Intro to Smooth Manifolds, Lee -- sweeping intro to geometry with minimal prereqs, great at balancing the nitty gritty details with conveying intuition
A Course In Arithmetic, Serre -- classically terse and elegant intro to algebraic and analytic number theory. Goes from quadratic forms to Dirichlet's theorem to modular forms in a mere 100 pgs!
Princeton Lectures in Analysis, Stein & Shakarchi -- 4 books covering much of classical/modern analysis, they really shine in their discussion of applications
The large scale structure of space-time, Hawking & Ellis -- The most mathematically satisfying treatment of general relativity I've found. High points include proof of singularity theorems!
Spin Geometry, Lawson & Michelson -- Deep dive into the enigmatic "spin groups" and their applications in geometry. Also the only good (book) reference I could find on the index theorem
Lee is a good book but Loring Tu's An Introduction to Manifolds is a masterpiece in clarity, conciseness, and notation. I'd heavily recommend it over Lee, which can be a bit meandering, for a first course, with a follow-up with Lee and others for material Tu leaves out. Tu's followups in differential geometry and algebraic topology also follow smoothly from it.
> I'm not sure how useful category theory actually is in the example cases.
It's hard to say that category theory is "applied" to this or that problem. you'll hear many mathematicians call it "abstract nonsense" half-jokingly.
More than anything it's a unified way of talking about mathematical structures that gives you a certain point of view (which is where it might be useful).
In Theorem 1.1, f is a function of random variables, which might be where you're confused.
> doesn't that mean y is a predictable function of x
Sort of: as function of real numbers, sha256 is just some deterministic function.
But point is its output "looks like" a uniform random variable for any reasonable input distribution i.e. as a function of random variables the input and output variables should have 0 correlation
I'd be interested to see if these models are robust against algorithms like TextFooler [0]. I'm skeptical this trend of 10x'ing the parameters will solve the "clever hans" problem.
This article is a great example of what I'd call "mindless problematization"
From what I've seen, this is you're trained to do in modern humanities departments. Take any seemingly obvious claim and "problematize" it.
It's telling that they do not say "nuance" it, rather it has to be made a _problem_. e.g. if you think "wilderness" is a thing you subscribe to a racist idea for "white male elites", if you think social media is affecting childhood development you're just caught up in the religious fervor of "scientized version of the biblical story of the Fall".
A whole lot of BS gets written this way, because these arguments have the superficial air of being subversive and contrarian.
In some cases it reminds me of my fundamentalist religious upbringing where everything was “of the world” and even trivial things like Pokémon, Christmas Trees, and Teletubbies were viewed as problematic and sinister
In Mary Roach's book "Packing for Mars" [0], in the section where they talk about the psychological impact of going into space, there is an interesting anecdote about the history of trains. As trains began to become more commonplace, there were concerns in England that the speed at which trains traveled would cause passengers to panic given that humans had never experienced both the speed and the motion parallax.
This of course, all turned out be for nought and was best summarized by a Russian cosmonaut in the book "This is problem only concern for psychologists".
"If you do not believe that mathematics is simple, it is only because you do not realize how complicated life is." -von Neumann