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!)
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.
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.
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.