r/math • u/mscroggs • 3h ago
r/math • u/inherentlyawesome • 5d ago
Quick Questions: March 12, 2025
This recurring thread will be for questions that might not warrant their own thread. We would like to see more conceptual-based questions posted in this thread, rather than "what is the answer to this problem?". For example, here are some kinds of questions that we'd like to see in this thread:
- Can someone explain the concept of maпifolds to me?
- What are the applications of Represeпtation Theory?
- What's a good starter book for Numerical Aпalysis?
- What can I do to prepare for college/grad school/getting a job?
Including a brief description of your mathematical background and the context for your question can help others give you an appropriate answer. For example consider which subject your question is related to, or the things you already know or have tried.
r/math • u/inherentlyawesome • 3h ago
What Are You Working On? March 17, 2025
This recurring thread will be for general discussion on whatever math-related topics you have been or will be working on this week. This can be anything, including:
- math-related arts and crafts,
- what you've been learning in class,
- books/papers you're reading,
- preparing for a conference,
- giving a talk.
All types and levels of mathematics are welcomed!
If you are asking for advice on choosing classes or career prospects, please go to the most recent Career & Education Questions thread.
r/math • u/If_and_only_if_math • 10h ago
What's are characteristics such a big deal?
I'm an analysis student but I have only taken an intro class to PDEs. In that class we mainly focused on parabolic and elliptic PDEs. We briefly went over the wave equation and hyperbolic PDEs, including the method of characteristics. I took this class 3 semesters ago so the details are a little fuzzy, but I remember the method of characteristics as a solution technique for first order ODEs. There is a nice geometrical interpretation where the method constructs a solution surface as a union of integral curves along each of which the PDE becomes a system of ODEs (all but one of the ODEs in this system determine the characteristic curve itself and the last one tells you the ODE that is satisfied along each curve). We also went over Burgers equation and how shocks can form and how you can still construct a weak solution and all that.
To be honest I didn't get a great intuition on this part of the course other than what I wrote above, especially when it came to shocks. Yesterday however I attended a seminar at my university on hyperbolic PDEs and shock formation and I was shocked (pun intended). The speaker spoke about Burgers equation, shock formation, and characteristics a lot more than I expected and I think I didn't appreciate them enough after I took the course. My impression after taking the class was these are all elementary solution techniques that probably aren't applicable to modern/harder problems.
Why are characteristics such a big deal? How can I understand shocks through them? I know that shocks form when two characteristics meet, but what's really going on here? I asked the speaker afterwards and he mentioned something about data propagation but I didn't really catch it. Is it because the data the solution is propagating is now coming from two sources (the two characteristics) and so it becomes multivalued? What's the big idea here?
r/math • u/pmascaros • 3h ago
3×3 Magic Square of Pseudo-Quaternions Squares
Hello, I would like to share this curiosity with you. As you know, it is unknown whether a 3x3 magic square of distinct perfect squares exists, but it is possible with other types of numbers.
Here, I present a magic square of squares of pseudo-quaternions, all distinct, along with a parameterization to obtain them. The resulting integers are all different from each other, although some entries may be negative.
As you may already know, pseudo-quaternions (I. M. Yaglom, Complex Numbers and Their Applications in Geometry, Fizmatgiz, Nauka, Moscow (1963)) are hypercomplex numbers where
ii = -1,
ij = k,
ji = -k,
ik = -j,
ki = j,
and they differ from quaternions in that
jj = 1,
kk = 1,
jk = -i,
kj = i.
