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< Another thing is that they're specific kinds of tensors on the manifold. Tensors represent basically all the information we might be interested in about a manifold (for example, its curvature). Differential 1-forms are "dual" to vectors and are therefore important to building more complex tensors (higher tensors take in some number of vectors and 1-forms and output a value).

This was how differential forms were introduced in the course. I understood all of this from an algebraic standpoint, but I was lacking any geometric intuition for differential forms whatsoever. Say you have a k-form on some manifold and you evaluate it at some point which gives you an alternating covariant k-tensor. Then when you evaluate that at k tangent vectors at the point you get a scalar, does this scalar have any geometric meaning? Does it measure anything? Later when we did Riemannian manifolds and introduced the volume form that was at least a little more intuitive, as far as I remember, but general differential forms were intuitively a complete mystery to me. Also I kind of got their usefulness in an algebraic sense when we did some typical vector calculus calculations using the concepts of divergence and curl, but I didn't have much intuition for these concepts since I don't have a physics background and only worked with vector fields in this abstract setting. Unfortunately we did not cover De Rham cohomology. Thanks for your answer, I will take a look at planimeters.

I took a Differential Geometry course in university. The course covered basic theory on smooth manifolds and Riemannian Geometry. While I did appreciate the beauty of the subject from a theoretical perspective, I have to admit that I somewhat lacked intuition for the material, especially with regards to differential forms. I understood that their properties make sense to define the notion of integration on a smooth manifold, but never really saw whether they had any purpose besides using them for integration. Even in their use in integration they were kind of a mystery to me, if I recall correctly we just defined integration on subset of R^n, where differential forms played the same role as d_{x_1}d_{x_2}...d_{x_n}, which is known from the riemann integral, but then to define integration on manifolds, we just used the pullback of the charts, which didn't alleviate any of the mystery and didn't add anything about why differential forms themselves are supposed to be important. I wish I had taken some physics courses, where vector calculus is used heavily and that might have helped some, but in the end I was somewhat disappointed, because even though taking the course certainly did make me a better mathematician, I still didn't have the feeling of really completely grasping the concepts, even though I could prove statements.

I didn't intend to contradict you. Granted it's totally valid to assume that many people who are tested positive and show no symptoms might develop some later, but I just thought it might be of interest to some people to have numbers where they tested patients twice over a long enough interval to conclude that they really were asymptomatic.

>For example, there seem to be multiple events where people get sick even after it has been established that they have antibodies.

Do you have a source for that?

In a region in Italy they tested [1] people twice, 2 weeks apart, and found that 43% were completely asymptomatic over the course of the infection.

[1] https://www.medrxiv.org/content/10.1101/2020.04.17.20053157v...

Probably by implementing strict quarantine measures at the start of the pandemic. They likely had many undetected cases in the beginning as well, the quarantine reduced the pool of infected people a lot. Given that the number of infected people is relatively low right now, their testing capacity probably allows for fairly good tracing of cases, which is one of the primary reasons why their numbers remain low, in addition to factors such as social distancing and high hygiene standards. But the large number of undetected cases in the beginning would would reduce the ifr significantly.

South Korea apparently has conducted 11500 tests/ 1m population. It's ridiculous to claim that they detected every covid case and to suggest that their cfr is a lower bound for the ifr.

It does. The german study was conducted in a town that was relatively heavily tested during its outbreak, and only recorded 2.5% official infections in the population of that town, so the german study showed the actual number to be 7 times as high. This Austrian study found this number to be only ~2.5 times as high. Germany and Austria overall seem to test fairly similarly.

This goes contrary to what other studies have found. In the Stockholm metropolitan area they found that 2.5% percent of the population was infected [1]. In in a community in Germany, which was the epicenter of the outbreak for a time, they found that 15% of the population tested positive for antibodies [2]. In Iceland they found that 88-93% of the infections are undetected [3].

[1] https://www.folkhalsomyndigheten.se/nyheter-och-press/nyhets...

[2] https://www.land.nrw/sites/default/files/asset/document/zwis...

[3] http://www.igmchicago.org/wp-content/uploads/2020/04/Covid_I...

Take a look at the Economics & Finance videos on Khan Academy [1]. I have only watched some of them, but those I watched were good.

