I spent 20 minutes trying to learn what this means as someone who hasn’t taken a ton of college math. Please feel free to correct me/treat this as a teaching opportunity. But I think this is just stuff that can be understood with just algebra, just the numbers are very very large.
It sounds like the big deal is you have at least 30 points in a graph that aren’t related and can be expressed as rational numbers (not necessarily integers so you get giant fractions). Being very humble here and invite people more advanced in mathematics to chime in where I’m wrong.
So like this is one of the x,y coordinate points that is two rational numbers.
So there’s at least 30 points on the graph like this, which is a big deal because before there were only 29 independent points verified in any elliptic curve equation.
“Independent” seems to mean something precise here too but I’ll let someone else explain that because I’m not quite sure, somehow the points have to be unrelated to one another.
Bumping the rank from 28->29 previously took over ten years, so it’s a big deal in mathematics apparently.
You got nothing wrong. To expand, you can add points on elliptic curves together. To add P and Q, you draw the line between them, find the third point on the line, and reflect it over the x-axis [1]. Then the idea is to find a generating set S of points so that every rational point on the elliptic curve is a sum of points from S. The size of S is called the rank of the elliptic curve.
As ducttapecrown said in their comment, you can define an addition on points on elliptic curves. (You can think of an elliptic curve as a cubic equation on the plane, so if you take a line that goes through two points, it will go through a third. There's more work to do to turn it into an addition, but that's the basic idea.)
There are different types of addition, though. A rank two addition would mean it looks like (x, y) + (x', y') = (x + x', y + y'). A rank three addition would mean it looks like (x, y, z) + (x', y', z') = (x + x', y + y', z + z'). Here they found the first example of an elliptic curve where the rank is 30.
The rank of a rational elliptic curve can be seen as a measure of how complex its arithmetic structure is. Roughly speaking, you can imagine that a curve of rank r has a substructure of dimension r. So a curve with a high rank is a pretty exotic object. This curve here has a 30-dimensional (or greater) lattice substructure. You can think of it as being possible to arrange the rational points on this curve into a predictable structure in a vector space of dimension at least 30. In that space, there would be at least thirty independent directions in space that could be combined together to produce distinct rational points on the curve.
To really quantify how exotic, it's conjectured that curves with rank 2 or greater have an asymptotic density of zero. That doesn't mean they don't or can't exist, but it does mean they become vanishingly rare, so finding even individual examples of high-rank curves has been absurdly hard.
The rank of an elliptic curve is expected to be at most one by a combination of a simple heuristic from Cohen and Lenstra and a deep BSD consequence (but I think the case needed is known). However we know of families with rank 15, and sometimes those ranks go up at particular points.
The record before this morning was 29 and people suspected that was as high as it got. When we learn how this curve was obtained that might change.
> The record before this morning was 29 and people suspected that was as high as it got.
Exactly this. A fundamental question in the subject is, whether elliptic curve ranks are bounded. Contrast with e.g. prime numbers, of which are known to be infinitely many. If you set a new record for the largest known prime, then that's cool but everyone knew there were plenty out there to discover.
made a significant impact in the field, coming up with a heuristic argument for why ranks of elliptic curves should be bounded. The same heuristic suggests, albeit more loosely, that we should perhaps be a little bit surprised to see a curve with rank at least 30. So it's mild evidence that the heuristic itself could be mistaken.
"When we learn how this curve was obtained that might change." IF they tell us. AFAIK Anthropic never released the reasoning chain for the Jacobian conjecture counterexample, and they might not release anything for this either.
BSD == Birch and Swinnerton-Dyer I think? I am also uninitiated. I enjoy learning about the millennium prize problems but BSD is one I don’t have a very good understanding of.
BSD is hard to appreciate because it connects a function that only now is known to have the right properties for the expression on the left hand side to a group we know next to nothing about, plus other mysterious factors each of which could be a few months to really understand. Proving the function exists? That went a bit beyond what was needed for Fermat's Last Theorem. Number theory demands breadth as well as depth. So no surprise it's hard to understand.
I wonder how much less useful it is if I use those models for open code or similar.
What are you really learning about me, other than the fact that I am a technical person, which you could know by the fact that I signed up for open router to start with.
Do you run your coding harness in a completely segregated sandbox? Even folder names of what projects you have, what projects you actually work on, etc are very useful for targeted advertising.
I think to some extent is that the right candidates and employers are not meeting.
Working at a ~1k employee company, we have been trying to hire both junior and senior devs. Out of ~10 interviews I conducted for junior, only 2 was able to program a linked list, and even that was with hints. It was not much better for senior(more system design question instead).
I understand that leetcode questions are (and have been) outdated, but linked list for somebody fresh out of school seems reasonable.
And this overlooks all the people using ai to cheat on the interview.
I would expect in this market we would have better candidates. But for us it was a challenge to hire.
And let’s not even get started with all the ai generated resumes and candidates that don’t actually exist, or have the experience they say they do. Really a waste of my time to interview.
The recruitment process is not made for a world where anybody can just copy paste the job req and generate a resume that fits everything.
A linked list is your litmus test? You did qualify with your "leetcode" blurb, but last I've used one was in college 30y ago and never again. We are not all kernel hackers / bare to the metal here.
I am just a hobbyist programmer, but asking to implement a linked list, IMO, is like asking to build a hammer - harvesting the wood for the handle, smelting iron, etc. We know what hammers look like and do, they haven't changed much in millennia.
I'd move the questions up the level of abstraction, as in - given this problem, how would you solve it? You have access to a hammer, a drill, a rotary tool. Solve it using any one tool or a combination and then explain why you chose that tool.
Depend what your company actual does. More relevant tasks I have seen involve 2D arrays or transforming data from a Python dict, or parsing JSON. The idea is to test the ability to understand and manipulate complex objects, which is a good generic bar.
linked lists aren't really emphasized anymore. languages have built in lists now. they had heard of them, knew what they were, but would probably struggle to do one from scratch.
of course i'm old school and kind of think that is basic, but colleges now aren't teaching it much anymore.
If someone can't implement the very simplest Graph Structure, Linked Lists, then I assume they can't implement any linked data structure. Learning variations of Linked Lists is where you start, e.g. https://rust-unofficial.github.io/too-many-lists/
I work in research and that’s how we’ve been doing interviews for years. Still difficult to find qualified candidates, but better than doing it all online and hoping for the best.
Could this be extended to lean into teaching quantum physics ?
Include seconds/ sub seconds hand in the watch, and people will realize the watch face time + time it takes to read will never equal the watch face time.
You can know the exact time, by looking at the analog watch face, or you can measure it (convert it) but it will not be the same anymore.
