ceh123

For now.

Context: I finished a PhD in pure math in 2025 and have transitioned to being a data scientist and I do ML/stats research on the side now.

For me, deep research tools have been essential for getting caught up with a quick lit review about research ideas I have now that I'm transitioning fields. They have also been quite helpful with some routine math that I'm not as familiar with but is relatively established (like standard random matrix theory results from ~5 years ago).

It does feel like the spectrum of utility is pretty aligned with what you might expect: routine programming > applied ML research > stats/applied math research > pure math research.

I will say ~1 year ago they were still useless for my math research area, but things have been changing quickly.

Exactly! It's n+1 points in n dimensions (when finite). Another way to think about it (the way that I know because it extends into general Banach spaces and not just n dimensional spaces) is that each point inside is the unique weighted average of the extreme points (corners). So in 2d, if you have a square you can get that middle point by averaging all the corners, or averaging two opposing corners, so it's not a simplex.

On the topic of simplices! I did my PhD in dynamical systems and the space of invariant measures [0] is (in the compact setting) always a simplex and the extreme points are the ergodic measures. It's because of this that you can kind of assume your system is ergodic do work there and frequently be able to generalize to the non-ergodic case (through ergodic decomposition).

But the real thing I wanted to mention here was the Poulsen Simplex [1]. This is the unique Choquet simplex [2] for which the extreme points are dense. This means that it's like an uncountably infinite dimensional triangle where no matter where you are inside the triangle, you're arbitrarily close to a corner. It's my favorite shape and absolutely wild and impossible to conceptualize (even though I worked with it daily for years!)

[0] https://en.wikipedia.org/wiki/Invariant_measure

[1] https://eudml.org/doc/74350

[2] https://en.wikipedia.org/wiki/Choquet_theory

This paper is a theoretical analysis showing that the ridge regularization that optimizes the source task almost never optimizes transfer performance. Interestingly, in high SNR regimes (low noise) the optimal regularization for pre-training is higher than the task specific optimal regularization, and in low SNR regimes (high noise) it’s better to regularize less than you would if you were just optimizing for that task.

Although the proofs are in the world of (L2-SP) ridge regression, experiments were run using an MLP on MNIST and CNN on CIFAR-10 and suggest the SNR-regularization relationship persists in non-linear networks.

I think my main point is just because an LLM can lie, doesn’t necessarily mean an LLM generated slide is fraud. It could very easily be correct and verified/certified by the accountant and not fraud. Just cuz the text was generated first by an LLM doesn’t mean fraud.

That being said, oh for sure this will lead to more incidental fraud (and deliberate fraud) and I’m sure it already has. Would be curious to see the prevalence of em-dash’s in 10k’s over the years.

US v Simon 1969, see [0] for a review.

Establishes that accountants who certify financials are liable if they are incorrect. In particular, if they have a reason to believe they might not be accurate and they certify anyway they are liable. And at this stage of development it’s pretty clear that you need to double check LLM generated numbers.

Obviously no clue if this would hold up with today’s court, but I also wasn’t making a legal statement before. I’m not a lawyer and I’m not trying to pretend to be one.

[0] https://scholarship.law.stjohns.edu/cgi/viewcontent.cgi?arti...

Presenting false data to investors is fraud, doesn't matter how it was generated. In fact, humans are quite good at "generating plausible looking data", doesn't mean human generated spreadsheets are fraud.

On the other hand, presenting truthful data to investors is distinctly not fraud, and this again does not depend on the generation method.

Even one of these topics I would say it would take most PhDs at least 2-3 years to “master”. I feel like at the end of my math PhD (5 years, 3 focused solely on my research area) I had just scratched the surface of mastery in my sub field, and that’s with 3 published papers.

I guess you’re right though, defining “mastery” is the key missing point here.

Right but for self improving AI, training new models does have a real world bottleneck: energy and hardware. (Even if the data bottleneck is solved too)

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I'm not sure if this really says the truth is more complex? It is still doing next-token prediction, but it's prediction method is sufficiently complicated in terms of conditional probabilities that it recognizes that if you need to rhyme, you need to get to some future state, which then impacts the probabilities of the intermediate states.

At least in my view it's still inherently a next-token predictor, just with really good conditional probability understandings.

To add some extra numbers here just to showcase how little energy usage this is.

This means it's adding about 0.012% additional energy consumption to those users energy consumption.

From another angle: Average US house energy consumption is around 30kWh per day. 0.012% of that is 3.75 watt hours of energy per day. This is the equivalent amount of energy as streaming HD video to your iPhone on a 4G network for 1.5 seconds. [0]

So in other words, a 15s youtube ad you are forced to watch on your phone before watching the video you were going to watch anyway takes an order of magnitude more energy than the average AI user according to this article.

[0] https://www.statista.com/statistics/1109623/electricity-cons...

We don't need first principals thinking every time, but having an understanding of why you can't just test 100 variations of your hypothesis and accept p=0.05 as "statistically significant" is important.