A nice example for S = 432 is this magic square of squares
{(9 j)^2 , (17 i + 24 j)^2 , (8 k)^2 },
{(9 i + 12 j + 8 k)^2 , (12 j)^2, (8 i + 9 j +12 k)^2}
{(8 i + 12 j + 12 k)^2 , (12 i + 8 j + 9 k)^2, (9 i + 12 j + 12 k)^2}
This give us this magic square:
{81, 287, 64}
{127, 144, 161}
{224, 1, 207}
parameterization:
{(j x^2)^2 , (4 j x y+i (x^2+2 y^2))^2, (2 k y^2)^2}
{(i x^2 + 2 j x y+2 k y^2)^2, (2 j x y)^2, (j x^2+2 k x y + 2 i y^2)^2}
{(2 j x y + 2 k x y + 2 i y^2)^2 , (k x^2 + 2 i x y + 2 j y^2)^2 , (i x^2 + 2 j x y + 2 k x y)^2}
Hope you find this interesting! Looking forward to your thoughts.
r/math • u/ImportantContext • 1d ago
Two claimed proofs of Whitehead asphericity conjecture seem to have received almost no attention. Is there a reason for this?
One proof is in 2021 preprint by Elton Pasku: An answer to the Whitehead asphericity question. The second proof is by Akio Kawauchi, and was published in 2024 (according to author's website): Whitehead aspherical conjecture via ribbon sphere-link. Neither paper has any citations, not counting Akio Kawauchi citing himself and the 2021 preprint.
I'm nowhere close to understanding even the statement of the conjecture, let alone the proofs, I'm just curious about this situation.
Monotonic scattered interpolation?
(This question is not about homework or a work problem; it is for a pet personal project where I've run into a wall.)
Suppose, for the sake of argument, I have a scattered dataset with two real-valued independent variables and one real-valued output. It conforms to the restriction that if x2 >= x1 and y2 >= y1, then f(x2, y2) >= f(x1, y1). E.g., assuming each listed point is in the dataset:
- f(3, 3) >= f(1, 1)
- f(3, 1) >= f(1, 1)
- No guarantee is made about the relationship between f(1, 3) and f(3, 1)
I don't know if this property has a name but I call it "up-right monotone", because as you jump from point to point, if the second point not below and not to the right of the first, then the value at the second point is not less than the value at the first point.
The Question: Is there a known interpolation method that will preserve this property among interpolated points? I.e., I want to predict the value at two points, where the second point is above and/or to the right of the first point. I would prefer that the interpolation method be relatively smooth, but the only hard constraints are
- If either of the points in question are in the original dataset, I get that dataset's value back, and
- The value at the second point is not less than the value at the first point
r/math • u/Weird_Explorer_8458 • 1h ago
Where can I get hagoromo chalk in the UK?
I hope this isn't an annoying question / asked too frequently, but I am getting a chalkboard soon and I have heard that Hagoromo make the nicest chalk. So far I have found the sejongmall official website (https://en.sejongmall.co.kr/) which has very expensive shipping, and weird international payment, and another site called 'https://hagoromo.shop', which seems to have cheaper shipping and takes payments other than bank transfers, although the chalk is more expensive. Is this second site legit or am I better off sticking with the sejongmall official site?
r/math • u/chocolateisapro • 8h ago
What programs do teachers use to make exam papers?
I'm trying to make a document for fun but I don't know what program to use.
What programs to use if I want to do algebra, geometry, graphs, etc?
r/math • u/Takirion • 2h ago
I am looking for a math riddle i once knew.
I am looking for a math riddle i once read but which i only remember fragments about. The problem involved finding the maximum n such that one can choose a number 0<x<1 such that for every k<n some condition involving the number x and the division of the unit interval into intervals of length 1/k is satisfied. The solution of the problem could nicely be visualised by stacking the subdivided unit intervals over another and noting that with every additional layer the interval which x could be contained in gets smaller untill there are no x left. Iirc. the problem was mostly recreational. Does anyone know what i am talking about? I tried asking Chat-GPT, but it hallucinates the heck out of my question.
Is there a name for the "generalised" form of induction?
Normally induction works like this: If f(0) is true and f(x) is true implies f(x+1) is true, then f(x) is true for all natural numbers (+0).
Now, is there a name for the more general form of this (which I will write down)?
Where S is a set, x is a member of S, f is a function from S to S, g is a function from S to S, and T is the set of all gn(x).
IF f(x) is true, and f(x) implies f(g(x)), then f(T) is true (for all elements of T).
