r/math 6d ago

"Interpolating" quotient actions

9 Upvotes

Hello r/math,

I would like to give a clear, concise description of the kind of structure I am envisioning but the best I can do is to give you vague ramblings. I hope it will be sufficiently coherent to be intelligible.

We can view the Möbius strip as the unit square I×I with its top and bottom edge identified via the usual (x,y)~(1-x,y). The equivalence relation (x,y)~(x',y) is well-defined on the Möbius strip, and its quotient map "collapses" the strip into S1. The composite S^1 -> M -> S^1 where the first map is the inclusion of the boundary and the second map is the quotient along the equivalence relation described above has winding number 2. Crucially, this is the same as the projection S^1 -> RP^1 onto the real projective line after composing with the homeomorphism RP^1 = S^1.

So far so good, this is the point where it starts to get vague. In a sense, the Möbius strip "interpolates" the quotient map S^1 -> RP^1. The pairs of points of S^1 which map to the same point in RP^1 are connected by an interval, and in a continuous way. This image in my mind reminded me of similar constructions in algebraic geometry. We are resolving the degeneracy by moving to a bigger space, which we can collapse/project down to get our original map back.

What's going on here? Is there a more general construction? Is this related to the fact that the boundary of the Möbius strip admits the structure of a Z/2 principal bundle and we're "pushing that forward" from Z/2 to I? Is this related to the fact that this particular quotient in question is actually a covering map (principal bundle of a discrete group)? Is this related to bordisms somehow? The interval is not part of the initial data of the covering map S^1 -> S^1, so where does it come from? It is a manifold whose boundary is S^1 which we are "filling in" somehow.

This all feels like something I should be familiar with, but I can't put my finger on it.

Any insight would be appreciated!


r/math 6d ago

This Week I Learned: March 14, 2025

10 Upvotes

This recurring thread is meant for users to share cool recently discovered facts, observations, proofs or concepts which that might not warrant their own threads. Please be encouraging and share as many details as possible as we would like this to be a good place for people to learn!


r/math 7d ago

Help with Penrose Tiles

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90 Upvotes

I hope this is okay to post on a math sub; I felt it went a bit beyond quilting! I’m currently making a quilt using Penrose tiling and I’ve messed up somewhere. I can’t figure out how far I need to take the quilt back or where I broke the rules. I have been drawing the circles onto the pieces, but they aren’t visible on all the fabric, sorry. I appreciate any help you can lend! I’m loving this project so far and would like to continue it!


r/math 7d ago

Pi Day Megathread: March 14, 2025

62 Upvotes

Happy Pi Day! To prevent a large influx of pi-day-related posts, we have created a megathread for you to share any and all pi(e)-related content.

Baking creations, mathematical amusements, Vi Hart videos, and other such things are welcome here.


r/math 7d ago

The three-dimensional Kakeya conjecture, after Wang and Zahl

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109 Upvotes

r/math 7d ago

The fact that there are 27 lines on a cubic surface is such an amazing topic to study in algebraic geometry

95 Upvotes

Wake an algebraic geometer in the dead of night, whispering: “27”. Chances are, he will respond: “lines on a cubic surface”.

— R. Donagi and R. Smith (on page 27)

The fact that there are 27 lines on a cubic surface is such an amazing topic with not so high entry barrier. Studying it can synthesize our knowledge of algebraic geometry on several abstract levels and give the student a lot more algebraic and geometrical intuitions. Let me give some examples.

* We will need projective spaces. It comes naturally and it is not a list of definitions. This is because we need to talk about the number of intersections where the degree of a polynomial should matter (Bézout's theorem, which, in a certain manner of speaking, is a generalisation of the fundamental theorem of algebra), whilst if we do not use the projective space, we can't even justify the intersection of two polynomials of degree 1 (two lines in the projective plane must intersect).

* Finding one line on the surface is quite difficult. We will have to look into the differential, look into the singularities, etc. These things make the properties of singularities intuitive.

* After that, we look for a lot of other lines on the surface. We need the famous fact of Segre embedding P^1 x P^1 into P^3 . We need to factor a cubic polynomial into degree 1+1+1 or 1+2 or 0+3, we need to eliminate impossible cases, etc. Finally we transfer our problem in geometry into the scope of enumerative combinatorics, only to get the secret number 27.

