If you think of this as a search, retrieval and “application” problem on the space of convex optimization proof techniques, it’s not a particularly striking result to a mathematician. Partly because: the space of results/techniques and crucially applications of those results and proof techniques is very rich (it’s an active field with many follow up papers).
On the other hand, I have a collection of unpublished results in less active fields that I’ve tested every frontier model on (publicly accessible and otherwise) and each time the models have failed to solve them. Some of these are simply reformulations of results in the literature that the models are unable to find/connect which is what leads me to formulate this as a search problem with the space not being densely populated enough in this case (in terms of activity in these subfields).
Agreed. Something else that might be driving this is that existing models essentially get the job done for most users, who —- unlike HN commenters (I promise this is human generated, em dash notwithstanding ;P) —- don't quite about the state of the art.
Sure Klartag isnt a sphere packing specialist by training, but he's one of the best problem solvers around. He just resolved the Hyperplane Conjecture earlier this year and has contributed to progress on related problems in convexity theory such as: KLS Conjecture, Mahler Conjecture, Central Limit Theorem for Convex Bodies, to name a few. His student Eldan's work on Stochastic Localization has also proven critical in log-concave sampling algorithms (related to the KLS conjecture, and he gave a talk at the ICM).
Also, the toolkit one uses in convex geometry, especially some of the harmonic analysis tools are quite handy in the study of sphere packing.
It does a decent job of conveying the essential idea for a broader readership: perturb a graph through its adjacency matrix just enough to make the universality conjecture hold for the distribution of eigenvalues -> analytically establish that the perturbation was so small that the result would carry back to the original adjacency matrix (I imagine this is an analytical estimate bounding the distance between distributions in terms of the perturbation) -> use the determined distribution to study the probability of the second eigenvalue being concentrated around the Alon-Bopanna number.
I haven't had a chance to read the paper and don't work in graph theory but close enough to have enjoyed the article.
> I wonder how helpful it would be for music and art education if AI could help us deconstruct some songs.
Was thinking along the same lines as I fell into the trap of Ghiblifying pictures earlier this week. As someone who spent countless hours in my childhood trying to copy the styles of my favorite comic books (Japanese and otherwise), at some point in this AI exercise I started rendering each picture in the artistic styles of each of my favorite artists and placing them side by side for comparison. Realized an exercise like this would have been very useful back when I drew in comparing different styles and what _exactly_ made them different. Maybe it speaks more to how I think —- lacking a true artistic intuition —- but simply comparing styles and giving words to their distinctions helped me appreciate them in a way I hadn't (of course the AI didn't produce a perfect representation but a crude enough approximation)
As a mathematician, I can't help but simmer each time I find the profession's insistence on grasping the how's and why's of matters to be dismissed as pedantry. Actionable results are important but absent understanding, we will never have any grasp on downstream impact of such progress.
I fear AI is just going to lower our general epistemic standards as a society, and we forget essential truth verifying techniques in the technical (and other) realms all together. Needless to say the impact this has on our society's ethical and effectively legal foundations, because ultimately without clarity on how's and why's it will be near impossible to justly assign damages.
You're on the right track. My comment alludes to sections, but there's a lot of essentially analogous math that explains the phenomena via shadows/"projections."
Something I worked on in my PhD was analyzing high dimensional bodies via their "sections."
Here's the Busemann Petty Problem:
Given two origin symmetric convex bodies K and L in n dimensions. Suppose for every linear hyperplane A (passing through the origin) we have vol_{n-1}(K intersect A) \leq vol_{n-1}(L intersect A).
Is it true that vol_{n}(K) < vol_{n}(L)?
[Here vol_k should be thought of as length when k = 1, area when k = 2, and volume in the traditional sense in k = 3.... generalizes quite well to arbitrary dimensions. And sections are these quantities L (resp. K) intersect A]
Turns out the answer is NO! In n \geq 10, it can be explained with the simple examples of K and L being the unit volume (vol_n) cube and a euclidean ball of volume (vol_n) slightly less 1 respectively. Comes from Keith Ball who, in his PhD thesis, established that {n-1}-section volume of the unit volume cube lies in [1, \sqrt(2)]. However for the euclidean ball of unit volume the section volume is at least sqrt(2). So you can start with the unit volume ball, decrease its radius infinitesimally so (the n-1 section volume falls less than the n-volume does) and generate a clear counterexample.
What this looks like is a ball with volume less than a cube but section volume seemingly leaks out of the faces of the cube. So a "spikey ball," if you may.
Not at all, the proof was a very elegant argument involving fourier transforms and an integral estimate going back to the study of controlling random walks (Khintchine's inequality). I say elegant in the manner of it being enviably so -- a proof a beginning graduate student could follow while capturing a fundamental, easy to state fact.
This work does however situate itself in/adjacent to that broad space of Brunn-Minkowski theory.
On the other hand, I have a collection of unpublished results in less active fields that I’ve tested every frontier model on (publicly accessible and otherwise) and each time the models have failed to solve them. Some of these are simply reformulations of results in the literature that the models are unable to find/connect which is what leads me to formulate this as a search problem with the space not being densely populated enough in this case (in terms of activity in these subfields).