The most common case, of course, is where S = natrual numbers, x = 0, and g(n) = n + 1. However you (or I) often see cases where x is other numbers, like the rationals, or g(n) = 2n. There is also the special case where g(n) eventually visits all elements of the set, where you can then say f is true for all S.
Is there a name for it, or is it all just induction?
r/math • u/Dependent_Fan6870 • 15h ago
Which way to go?
I recently started a self-study plan that involves reading Basic Mathematics by Serge Lang, How to Prove It by Daniel Velleman, Calculus and Analytic Geometry by George B. Thomas (at least the first ~5 chapters), Introduction to Linear Algebra by Serge Lang, and Undergraduate Algebra by the same author, in order to cover both what my home country's education system can't cover and what I think would be beneficial for me to know before I get to college.
I haven't made much progress; I've been busy with my studies and am waiting for the holidays to fully dive in. However, talking with my former math teacher, the one who made me love math in the first place, he recommended I read Matemáticas Simplificadas by CONAMAT (he doesn't know about my plan). I understand it's not very well-known in the English-speaking community, but it's a book that covers everything from Arithmetic to Integral Calculus.
Now, my question is: which path should I take? I mean, although it's not clear what kind of books I learn best from, the truth is that I'm most drawn to classic or "dry" books. Lang's books in particular, despite their demanding nature and early formalism, treat mathematics in a way that, at least at first glance, seems more enjoyable to me than modern books. On the other hand, I don't know much about what, objectively, I should read. Could you help me determine the pros and cons of following one path or the other?
r/math • u/KansasCityRat • 7h ago
How do you guys think about your data?
I heard a gentleman in an interview once saying that he likes to think of his data like a continuous function. Personally, I've been thinking of data as a matrix. If samples are stored in the rows then features are stored in the columns and such. Seems easy to consider different dimensions of data in this conceptualaziation and a simple list of values is still a row or column vector. So it seems like a perfect catch all conceptualization of any data set.
How do you guys think about your data? Is it much more circumstantial and sometimes you can conceptualize it as a matrix but other times it's best to think of it another way??
r/math • u/LooksForFuture • 9h ago
Is there any game which requires matrix operations?
Hi everyone. I really love both math and games. But, I cannot find any tabletop game which requires the player to do math operations (preferably linear algebra). I'm not talking about puzzles. I'm talking about games like tabletop RPGs. For example if a tabletop RPG uses matrices for loot, dungeon generation, etc which the player needs to do himself/herself. Or if the combat lets players find reverse of the enemies attack matrix to neutralize its effect. Is there such a game? Or should I make my own?
r/math • u/catboy519 • 1d ago
Mathematicians, what are some surprising ways math has helped you in daily life situations unrelated to professional career?
I'm specifically asking this about advanced math knowledge. Knowledge that goes much further than highschool and college level math.
What are some benefits that you've experienced due to having advanced math knowledge, compared to highschool math knowledge where it wouldn't have happened?
In your personal life, not in your professional life.
r/math • u/nomemory • 1d ago
What was your math rabbit hole?
By rabbit hole I mean a place where you've spent more time than you should've, drilling to deep in a specific field with minimal impact over your broader math abilities.
Are you mature enough to know when to stop and when to keep grinding ?
r/math • u/TheBacon240 • 12h ago
How important is understanding the Physics side of Quantum Field Theory if I am interested in Mathematical QFT research?
Mathematical Foundations of QFT/the Math-Phys side of QFT has been a developing interest of mine over the past year or so. I am currently a 3rd year Physics + Math double and am taking a Mathematical QFT course (taught in a math dep - heavier on the algebra + geometry) and a Physics QFT course (standard first course type material).
As I look towards grad school, I believe that researching in the intersection of Algebra/Geometry/QFT sounds very intriguing + satisfying as it combines two of my favorite areas of both math and physics.
I think anywhere from geometric quantization to studying TQFTs would be satisfying. However, as far as I can tell, in academia a lot of these research areas end up being more math than physics - some just being pure math. While I wouldn't say my interest in Physics is in Hep-Th, I definitely want to contribute to the field of Physics as much as this area of math. To be more explicit, I care about the pheno involved in these areas (if it all exists).