Additionally it's quite useful to have the background to understand the differences between Pearson correlation and Spearman rank, or why you might want to use Welch's t-test vs students, etc.

Not that you should know all of these things off the top of your head necessarily, but you should have the foundation to be able to quickly learn them, and you should know what assumptions the tests you're using actually make.

Important correction, it’s not DD/MM/YYYY, but DD Month YYYY. (At least what I saw in the article)

This format is common in heavily regulated industries and frequently a regulatory requirement since it’s fully unambiguous. I (American) worked in clinical research/pharma for a bit and still write my dates like 23Jan2024.

As someone that’s in the later stages of a PhD in math, given the title starts with “Mathematical Introduction…”, the notation feels pretty reasonable for someone with a background in math.

Sure I might want some slight changes to the notation I found skimming through on my phone, but everything they define and the notation they choose feels pretty familiar and I understand why they did what they did.

Mirroring what someone else said, this is exactly the kind of intro I’ve been looking for for deep learning.

Would love a link to this if anyone knows the paper?

Also a subscriber to Kagi.

In my experience it's at least marginally better, but one of the really nice features that Kagi has (and probably the main reason I subscribe) is you can extremely easily block domains. So whenever I hit a SEO garbage site, I just go back, block it, and I never worry about it again. In the areas you regularly search, this quickly gets you to a result page that is substantially higher quality than google.

They were caught not too long ago actually spreading environmental causes in Texas against rare earth mining. They were trying to leverage our political process to make us dependent on them.

Do you have a source for this? I did some quick searching but couldn't find anything concrete. Super curious to learn more about this case if it's true.

It's the first symbol referenced for symbols used in proof by contradiction to show contradiction [0]. I know that's not exactly "not" or "disproof" but I think that might be what the poster was getting at.

[0] https://en.wikipedia.org/wiki/Contradiction#Symbolic_represe...

Big ups for Bernstein, Against The Gods was a fantastic read and I can’t wait to read Power of Gold.

Another book I’d like to throw into the mix here: Central Banking 101 [0]

Fantastic overview of how the current central banking system works, written post-Covid so it has some context for the extensive QE that’s been happening over the last 2 years. Author worked on the Fed’s trading desk for over a decade in various capacities so he has a deep experiential knowledge of the plumbing in action.

[0] https://www.amazon.com/Central-Banking-101-Joseph-Wang/dp/09...

“Found in Britain in the last 100 years” is quite the qualifier to drop completely in this context.

Okay yup, turns out this is correct and it's slightly upsetting to me honestly haha. Just for someone else if they're interested where contrapositive and contradiction both use law of excluded middle (the ==* step requires it):

Contrapositive:

A-> B == A or ~B == ~B or A ==* ~B or ~(~A) == ~B -> ~A

Contradiction:

~(~A and B) == ~~A or ~B ==* A or ~B == A -> B

You could state (2) as a contrapositive where your re-phrasing of (2) is:

If A is the set of all primes, then A is an infinite set.

Contrapositive proof: Suppose A is a finite set containing only primes. By (1) we know there exists some p not in A and therefore A is not the set of all primes.

I don't believe contrapositive needs law of excluded middle but I'm honestly not sure. Logic is not my area

Yup good point. Thanks for the clarification

Under this perfect assumption you end up having a problem with the red shift caused by the expansion of the universe (which I didn't account for above) that lengthens the wavelength of the light you want to see, requiring an even larger lens.

On the lens side of things, as far as I'm aware (not a physicist, math PhD student who is just generally into this sort of thing) there isn't anything fundamental preventing you from collecting this light and building the lens, but from our current understanding of materials science I'm fairly confident it's currently impossible to construct a structure that will stay together that large.

That being said there may be ways around this problem, like I said not a physicist or engineer, but you are right that from an information theoretical perspective if you ignore dust and other things in the way then yes all the information is still there.

Check out my reply to a sibling comment. TLDR: huge

To add some back of the napkin calculations to this:

Angular resolution of the earth at 66 million light years away would be approximately 2e-17 radians. Using the Raleigh Criterion [0] for lens size for the visible light spectrum (700nm for the best case scenario), you would need a lens with a diameter of about 4e10 meters. That's about the radius of Mercury's orbit around the sun.

If you want to see dinosaurs, say 1m resolution, that's about 1.5e-24 radians at 66M light years, needing a lens of diameter 5E17. The entire solar system has a diameter of ~3e14. So even if your lens was the size of the solar system you'd still be off by a factor of 1000 trying to resolve the dinosaurs.

At these scales you start running into some pretty fundamental engineering and physics problems with building a telescope this big.

Warning: this math may not be totally right, I'm just procrastinating some PDE homework right now, but the scales should be roughly correct.

[0] https://en.wikipedia.org/wiki/Angular_resolution#The_Rayleig...

Mandated vaccines in healthcare settings aren’t a new demand though. It’s more like a certificate that takes an hour is added to the list of 8 other certificates you need to maintain.

Right but if you have no diagnosis of Covid then the argument above doesn’t apply since you don’t actually have any evidence of “natural immunity”