time to studying animal communication and behavior with some ML involved.

It's a back-of-the-envelope calculation not a precise one. It doesn't take foresight to invest in SP500. DCAing (dollar-cost averaging) into an index fund is actually the recommended savings strategy with a short-term cash balance of 6 months-2 years of cash savings depending on your plans (sabbatical etc.), especially when one is decades away from retirement age.

I only included meta because he works/worked at meta and it's not unusual for people to just leave their rsus in their accounts after they vested. I agree though that one shouldn't pick stocks that happened to explode (e.g. nvda).

There are several unrealistic assumptions I did make:

* Presumably when someone starts, they earn less than in recent years. He probably wasn't making huge amounts his first few years. Amounts invested in earlier years are smaller but have more time to compound and amounts invested in recent years are larger but have had less time to compound.

* Returns aren't constant.

* I pulled the $2 million/yr out of thin air. It could be $1 million/yr or even $10 million/yr. I have no idea what the overall head of a project like PyTorch would make.

* Everyone's expenses are different. In and around NYC, one can live on $80k/year, $120-150k/year as well as as on $1 million/yr. I assumed zero since I wanted a nice even $1 million/yr savings. Maybe it was $500k/yr of savings in which case all the numbers should be halved.

In any case, I can't see how one wouldn't end up with at least $10 million in a position like this with 10 years at meta. Unless one buys a $5 million unit in Manhattan and is burdened by a high mortgage.