krnsll

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On the other hand, I have a collection of unpublished results in less active fields that I’ve tested every frontier model on (publicly accessible and otherwise) and each time the models have failed to solve them.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.

So "unexpected"? Not quite.

Not sure I agree.

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.

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.

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.

If you're willing to pay, Interintellect might be it.

https://interintellect.com/

Previous discussion from 2018:

https://news.ycombinator.com/item?id=17873284

Ah yes, learning (imitating?) how not to imitate from a listicle that itself cites advice from a compendium of books and sources.

Sorry, couldn't help myself.

I don't think software developers/professionals have acquired the cultural capital to be viewed as elites quite yet. I think much of the antagonistic Silicon Valley coverage in publications like NYT, which one could say are more traditionally elite, is driven by a difficulty with reckoning with their material decline and the emergence of these new candidate elites.

Great recommendation. When I first listened to this podcast, I was impressed by how the conversation managed to hit the sweet spot of engagement for both non technical (a friend I shared it with) and technical/mathematical (myself) audiences. Later learned that Melvyn Bragg hosted two of guests here (Gowers and Roney-Dougal) for another good conversation on the notions of randomness and pseudo-randomness.

https://www.bbc.co.uk/programmes/b00x9xjb

Thanks for pointing that out. My apologies for missing Tomonaga. Was going on my anecdotal memory, should have researched.

Freeman Dyson is a glaring miss here.

Feynman and Schwinger shared the 1965 Nobel in Physics for their respective perspectives on Quantum Electrodynamics. But it was Dyson who unified Feynman's diagram approach with Schwinger's field model.

Seconded. Read Lasch (Revolt of the Elites) to make sense of what dyed-in-wool Marxists think of the Woke "ideology" (if we can even be charitable enough to use that word, for most of what they say lacks coherence). In fact, even the widely recognized coiner of the term "intersectionality," Dr. Kimberle Crenshaw, wouldn't explicitly subscribe to many of their positions.

1) These two single volume histories were good starts for me: a) John Keay's "China: A History" and b) Fairbank/Goldman's "China: A New History".

2) HUP's History of Imperial China six volume series edited by UBC Sinologist Timothy Brook is a deeeep dive.

3) The People's Trilogy by Dikotter is heralded for its presentation of Mao's China. Julia Lovell also has a recent work on Maoism that's been lauded.

4) Finally, something more strategic/contemporaneous and forward looking is Martin Jacques' "When China Rules the World".

The point about your mom (and more broadly older faculty) is important. I hope it works out for her.

Vulnerable faculty will likely be forced to hold "in-person" lectures (ergo worsening their risk profiles) for the universities to create quasi-in-person courses: in that, videos will still be recorded and students probably won't show up but, on paper, it wouldn't be classified as an online course. This would likely be coupled with batch-scheduled exams being the only occasion the students need to show up in person for (or perhaps, those too are obviated via take-home exams?). This is essentially identical to some large enrollment, advanced CS theory courses I've taken.

John Cochrane, over at Hoover wrote about university finances in light of Covid19 a few months back. Worth a read.

https://johnhcochrane.blogspot.com/2020/04/university-financ...

I've always found the mutual knowledge vs common knowledge dichotomy to be a good way to conceptualize this distinction between kayfabe and propaganda (although it is not strictly speaking the perfect analogy here, it does offer some insight). The famous blue-eyed islanders puzzle is a great illustration of the concept (and perhaps something to occupy one's time with in quarantine).

Would be interested to see what Judea Pearl says about this. Not on twitter any more but back when I was, I recall him regularly tweeting about causality and AI's shortcomings in that regard.

Thanks for sharing this. The piece makes an urgent point that needs to be appropriately acknowledged. It falls in the vein of arguments I've come to term as "relational poverty" or "relational inequality" arguments (what Popper is presented as terming "moral inequality" in the piece) that most social science research I've come across has failed to or is ill-equipped to account for. I do recall Amartya Sen making a definition of poverty along these grounds (in the sense of dignity, or lack thereof, in what one perceives as their social environment) some years back but haven't been able to trace it down since or see if any further work was done on it. (If someone reading this happens to know more about such work, I'd appreciate being directed towards it).

The point made about Trump expressing a liking for poorly educated folk is also noteworthy. Even if the more sensible in the political class come to acknowledge the paradox we face, the matter of articulating and expressing it appropriately arises. Currently, as the author notes, the "college for all" approach doesn't appear to be resonating electorally (that said, there are a number of confounding factors one could put forth) but rather is serving as a reminder to the once robust (perhaps this is romantic nostalgia) working- and lower-middle-classes of their lowered social standing and relational dignity.

Update: I have found a presentation of the relativistic perspective on poverty by Sen (1983). Apologies for the earlier bit, that was laziness on my part. Worth a read.

https://are.berkeley.edu/courses/ARE251/fall2008/Papers/sen8...

Thanks for sharing this. I'm inclined to agree with your observations/reflections. Moreover, I suspect this might be common to the post-colonial experience [fn0] , if we account for: (a) what life was like under colonial rule; (b) how stratified such societies were before/after independence; (c) the forms of government (statist or plutocratic/military regimes) that replaced the colonial structures. These points and your overall observations ring particularly true when I consider the case of people (usually older generations) I've met from India (and South Asia generally) with the risk averse, at times counterproductive frugal, mindset that the Nehruvian state encouraged. [fn1]

[fn0]: Here, I may be incorrectly inferring your origins to be in a former colony.

[fn1]: For Indian readers, I hope this comment doesn't initiate a partisan discussion. :-)

An afterthought: I suspect this is done subconsciously too quite often by people in such fields as part of their evidence seeking (hypothesis confirmation) procedures which are admittedly more adhoc than those sort of naturally built into many STEM fields (I'm sure someone could provide numerous counterexamples to this latter point though, especially with the replication crisis).