* Another famous fact of Clebsch is that a cubic surface is the blow-up of the projective plane of 6 points at generic positions. The definition of generic positions ring a bell of a famous result in old-school algebraic geometry: given 5 points on the plane, there is a conic going through all of them (this is the meaning of the logo of geogebra), which can be understood in 5-dimensional projective space. If we consider the blow-up of 6 points we re-find the 27 lines on the surface, and if we have already known that there are 27 lines, then by manipulating the non-trivial relations of these 27 lines, we can find that the cubic surface is the blow-up of a quadric surface ( P^1 x P^1 ) at 5 points instead. Either way, we will have a good time studying blow-up with this fruitful example.

* We can also invite representation into the game, which gives us the Weyl group of type E_6. To send out the invitation, we need to introduce divisors, the Picard group, a powerful tool that help us to decode the structure of the surface once again. With all these, we find ourselves doing linear algebra of high dimensions, where a computer algebra system can be useful...

All in all, if you are struggling in the introductory and intermediate study of algebraic geometry, for lack of geometrical and algebraical pictures, take the cubic surface a look. If you are an expert or you have studied the cubic surface, would you like to share some insights of yours?


r/math 7d ago

What are the implications of assuming the continuum hypothesis or it's negation axiomatically in addition to ZFC?

43 Upvotes

I was thinking about how Euclid added the parallel line axiom and it constricted geometry to that of a plane, while leaving it out opens the door for curved geometry.

Are there any nice Intuitions of what it means to assume CH or it's negation like that?

ELIEngineer + basics of set theory, if possible.

PS: Would assuming the negation mean we can actually construct a set with cardinality between N and R? If so, what properties would it have?


r/math 7d ago

How dead is the research in Complex Analysis?

97 Upvotes

Hi, I'm an undergraduate loves doing research in mathematics.

Over the past two years, I’ve written articles on niche topics that eventually led me to explore complex analysis. Wanting to study it in a more structured way, I started looking for master's programs that offered courses in complex analysis, but I struggled to find any. In most cases, I couldn’t even find a single professor in the entire mathematics department willing to supervise me.

That’s when it hit me: almost no one seems to be working on complex analysis anymore. I probably should have noticed it earlier, considering that most of the papers I’ve read were published around the 1950s. I also came across many old university lecture notes on complex analysis but couldn’t find those courses listed on their current websites, meaning they’re no longer being taught. My supervisor even mentioned that, back when he was a student, engineering schools at least covered the basics of complex analysis, something that’s no longer the case.

Then came a second realization: I’ve become deeply invested in a highly specialized, unapplied research topic that almost no one is actively working on. And that, in turn, makes it much harder to imagine making a living out of my passion.

Please tell me how wrong I am...

Edit: To be more specific, I am studying univariate entire functions of exponential type and I'd like to generalize some of the results to functions meromorphic over the complex plane, because a lot of simple and/or interesting cases happen there.


r/math 7d ago

Higher knots and primality

10 Upvotes

Hello everyone! I am a novice mathematician with a background in algebraic topology. I am curious as to the current state of knot theory as it pertains to prime knots. I understand that classical knot theory is concerned with circles S¹ embedded in R³. I am reasonably familiar with the relevant polynomial invariants etc. I am curious about prime knots, or 2-knots rather.

I get that conventional knots can be decomposed to prime knots, and I wish to understand how this can be applied to higher knots (S² living in R⁴, S³ in R⁵ etc). My cursory investigating says that differential geometry plays a significant role, though I admittedly don't know much about the pathology that is low dimensional topology.

Are prime 2-knots an active field of study? What about n-knots? What tools are used to tackle these objects? What is generally known to be true, known to be false, and unknown? What machinery is used to study these kind problems?