You could read some standard macroeconomics textbooks, but I'd advise against doing that, since they are expensive and unnecessarily formalized using elementary math, which doesn't really add anything to the material itself. There's however a good book on monetary economics by Mishkin called Economics of Money, Banking and Financial Markets, which is really good, but the Khan Academy videos on banking mostly cover the same material.

Personally I've learned a lot from reading economic history. Some books I can recommend are: Eichengreen - Globalizing Capital: A History of the International Monetary System

Frieden - Global Capitalism: Its Fall and Rise in the Twentieth Century

Soros - Alchemy of Finance (skip the first couple of chapters on his reflexivity theory, but the 'diary' chapters are a gold mine)

Ferguson - The Ascent of Money (probably a good book to start, the others might be harder to completely grasp for a beginner)

Some other good book I can recommend (not history books) are: The Secrets of Economic Indicators: Hidden Clues to Future Economic Trends and Investment Opportunities by Baumohl

Inside the House of Money by Drobny (mostly about global macro trading, but you would learn a lot about economics along the way, would only recommend this after having built up some basic knowledge though)

Most importantly you should start reading the Economist, which has really high quality economics and finance articles, but is released only weekly and offers a good trade off with regards to time commitment and knowledge about what's going on in the world. Note that you will probably need to know some basics in order to understand everything, but you can look up things along the way. Economics really isn't hard to learn if you have a technical background.

[1] https://www.youtube.com/user/khanacademy/playlists?view=50&s...

I'd recommend learning about markets and economics. My suspicion is that states will continue to implement all steps necessary to contain the virus over the next 18 months and bail out all businesses and consumers in order to keep the productivity of the economy close to the level prior to the outbreak. Depending on how difficult it will be to contain the virus, this might require funds on an unprecedented scale in modern times. It's difficult to predict how eager investors will be to finance these deficits. Obviously something like a hyperinflation scenario is completely out of the question for a developed state like the US and there will likely be more deflationary pressure over the next few years, but at some point states will have to deleverage and I'm inclined to say that they will probably do this using higher inflation rates, in which case you can probably profit a lot from knowing about markets and economics.

If the availability of testing kits is low and the ratio of infected to non infected people is extremely low, it makes absolutely no sense to sample randomly from the population.

They were economists... Which might explain the outcome.

Computer Systems - a Programmer's Perspective

"The widespread misimpression that data + neural networks is a universal formula has real consequences: [...] in what students choose to study, and in what universities choose to teach."

I think this is a problem that deserves more attention. A large portion of CS students nowadays choose to focus on Machine Learning, many brilliant CS students decide to get a PhD in ML in hope of contributing to a field that will continue to develop at a pace similar to the progress we have seen in the last 7 years. This view, I think, is not justified at all. The challenges that AI faces such as incorporating a world model into the techniques that work well will be significantly harder than what most researchers openly admit, probably to the detriment of other CS research areas.

Thinking through potential race conditions is certainly the right thing to do, but we still make mistakes and concurrency bugs are hard to find and sometimes even to notice. I'd still rather have the borrow checker keep an eye on my code and if its unnecessarily conservative in a situation that you know is thread-safe you can still use unsafe code.

The programming language itself is irrelevant. After you have learned the concepts sicp introduces you can learn any language much faster.

> At the far end are propositions where you have to cook up a really abstruse counterexample to show they aren't true. But that's tantamount to saying that they almost are true. And in fact a lot of the time you can treat them as true and not get into trouble.

It seems as if you totally didn't get the point of proof based mathematics. Falsity is not a continuum, either something is false or its true. If you can produce a counterexample the proposition is false.

The whole article just seems like an attempt to steal money from uninformed people. He starts by giving vague information about trading strategies in general, then linking to an article about Renaissance Technologies as an example for successful algorithmic trading, then stating that most trading bots aren't successful and that the crucial differentiator for deciding whose bots to trust is the person's professional experience, which is obviously a reasonable thing to do, however the picture at the beginning of the article of him at a trading desk and him repeatedly mentioning his 7 year experience as a trader combined with the complete lack of any actual proof that his bots are actually profitable, make it seem as if he just tries to profit off his previous experience.

