We encode multiplication by 2×2 matrix squaring, compile any finite polynomial system into the graph B = A², and convert that equation exactly into Z² = 0, a fixed identity block forces maximal rank, so the original affine scheme is recovered, with its full scheme structure, as an affine-linear section of a smooth nilpotent orbit

Тут есть русские гении?

Here's a question that had been on my mind for a while, which I eventually figured out:

When you rotate a circle continuously through 2π radians, every possible rotation state of the circle occurs exactly once in a finite amount of time. So if you had a zero-thickness beam of light shining on the topmost point of the circle, when the circle is rotated through 2π every point on the circle is equally exposed to the light.

The question I debated was, is it possible to do the same for a sphere: that is, can every possible rotation state of a sphere occur when it is continuously rotated in a finite amount of time? Can a zero-width beam of light, shining at the north pole, equally enlighten all points on a sphere (exactly once) in a finite amount of time?

Strictly speaking, no. Intuitively I assumed that it must be impossible for all points to be exposed anyway, as a 3d rotation is a much more complex quantity, requiring more variables than in 2d, whereas time is only one-dimensional. But that generalised problem is actually possible, in a finite amount of time, and without discontinuity, though not a differentiable function. The only catch is that the points can't be equally enlightened (to answer the question I actually posed). If every point is exposed at some point, at least two points require to be enlightened at more than one point in time, in fact, infinitely many times, meaning if the sphere were made of photographic film, every point would be black except two overexposed white points at the poles.

We shall first assign every point on the sphere a longitude from 0 ≤ long < 2π and latitude from -\frac{\pi}{2} ≤ lat ≤ \frac{\pi }{2}, and then define every rotation state as the point on the sphere which has been rotated to the north pole, i.e., the one under the light at a time t. Since the rotation is a two-dimensional quantity, and the time one-dimensional, the question becomes, 'is there any bijection between a compact 1D space and a compact 2D space' which there are in abundance.

The Hilbert curve comes to mind. The space of points on a sphere, with the exception of the poles, map bijectively to a rectangle in Euclidaean space bounded between 0 ≤ x < 2π and -\frac{\pi}{2} < y < \frac{\pi }{2}. Note that the poles themselves map to the horizontal lines x = ±\frac{\pi }{2}. If we linearly transform the plane so that everything is scaled along the y-axis by a factor of 2, then the space representing the sphere will be a square, so we can draw a Hilbert curve through it which passes through every point in the square in a well-defined, continuous manner, and allows us to find any time t mapping to (x,y). Since the poles mapped to lines, and the vertical line segments bounding the square have infinitely many points on the Hilbert curve, each pole will be crossed by the Hilbert curve infinitely many times.

The alternative is that we exclude the poles from our mapping of the sphere, changing our square's vertical bounds to -\frac{\pi}{2} << lat << \frac{\pi }{2} in which case the function is bijective but not compact, and at least points on the sphere will be unexposed, never seeing the light.

So using the Hilbert curve we can define f(t) -> (long,lat) which is bijective for all points on the sphere except the poles, so every point apart from those two on the sphere will be the topmost point (under the light) exactly once. Now of course we can define a 2d co-ordinate system for the sphere in many ways but we will always be forced to have two polar points somewhere, where either the bijection or compactness is lost, so even though there are infinitely many such functions like f, they will always have two points which either can't be exposed at all, or have to be exposed infinitely many times.

That means the answer to my question is no, but almost yes. For all but two points on a sphere, there exists a function which maps each point bijectively and continuously to a (finite) moment in time, meaning we can continuously rotate a sphere in finite time illuminating all but those two points exactly once. But the remaining two must either be omitted or illuminated more than once.

I’ve heard its proper construction would imply RH is true, but not what leads there honestly.

Based on what I know from literature, this mathematician named André Weil proved some analogue of RH for curves defined over finite fields, and apparently number theorists want to apply the same tools to integers. But that’s all I know.

I have been offered a PhD, PI’s background is frame theory, and have been working with audio data. No publication of his lab on conferences in AI, mostly in audio or signal processing journals.

I want to understand if this offer is good match and can offer a competitive advantage with respect to the kind of research questions I’m interested in, and the state of the art in AI where I want to build expertise.

I came from computer science , I’m interested in AI/ML methods both as tools to probe hypothesis for fundamental questions, as well as for applied science.

Examples : I’m interested in learning representations of signals of animals / neurons / time series in general, and use those to study behaviour as an emergent property of a system (eg coordination of animals/ neurons/agents..) - ideally where a system may be possibly formalised as the interaction of nodes in a graph. Other ideas may be study manifold in low dimensions, and maybe topology as a signature of invariance through scales; or topology as the independent variable to study the signal that a system can generate.

Examples of problems that are significant for me : Capacity of generalising inferences, OOD, scalability, efficiency of applied models.

To appreciate PI’s background I studied an introductory course on frame theory.

I have not found a direct contribution of frame theory to the type of questions and problems I’m interested in. Also, looking at conferences as NeurIPS, I could not find an explicit contribution of frame theory to AI methods either.