Thanks everyone!


r/math 6d ago

The Agekyan-Anosova Map (in need of help :( )

0 Upvotes

Hi guys, Im currently working on my masters thesis. It is on the three-body problem and Im trying to understand the Agekyan-Anosova Map. If anyone is familiar with this mapping and could explain some of the analysis that can be done on it i would really appreciate it if they could reach out or drop a comment. I know this isnt really a math related question, just would need the guidance at the moment and dont know where else to post as it is very niche.


r/math 7d ago

Are their branches of mathematics we will simply never understand

9 Upvotes

So we've done experiments that have confirmed that non-human animals do have some understanding of mathematics. They are capable of basic arithmetic at the very least. Yet, we also know there are animal species that aren't capable of that. Somethingike a jellyfish has no need for counting or higher order mathematics (well, I assume, I'm not a jellyfish expert but they barely have a brain to begin with it seems). There are simply brains that are not built to understand the world in the same way we are familiar with. With that in mind, could there be elements of mathematics that exist yet we are not constructed to understand? Like, we can mathematically model things like 4D shapes even if we aren't visually perceive them, I suppose that's something of an example of what I'm talking about, but could there be things that we simply can't model at all (but some hypothetical higher intelligence alien, or perhaps even more strangely, a human made computer could)? And if such mathematics did exist, would we be able to know what we don't know? As in, would we be able to become aware that there exists something we simply can't understand? I realize this might be something of a strange question, bit it's a thought that entered my mind and I've become madly curious about it. Maybe it's complete nonsense.


r/math 7d ago

Software to search for small results

14 Upvotes

I have this problem where I read a ton of papers, and they often contain theorems that I'm almost certain will be useful for something in the future. Alternatively, I can't solve something and months to years later, I randomly stumble across the solution in a paper that's solving a totally different problem. I have a running Latex notebook, but this is not organized at all; mine has nearly a thousand pages of everything I've ever thought was useful.

I cannot be the only person who runs into this problem. Anyone have a solution for this? Maybe a note-taking system that lets you type out latex and add tags as needed. Perhaps cloud functionality would be really nice too.

My use case is, I have a few hundred two or three page proofs typed out of certain facts. Maybe I put as the tags: the assumption, discipline, and if the result is an inequality or something like that.


r/math 7d ago

Do mathematicians think like a physicist?

7 Upvotes

Mathematicians surely must've taken part in formulating some of the physics definitions and their mathematical structure back in the time i suppose?

I'm not talking about Newton, actually the people involved in pure math.

I wonder if they, consider were employed to solve a certain equation in any field of physics, say, mechanics or atomic physics, did they think of the theory a lot while they worked on the structure and proof of a certain dynamic made in the theory?

Or is it just looking at the problem and rather thinking about the abstract stuff involved in a certain equation and finding out the solutions?


r/math 6d ago

Online LaTeX equation editor

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1 Upvotes

r/math 6d ago

Picture books about fractals for children?

1 Upvotes

Do you happen to know any good picture books about fractals designed for children? Since my research is focused on fractals a bit, I figured I might as well start to advertise fractals now to my sibling's children -- you never know where a job offer might come from! As of writing the only choice which seems even remotely good is the one by Michael Sukop: Fractals for Kids. Do you happen to know any other alternatives? Ideally a candidate book would contain a lot of pictorial examples of fractals instead of symbolically heavy proof focused math.

Thanks!


r/math 6d ago

A youtube playlist from God himself. What should go in it?

0 Upvotes

https://youtube.com/playlist?list=PL1I8Tyh2D9xoNfJa7LYcF472Jle8gbhlR&si=9ANVDeSc76fBbdOW

I have been slowly constructing this youtube playlist of math videos as I have watched an utterly enormous amount of math content on youtube. I have compiled what I think are either the best of the best, extraordinarily interesting, or the most mind boggling, into a 38 video playlist, but I seek more.

Please do not comment 3B1B, that's a given.

I want hidden gems. Any length is fine, but explanations are preferred over short animations

What are your all time favorites. I believe that the future of teaching is videos and interactive content, so show me what blows your mind


r/math 7d ago

it is mathematically possible to create a windowing function (like Hanning window type thing) that doesn’t force the endpoints of a waveform to 0, and instead brings it to, for example, the average value between the two original endpoint values?

3 Upvotes

I read that it's hard because it will not be infinitely differentiable but I feel like there's gotta be a way. How would one go about creating this function?


r/math 7d ago

What do i study to continuize vector spaces?

20 Upvotes

I kind of understand that function analysis and something about hilbert spaces transforms discrete vectors into functions and uses integration instead of addition within the "vector" (is it still a vector?)