He ends the article writing this:

"All you’ll need to get started is: 1. $1000 2. To press a single button to get the bots started"

Furthermore in the reddit comments in response to the following question: "130 papers re-implemented in 7 months? I'm blown away. Write a software engineering book about how you did it so quickly. Then write a self-help book about having enough motivation to see it through." he writes:

"100-hour weeks and a desire for a better life for the ones you love will get you there pretty quickly"

This guy seems like a complete fraud, I find it sort of sad that this has landed on the frontpage of HN with that money upvotes.

I'm not an expert and I would love for someone to tell me I'm wrong. Having said that I think what I wrote in the previous post is wrong, Amazon going to $0 would clearly entail some wealth destruction, namely at least the amount that was raised in the IPO and any further issuing of stock (options not included since they only dilute). But besides that any temporary increase in the stock's value, is not equivalent to each stockholder getting an equivalent amount of cash (which was what the OP claimed and what I assumed he meant with wealth creation/destruction) just as a decrease in the stock price is not the same as an overall decrease in money in the population. Some people sell at a high or at any price above the price of the IPO/ price at which money was raised, which captures some of the temporary increase, any loss that includes the money that was actually raised by Amazon would be actual wealth destruction (cash losses), but I don't see how the rest of those losses would also correspond to actual cash losses.

> When I write code, the computer forces me to be 100% precise and will spit errors at me as soon as I do something wrong.

This is similar to math though, with the exception that you don't have something or someone telling you that you made a mistake, but you have to seriously question every step in your proof by applying the rules of logic and your knowledge. At a certain point you develop sufficient intuition to spot steps that might be wrong. Most mathematicians ignore steps, which they are not completely confident of being true for the time being, assume the step is true and continue to see whether their derivation leads to what was to be proved, only later checking the steps which they had doubts about. It's more similar to programming than you think. If you like the logical part and algorithmic thinking involved in programming, I'm sure you would also enjoy math if you'd give it a real chance.

I don't think Amazon's stock going to $0 would necessarily entail wealth destruction. The money doesn't just disappear, people who have sold at the high have profited. Wealth destruction only happens when credit is involved.

You can e.g. do this using semantic parsing, I remember some older papers but cannot find them right now. For a more recent paper see [1] (haven't read it though). You can also use Deep Learning for this. There are some interesting approaches using Reinforcement Learning [2]

[1] https://www.aclweb.org/anthology/P18-1034 [2] https://arxiv.org/pdf/1709.00103.pdf

I use Anki, but I have the problem that I tend to associate the specific form of the question, a particular latex formula etc. with the answer. As soon as I look at the question or read the first couple of words, the answer already pops up, so I'm afraid that I don't actually learn to remember the concepts but rather question/answer pairs. Does anybody have some recommendations for dealing with this problem?

I completely agree that wanting to make a positive impact on the world is important, although, and I don't want to sound too nihilistic here, the actual magnitude of that will probably be small for most people, therefore I think that personal satisfaction in life should be important and isn't meaningless, which was what OP claimed and the reason why I asked that question.

Intentionally being provocative here, but by that logic, why does your effect on other people matter? You are unlikely to leave a lasting legacy, and the generation you do affect, will die as well.

I didn't say anything that would imply that. However, if gut microbes, who act in their self interest, do cause cravings, then cravings cannot be used to determine what your body needs.

It's strongly hypothesised [1] and currently a very active research question that gut microbes are able to send signals to your brain to influence your "cravings". If that should be the case, listening to your cravings wouldn't be what your body naturally requires or what's healthy for your.

[1] https://www.smithsonianmag.com/science-nature/gut-bacteria-m...

How do you know your body naturally requires more salt?

I'm not an expert on experimental psychology, but I'm fairly certain that the principles of how experiments are conducted in the sense that they control for other factors, choose sufficiently large samples etc., are pretty sound. An often mentioned problem in psychology research is "p-hacking", where researchers manipulate the data in a way that validates their hypotheses statistically, which seems like a more logical reason for a reproducibility crisis than the reasons you mentioned.

Also just because there are some problems in the field should not cause you to dismiss science completely in favor of completely subjective speculating about how the mind works, which certainly also has value, but I don't think it's warranted to dismiss actual science in favor of it.