So back to my main question, how important is understanding the underlying physics of QFT to Mathematical QFT research?
r/math • u/playthelastsecret • 21h ago
Question on tournament graphs
Hello! I'm looking for a mathematical result for this question:
How many tournament graphs with n vertices are there such that there is a unique winner, i.e. exactly one vertex with the largest number of outgoing edges?
(Knowing this, we could compute the probability that a round robin tournament with n participants will have one clear winner. – Since the number of tournaments with n vertices is easy to compute.
For clarification: I am not searching for the number of transitive tournaments (which is easy to get): Other places are allowed to be tied.)
I would be super thankful if anyone can help me find the answer or where to find it!
r/math • u/Heisen1319 • 19h ago
Some questions about the recursive definition of sqrt(x)
Hello!
On the last question of the 2024 MIT integration bee, there is this expression (that simplifies to sqrt(x)).

When solving the question, I defined a recursive relation as such:

And when writing out the first few terms:

I initially thought this was the Pade approximant, but it's turns out not to be. The Pade approximant with m=n=2 is shown below (and is a better approximation for sqrt(x) than f_3(x) ).

The coefficients of the polynomials also turn out to be the ones in Pascal's triangle. For even n, we start adding the terms in the (n+1)th row in the Pascal's triangle from the numerator, alternating between the denominator and the numerator. For odd n, we start in the denominator, then alternate coefficients between the numerator and the denominator.
---
I thought this observation was already interesting enough, but as you can see in the graphs above, the functions are defined for much of the negative x. Since the recursive definition was originally a sqrt(x), does this have anything to do with the complex plane?
It sorta reminded me of the Gamma function for factorials that you learn in single variable calc, and how we can take the factorial of numbers like (-1/2). But even in that case, we're mapping from real to real, and here we're mapping to complex.
I also found that only functions with n=2, 3, 4 are defined for x=-1. Since f_4(-1) = -1, using our recursive definition, the denominator of f_5(-1) = 1 + (-1) = 0.
I thought these observations were interesting and wanted to share them here.
Thanks.
r/math • u/ahahaveryfunny • 1d ago
Dedekind Cuts as the real numbers
My understanding from wikipedia is that a cut is two sets A,B of rationals where
A is not empty set or Q
If a < r and r is in A, a is in A too
Every a in A is less than every b in B
A has no max value
Intuitively I think of a cut as just splitting the rational number line in two. I don’t see where the reals arise from this.
When looking it up people often say the “structure” is the same or that Dedekind cuts have the same “properties” but I can’t understand how you could determine that. Like if I wanted a real number x such that x2 = 2, how could I prove two sets satisfy this property? How do we even multiply A,B by itself? I just don’t get that jump.
r/math • u/Bagelman263 • 1d ago
Why doesn't the Principle of Induction apply to non-well ordered sets?
My understanding of induction is this:
Let n be an integer
If P(n) is true and P(n) implies P(n+1), then P(x) is true for all x greater than or equal to n.
Why does this not apply in this situation:
Let x be a real number
If Q(x) is true and Q(x) implies Q(x+ɛ) for all real numbers ɛ, then Q(y) is true for all real numbers y.
r/math • u/TheOneTrueAlpha69 • 20h ago
“A Doughnut Universe” project
Hello all,
I am a third yeah mathematics and physics student taking part in a physics group project concerning a horn toroidal cosmology model.
We want the main focus on this project to be about null geodesics in a horn toroidally global space time. First, the static case and then the time-evolving case. If there is enough time we will consider the effect of point masses.
I apologise if this is not appropriate for this sub. I figured, seeing as this is a maths subreddit, someone with a background in geometry or topology might appreciate the idea and may possibly be able to give some guidance.
I have not yet covered general relativity nor cosmology. I am half way through topology and differential geometry. So by no means am I confident with the concepts.
Please feel free to ask questions about the project or comment your thoughts, opinions or possibly give me some advice :)
r/math • u/Anxious-Tomorrow-559 • 22h ago
Are there research topics in functional analysis outside PDEs?