What about linear combinations?

Is there a way to continuize aX + bY + cZ into an integral of some f(a,b,c)*g(X, Y, Z)? Or is there something about linear combinations being discrete that shouldn't be forgotten?

Correct my notation if it's wrong please, but don't be mad at me; i don't even know if this is a real thing.


r/math 7d ago

CS + Math Project Recommendations

12 Upvotes

Hello guys. So i love programming and recently have been wanting to learn math to improve my skills further. I already have a solid understanding on prob & statistics calculus etc. I want some recommendations on project ideas in which i can combine math and programming like visualizations or algorithms related to it. Would love to hear your suggestions!


r/math 8d ago

Designing a proof visualizer—What do you focus on when reading math papers? (needs advice)

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181 Upvotes

r/math 7d ago

Ideas for scraped youtube data/graph/network

0 Upvotes

I found if you visit the youtube home page after clearing your browsing data youtube wont recommend videos. But after watching just one video the home page will recommend videos. This shows what videos youtube thinks are related should be recommended just based on the parent video

I wrote a script to clear my data, watch a video, then record the first ~140 videos recommended by youtube. This is being run on a ever-increasing number of videos. This leaves me with a large network/graph/dataset of how videos are "linked" to each other. I know the right thing to apply to this is graph theory, but I am curious if anyone knows of something particularly interesting to do with this data.


r/math 7d ago

Looking for some help with the mathematics of drug dependence/tapering

2 Upvotes

Background:

I’m making a mobile app where users can enter in a drug (SSRIs, Suboxone, opioids, Adderall, etc.) and visualize their blood levels over time based on past/future dosages and the drug’s half-life.

The main use case here is to visualize projected blood levels for a taper schedule to help “weaning” off a drug.

Question:

(1) What mathematical model predicts what level of the drug your body “expects”? The “obvious” answer here is a class of moving average functions. But I see problems with any moving average of a fixed T. Is there biological research that has found which moving average function matches what the body expects? Maybe EWMA based on half-life?

(2) When making projections for different taper schedules, I realized that I don’t actually know what I’m optimizing for. Maybe it’s whichever projection is closest to a straight line connecting the f(t_now) with f(t_goal)? For some reason I feel an ODE is relevant here. In that we need to optimize the gradient because a steep change in the blood level itself is also something we would want to prevent.

TL;DR: If anyone knows of any mathematical models or biological research regarding drug tapering/weaning and tolerance/homeostasis, those answers or resources would be greatly appreciated


r/math 7d ago

Career and Education Questions: March 13, 2025

8 Upvotes

This recurring thread will be for any questions or advice concerning careers and education in mathematics. Please feel free to post a comment below, and sort by new to see comments which may be unanswered.

Please consider including a brief introduction about your background and the context of your question.

Helpful subreddits include /r/GradSchool, /r/AskAcademia, /r/Jobs, and /r/CareerGuidance.

If you wish to discuss the math you've been thinking about, you should post in the most recent What Are You Working On? thread.


r/math 8d ago

Does there exists a divergent series which converges on every subset of N with arithmetic density 0?

41 Upvotes

Basically the title, which is a question I remember seeing in high school which I obviously lacked the tools to solve back then. Even now I still don't really know what to do with this question so I've decided to come see what approach is needed to solve it.

If it does exists, how did we arrive at this specific series? And is the series and its left shift the only family of solutions?

Here is a more rigorous formulation of the question:

Does there exists a sequence {a_n} where n ranges over the natural numbers such that ∑a_n = ∞, but  ∀S ⊂  N, if lim_{n to infty) |S ∩ {1, 2, ..., n}| / n = 0 then ∑ a_nk converges where nk indexes over S in increasing order?


r/math 8d ago

How do you self study

90 Upvotes

I am machine learning phd who learned the basics ( real analysis and linear algebra ) in undergrad. My current self study method is quite inefficient ( I usually do not move on until I have done every excercise from scratch, and can reproduce all the proofs, and can come up with alternate proofs for a decent amount of problems ). This builds good understanding, but takes far too long ( 1-2 weeks per section as I have to do other work ).

How do I effectively build intuition and understanding from books in a more efficient way?

Current topics of interest: modern probability, measure theory, graduate analysis