qmalzp

First project sounds like Coinbase.

Him, Tao, Emerton, Kisin, Coates, Calegari Bros... all Australian. Must be something in the water.

I'm not sure how many people care. It's like finding out Pepsi owns Aquafina.

N.b. the article uses d for the dimension, and n for the moment index. I haven't cranked out the details, but I believe that this is already true just assuming the fourth moments of the distance are integers. That is, for n != 1, 2, or 4, the fourth moment is never an integer. Idea of the "brute force" proof:

Take the formula in the article for the 4th moment of the d-dimensional sphere, which is always a rational number. Basically, for n=2^k, the denominator should be divisible by a larger power of two than the numerator (specifically, if I crunched the numbers right, the gap should be k-2). When n is not a power of two, then for any odd prime p dividing n, I believe the denominator should be divisible by a larger power of p than the numerator. This requires calculating exactly how many powers of p divide various factorial expressions, but you get the idea.

If you pass the interview as a generalist, Google absolutely prefers for people to work in their Mountain View office.

1. The cost of relocation is minuscule compared to the total cost of employing someone, so I don't buy that argument.

2. Do you have data to back up that Google engineers in Denver get paid less than their Mountain View counterparts? I believe this is true when comparing US to non-US salaries, but at least where I've worked, engineers get paid the same everywhere within the US (and indeed, you can relocate from the Bay Area to cheaper locales without taking a pay cut).

When talking about chess: "But even the most powerful program can be defeated by a skilled human player with access to a computer—even a computer less powerful than the opponent."

Is that true? Can a state-of-the-art chess engine plus a grandmaster really outperform just the state-of-the-art chess engine?

Google doesn't open a 500-engineer research lab in Grand Rapids, MI because there are not 500 Google-caliber engineers already living in Grand Rapids, MI. This is not a statement about the average talent there, but simply that the starting pool is not sufficiently large; hence the need to build offices in high density urban areas.

Google doesn't open an office in Denver so that Bay Area engineers have a slightly cheaper place to work, they do it to tap into the talent of folks who already live there.

Edit: I am basing these claims off of my own experience. I work at a satellite office of one of the big tech co's, and the vast majority of my coworkers are people who are either fresh out of a local university, or already lived here for many years before joining the company.

What if you replace it with

p(x|y) = 1, p(x|!y) = 0, and time(y) < time(x)

That rules out the rooster counter-example. If y is a boolean, I guess the only thing you can "do" to it is negate it.

As a former mathematican, I was at first a little offended and dismissive of his claim. But, perhaps what one can say is that mathematicians don't seem to distinguish "causation" with "implication". After all, if the barometer goes down, that does imply a storm is coming (perhaps with some increased probability), but it still doesn't cause the storm to come (even with increased probability).

In a simplified closed system, where all you have are barometers and storms, maybe there is no difference between implication and causation; all you know is these variables are correlated. Perhaps once you take every atom in the universe into account, the two start to look the same.

I think the first point is only true for symmetric matrices (which includes those that show up in multivariable calc). In general, the eigenvectors need not be orthogonal.

I completely disagree. One of the main reasons I prefer Instagram is precisely that it's not packed to the brim with features I don't use.

It does seem reasonable that there would be a cap, but I wonder how much cutting the pensions of a few thousand highly paid retirees will impact the budget.

Perhaps a more apt analogy is that mental arithmetic tricks are to math what stenography is to writing.

Category theory is basically a language. If one wanted to formulate an explicit conjecture corresponding to the original Langlands program, it could possibly be phrased as some kind of equivalence of a category of automorphic representations and a category of motives. Precisely defining those two categories is the hard part.

So even with this language, one still has to do the work of actually proving these things. It's like having a nice programming language; you still have to write the code to do the thing!

3. There are ways to make money online without ads begging you to click on them, and they involve real-world goods and services that your website can help connect people for?

Yeah, I could even imagine a scenario where you help connect people to real-world goods and services for free; the provider of those goods and services would be more than happy to help keep your bills paid, assuming there was some kind of mechanism that would drive people to their goods and services and not a competitor.

What would such a mechanism look like?

The average gap is log(n), but the maximal gap is not well understood.

According to https://en.wikipedia.org/wiki/Cram%C3%A9r%27s_conjecture, current results are

Unconditional: n^0.525

Conditional on RH: log(n)*sqrt(n)

Conjecturally: log(n)^2

Fun fact: compositions of the two operations

x -> x + 1

and

x -> (x==0) ? 1 : (x == 1) ? 0 : x

generate all possible operations.

Consider it something like a benchmark of human progress.

Nit: the largest prime discovered is

2^(277,232,917)-1

not

2^(277,232,917-1)

That top pay grade is less than the starting compensation for PhD's at the big tech co's. Not to mention free food, potential for career growth, and an open non-militaristic culture where you can wear a hoodie instead of a tie.

I honestly don't understand why someone would work for the NSA given the choice.

Probably because at the end of the day, he doesn't want to fight some legal battle, he wants to do mathematics.

I have never seen a theorem prover applied to even basic 100-year-old results in Algebraic Number Theory. I think you underestimate the difficulty in translating a mathematical idea into a format a computer program can understand.

You can try requesting as desktop from the mobile browser. I've gotten it to work before, though now when I try it, it crashes the browser?!

So brave.

Use messenger.com?

Tl;Dr

It's slightly more accurate to say if

F(s) = 1^s + 2^s + 3^s + ...

Then

F(1) = -1/12.

This is still a lie, but the truth is that F(s) does make sense for a large domain of s, and there is a unique and canonical way of extending F(s) to a larger domain which contains 1. And then that new function evaluated at 1 is equal to -1/12.

Snapchat would like a word with you. (No pun intended)

I'm happy with my path:

Youthful idealism -> number theory Mature pragmatism -> pivot to CS

It was entirely luck that my passion was so closely related to the current gold rush.

I concur.

If you're talking about Scholze's lectures about his cutting-edge work, there is not a chance they are accessible to undergraduates; when I attended one as a fourth-year PhD student specializing in algebraic number theory, I would say I only kind of got the gist of it.

Re: your skepticism... The guy is a once-in-a-generation talent; his constructions were able to vastly simplify multiple very long, very complicated proofs that groups of the top people in this field were working on. This is in a field (algebraic number theory) which is considered one of the more saturated and technically difficult within all of mathematics (admittedly, I am likely biased on this point). That being said, all of his work so far has been in the ballpark of Langlands/p-adic/arithmetic geometry, so I would be surprised if he achieved significant results that strayed too far from this stuff.

I'm not sure what you mean about Feynman; Peter's genius is not so much his ability to communicate complex ideas in a simple way, but rather he was able to come up with constructions (or if you like, abstractions) which compartmentalize the complex ideas in the right way so that they are easier to deal with. To make an analogy with computing, think of the concept of a "thread". Without the concept of a thread, you'd have to do so much manual maintenance that you could never dream of building say Google. Scholze's perfectoid spaces are analogous; their definition would have been understood by mathematicians 50 years ago, but no one really got that this was the right thing to consider.