I was in a similar situation as the parent post and skipped/was moved up by two years of high school.
I think it was very beneficial to have to work hard to catch up with more advanced classes. I feel flexibility around this is something parents and schools should take seriously.
(Tbf I was also super lucky to find a very accepting group of nerdy friends in the new year that would tolerate someone younger.)
There’s also a vague argument around hedging some actual risks that some market participants genuinely want to hedge… which depends a lot on the specific bet. Eg hedging exposure to specific political events, wars or even company announcements can be relevant and worth a premium for non-insiders. Where there’s a premium to be collected there are speculators to do so.
> “Hedge funds invest a ton in "alternative data", like credit card transaction data or satellite-imagery (are Walmart's parking lots full?) and need to process as much relevant information as possible to make predictions that are relevant to investments. “
Ah yes the famous credit card data and Walmart parking lots example that hedge funds were giving a few years ago in every interview and news article. Safe to assume that specifically these data sets are not what you should look at to make money.
"High variance, slightly-negative EV shots on a long time horizon" is a gambling addict's way of justifying the old adage, "sure, we're losing money on each sale, but we'll make up for it in volume!"
It's possible but unlikely given the short timeline, diverse questions that require multiple matheamticians, and low stakes. Also they've already run preliminary tests.
> It's possible but unlikely given the short timeline
Yep. "possible but unlikely" was my take too. As another person commented, this isn't really a benchmark, and as long as that's clear, it seems fair. My only fear is that some submissions may be AI-assisted rather than fully AI-generated, with crucial insights coming from experienced mathematicians. That's still a real achievement even if it's human + AI collaboration. But I fear that the nuance would be lost on news media and they'll publish news about the dawn of fully autonomous math reasoning.
Interesting topic and seems like a reasonable thing to study.
It would be cool to see a study where participants behave in a way that leads to more light exposure in month A and the opposite in month B, randomising the time ordering etc.
Despite power analysis and all the “50 people convenience sample, mostly observing based on what people do anyway” seems a bit like it won’t really lead to any action guiding outcomes beyond vaguely confirming people’s priors? As in, perhaps this style of research is not ambitious enough in a way?
I like quaternions as much as the next guy (I’ve used them in numerical computations etc), but what is it about them that makes them show up on the front page every few weeks?
Lots of engineers are lapsed physicists who have hobbyist interest in this stuff.
Also lots of HN readers are actual physicists or mathematicians. It's not all techies.
Also, lots of engineers have at some point learned some computer graphics and so been exposed to quaternions in that setting. Since they're mysterious and hard to wrap your head around most people don't really 'get it', leaving a sort of standing curiosity that articles like this tap into.
Baez wrote some ideas in [1], one I'm liking connects Lorentz group in dimensions 3,4,6 and 10 with the modular group SL(2,Z) that is at a crossroads of several hardcore math themes. For Lie algebras:
sl(2, R) ≅ so(2,1)
sl(2, C) ≅ so(3,1)
sl(2, H) ≅ so(5,1)
sl(2, O) ≅ so(9,1)
Dirac equation is the C case, the other cases have their uses.
I mean, there are practical reasons too (which are mostly just isomorphic to the stuff in the paper). But really that's why. It's part of our cultural history in ways that more esoteric math isn't.
Because people like me use quaternions but have never attained a full understanding like 3x3 rotation matricies. I will be reading the above link since its only 12 pages and someone indicated its an easier read.
That quaternions also solve for what we normally have 3D+time for.
And Lewis Carroll (Oxford (Math)), preferred Euclidean geometry over quaternions, for "Alice's Adventures in Wonderland" (1865).
Quaternions:
q = a + bi + cj + dk
-1 = i^2 = j^2 = k^2
Summarized by a model:
> In a quaternion, if you lose the scalar (a) — the "real" or "time" component — you are left with only the three imaginary components (i, j, k) rotating endlessly in a circle.
(An exercise for learning about Lorentzian mechanics, then undefined: Rotate a cube about a point other than its origin. Then, rotate the camera about the origin.)
4D Quaternions (a + bi + cj + dk) are more efficient for computers than 3D+t Euclideans. Quaternions do not have the Gimbal Lock problem that Euclidean vectors have. Quaternions interpolate more smoothly and efficiently, which is valuable for interpolating between keyframes in a physical simulation.
Why are rotations and a scalar a better fit?