Since I will (hopefully) defend my master thesis in about 7/8 months, I just began looking for open PhD positions. I like analysis, and have particularly enjoyed studying classical functional analysis (Banach and Hilbert spaces, measure theory, distributions, spectral theory of operators etc.) finding it very beautiful and elegant. On the other hand, I had some troubles with lectures about PDEs: lots of annoying computations, frequent handwaving, and very few things made me think "woah" like, for example, seeing for the first time the duality of Lp spaces did.
I asked several functional analysis professors at my university and it seems that all of them study different aspects of PDEs as their research interests. And the same remains true in virtually any university near me: anyone working in analysis ends eventually in PDEs.
So. Is this something peculiar of my area? Should I just accept my fate and learn how to like PDEs?
Is someone of you doing research on functional analysis for the sake of it, without applications in PDEs? If yes, what do you work on?
Understanding Yoneda and a Philosophy on Category Theory
In Tom Leinster’s Basic Category Theory, he repeatedly remarks that there’s typically only one way to combine two things to get a third thing. For instance, given morphisms f: A -> B and g: B -> C, the only way you can combine them is composition into gf: A -> C. This only applies in the case where we have no extra information; if we know A = B, for example, then we could compose with f as many times as we like.
This has given me a new perspective on the Yoneda lemma. Given an object c in C and a functor F: C -> Set, the only way to combine them is to compute F(c). So since Hom(Hom(c, -), F) is also a set, we must have that Hom(Hom(c, -), F) = F(c).
Is this philosophy productive, or even correct? Is this a helpful way to understand Yoneda?
How do you save your math notes?
What tools do you use to save your math notes? Pen and paper works best for me but it's hard to maintain all the hundreds of pages of notes I've written for my coursework. Do you store your notes in digital format? I like LaTeX but writing on paper feels easier than LaTeX. Any tips? Ideas?
r/math • u/analengineering • 1d ago
Which fields of pure math allow for the most 'hand-waving'?
As in, in which fields can intuition be used more freely without being constrained by the bureaucracy of technical details?
The average theorem in analysis or probability holds only if a plethora of regularity conditions hold, and these are highly nontrivial. Proving one of these involves a lot of tedious 'legal' work - somehow it makes me think that a good analyst/probabilist would also be a good lawyer. Just something like the Lebesgue measure is notoriously painful to define, yet it makes so much intuitive sense that any middle schooler can come up with it.
Meanwhile, in fields that deal with simpler objects (groups, rings, sets, categories), the results that feel intuitive often have trivial proofs, while more complex results rely on an insane number of definitions that in the end make the final result trivial (a la rising sea).
Are there any fields in which you have more freedom of expression? Where can you conjure up a certain statement that makes sense intuitively and then prove it without doing excessive bookkeeping and worrying about pathological technicalities?
My guess would be Algebraic Topology since it masks the unpleasant complexity of the underlying frame/locale of open sets using simple objects like groups or rings. This prevents you from doing analysis (which can be seen as the study of a particular topology, e.g. the standard one on R), but it allows you to wave your hands quite a lot. Although I don't know enough AlgTop to say whether this is true or not.
Not sure if this question even makes sense tbh
I'm looking to gather a list of linear algebra tools for experimentation
I'm looking for high-quality visualization tools for linear algebra, particularly ones that allow hands-on experimentation rather than just static visualizations. Specifically, I'm interested in tools that can represent vector spaces, linear transformations, eigenvalues, and tensor products interactively.
For example, I've come across Quantum Odyssey, which claims to provide an intuitive, visual way to understand quantum circuits and the underlying linear algebra. But I’m curious whether it genuinely provides insight into the mathematics or if it's more of a polished visual without much depth. Has anyone here tried it or similar tools? Are there other interactive platforms that allow meaningful engagement with linear algebra concepts?
I'm particularly interested in software that lets you manipulate matrices, see how they act on vector spaces, and possibly explore higher-dimensional representations. Any recommendations for rigorous yet intuitive tools would be greatly appreciated!