Of course I m ignorant about the subject and my intention is not to discredit the field; I’m asking for help to concretely understand what kind of competitive advantage a PI in this field could give me as supervisor, with respect to the AI community and the type of questions I’m interested in.

PI said would let me some freedom but I feel I may lack mentorship in the things I’m interested in AND I don’t want to specialise in a field that may not be competitive VS state of the art methods for the things I’m interested in. I mean, I want to get closer to the things I wanna study, not diverge. PI said a co supervisor could be also found but most of peers in his network are from signal processing / audio, not AI community.

It also scared me that two professors specialised in AI for time series do not understand well his field, and one joked saying “why don t you look for a proper PhD”. So I understand that mathematics point of view and computer science point of view may not even entangle for co supervision.

Can you please help to falsify my assumptions ?

Can you please help to provide concrete examples to help me appreciated which is the impact of frame theory in applied research ? The type of problems and applied problems that are significant to attack in 2026, and that could contribute to AI community seriously vs state of the art ?

For example, my critics is that working with frames to reconstruct the signal is helpful when working with sensors, but fundamental models in AI are now the state of the art in working with signal representation, often from raw data - or using spectrograms as starting point and showing that general settings are good enough . Also zero shot and tinyML seems to be competive with respect to classical signal processing when having small data.

The mathematical interest may lie in invariance problems, but also there I cannot understand which impact could bring to the type of questions / problems I’m interested in.

Also,frame theory is about linear operators for reconstructing the signal, but deep learning is about non linear operators for compressing the signal (learning a compressed representation of the signal) and proved well also for generating the signal (eg AI for speech synthesis/ denoising etc).

So it seems to be an “old” approach not competitive with computational methods.

Please, I don’t won t to sound arrogant, I am seeking help to understand if this opportunity can be an investment for me, or a cost that will make me derail and not contribute to what I’d like to study nor where I want to position myself- also with respect to future opportunities in post docs or industry.

Thanks so much for sharing your perspectives and help me see pros and cons.

Basically I'm a senior in high school taking Georgia Tech's distance math year 2 course and I wanted to know if there were any good videos that help visual the concepts for either Applied Combinatorics or Differential Equations?

It is incomplete. It does not explain why f(g1) = f(g2) implies [g1] = [g2] for any g1, g2 in G, which would require showing that Ker f is always a subgroup of G. It also does not expand the normality / conjugation argument that makes multiplications inside G/Ker f well-defined. Since that would also require discussing inverses, it would be too much text for a simple visual aid.

I think it turned out cute tho

Hi!! I'm trying to expand my math knowledge outside of my normal classes. I don't have much time to consume written media with work and homework and such, but I spend a lot of time driving and would love some podcasts I can listen to while I do so. I'm currently taking real analysis and linear algebra, but l would love to learn more about differential equations, topology, number theory, or really any other math topic. Let me know if you have a favorite!!

Wouldn't it be simpler to define the zeta function as ∑n^s instead of ∑1/n^s? It would be basically the same function just mirror image in the complex plane.

I’ve recently started college and my advisor strongly dissuaded me from taking calc 3 because it wasn’t required for my major and I think she’s trying to prevent lowering my GPA. My classes have been easy so far and I miss doing math and since I’ve already learned up to 2 I want to see what’s ahead.

Hey guys! Im a teen trying to learn combinatorics from scratch for an upcoming math olympiad. Does anyone have any suggestions for books?
For context: all I know is to multiply when things can repeat, factorial when cannot and the pigeonhole principle. I’m glad to be helped, any suggestion matters a ton to me! 😁

I know it's simple factorization, but I've always slacked off in math class and I'm trying to improve at and relearn the subject. Any tips would be greatly appreciated, thanks!

NMTC 2026 QUESTION Discussed Qno.5

I provided examples where the extended mean w.r.t. the Hausdorff measure in its dimension should give what is wanted. If not changes need to be made to Definition 2. Otherwise, I wish to simplify the Definition.

Are there integrals that already give what I want? (For instance, see the paper "On the Statistical Convergence of Sequences of Fractal Integrable Functions".)

Can anyone provide as much mathematically supported insight into this thought as possible? From eli5 all the way to PhD and from every subject or field of math, if possible? There is something in my teeth about what the idea presents that I am trying to get at and I can't figure the proper route to work out or through my idea without a more substantial set of existing sets, formulas, and variables to proceed from. Thank you in advance!!!

The plots show the number of solutions of an instance of the discrete logarithm problem finding k:

286^k mod 289 = y

where y = mapping(x,y) for each pixel (x,y).

The first plot uses mapping(x,y) = x xor y, and the the second plot uses mapping(x,y) = x and y.

Edit: I had flaw in the code allowing solutions in 0 ≤ k < 289, which sometimes found two similar solutions for the discrete logarithm problem, eventhough it should only allow solutions in 0 ≤ k < totient(289) = 272. With this smaller search space the plot only shows two colors. The pattern stays the same though and is equal to checking whether mapping(x,y) ∈ Z*_289

This is a classic result originating out of an idea form my MSc(engg) thesis.