Quaternions were published by William Rowan Hamilton (Trinity,) in 1843, in application to classical mechanics and Lagrangian mechanics.
Maxwell's (1861,1862) original ~20 equations are also quaternionic; things are related with complex rotations in EM field theory too. Oliver Heaviside then "simplified" those quaternionic expressions into accessible vectors.
Is there Gimbal Lock in the Heaviside-Hertz vector field reinterpretation of Maxwell's quaternionic EM field theory? Maxwell's has U(1) gauge symmetry.
And then quantum has complex vectors and some unitarity, too
Shouldn't there be symmetry and unitarity given energy conservation? And quaternions express this with rotations in SO(3), but is there a better model than quaternions for EM field theory since 1861?
-1 = i^2 = j^2 = k^2
q = a + bi + cj + dk
q = a + xi + yj + zk
> Mathematically, QED is an abelian gauge theory with the symmetry group U(1), defined on Minkowski space (flat spacetime). The gauge field, which mediates the interaction between the charged spin-1/2 fields, is the electromagnetic field. The QED Lagrangian for a spin-1/2 field interacting with the electromagnetic field in natural units gives rise to the action [...]
And QED is the basis for the Standard Model of particle physics and for some theories of n-body quantum gravity.
I know you know, just practical intuition for 3D graphics in case someone finds it useful:
There's a 1-1 mapping between complex numbers and 2D rotation matrices that only do rotation and scaling. The benefit is that the complex number only has two coefficients, not four like the matrix. Multiplying these complex numbers is the same as multiplying the equivalent matrices. Quaternions are the same idea just in 3 dimensions (so with 3 imaginary units i j k, not just i, one per plane).
> Quaternions are the same idea just in 3 dimensions (so with 3 imaginary units i j k, not just i, one per plane).
I justify quaternions to myself with the intuition from [1]. In essence quaternions represent rotations in 4D, where multiplying by a "unit" (i,j,k), rotates two distinct planes by 90 degrees. The reason introducing a single unit j doesn't work is the same reason this rotation-is-multiplication trick doesn't work in 1D (or really any odd-number of dimensions). Anyways if we call this 4th axis w and pick a simple rule like ij = k then we get some nice properties like
- multiplying by i rotates xy + zw planes by 90 degrees
- j rotates xz + yw
- k rotates xw + yz
- 1 rotates nothing
Notably this definition covers all 6 unique planes. But if we want to rotate only a single plane, we have to make up a new property, something that lets us rotate say xz by 90 and yw by -90. So we make up another rule that multiplying by a unit on the right does this, which algebraically looks like ij = -ji. This is incidentally why the rotation formulas have 1/2 everywhere, because if we want to rotate xy by 90, we multiply on the left by i/2 then on the right by -i/2.
"Quaternions" definitely sounds shinier and more mysterious than "geometric algebra". Indeed I can’t immediately come up with any math term more shiny and mysterious, except maybe "transcendental", but as a concept transcendentals are much more familiar to most than quaternions.
true on the naming, but i think geometric/clifford algebra has its own mysterious aura precisely because it can be framed as "suppressed" or "overlooked".. plus it genuinely does have elegant mathematical structure backing up the hype
funny thing is quaternions had that exact same energy in the computer graphics community for years. after ken shoemake introduced them to CG in 1985, there was a long period of "why are we using euler angles like cavemen when this exists??". now quaternions are well known tooling for people in graphics and the mystique has worn off at least in that community.
In the case of quaternions, there is called double-covering, which turns out (rather than being an artefact), play fundamental role in particle physics.
Yeah sure SU(2) up to sign is isomorphic to SO(3) and whatnot…
I think it’s probably mostly the computer graphics history and the cool name that gets people excited about quaternions?
Honestly, with all my love for the HN community, I think we have a couple of topics that just get upvoted without reading because they signal that you're in the ingroup. Few years back, another reliably upvoted thing was anything with "Bayesian" in the name. In the past couple of years, "busy beavers" would also get upvotes even though they have no practical use, their mathematical significance is dubious, and few people understand them in the first place.
Imo the fun quant stuff these days is about predicting returns and not pricing derivatives. As others have said here, the pricing part is mostly commoditised and more about managing software.
I think it was very beneficial to have to work hard to catch up with more advanced classes. I feel flexibility around this is something parents and schools should take seriously.
(Tbf I was also super lucky to find a very accepting group of nerdy friends in the new year that would tolerate someone younger.)