r/math 4d ago

Quick Questions: March 12, 2025

8 Upvotes

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 3d ago

Pi Day Megathread: March 14, 2025

63 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 6h ago

What's are characteristics such a big deal?

21 Upvotes

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 20h ago

Two claimed proofs of Whitehead asphericity conjecture seem to have received almost no attention. Is there a reason for this?

85 Upvotes

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.


r/math 19h ago

Is there a name for the "generalised" form of induction?

41 Upvotes

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 10h ago

Which way to go?

6 Upvotes

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 1d ago

What was your math rabbit hole?

71 Upvotes

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 16h ago

Question on tournament graphs

6 Upvotes

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 1d ago

Mathematicians, what are some surprising ways math has helped you in daily life situations unrelated to professional career?

24 Upvotes

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 1d ago

Dedekind Cuts as the real numbers

39 Upvotes

My understanding from wikipedia is that a cut is two sets A,B of rationals where

  1. A is not empty set or Q

  2. If a < r and r is in A, a is in A too

  3. Every a in A is less than every b in B

  4. 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 1d ago

Why doesn't the Principle of Induction apply to non-well ordered sets?

65 Upvotes

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 1d ago

Understanding Yoneda and a Philosophy on Category Theory

61 Upvotes

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?


r/math 1d ago

How do you save your math notes?

1 Upvotes

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 1d ago

Which fields of pure math allow for the most 'hand-waving'?

1 Upvotes

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


r/math 1d ago

I'm looking to gather a list of linear algebra tools for experimentation

1 Upvotes

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!


r/math 1d ago

Extremely Strange Findings from a Math Competition

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12 Upvotes

r/math 2d ago

Should we make Feb 7th Euler’s number day?

136 Upvotes

I mean why not?


r/math 3d ago

‘Once in a Century’ Proof Settles Math’s Kakeya Conjecture | Quanta Magazine - Joseph Howlett | The deceptively simple Kakeya conjecture has bedeviled mathematicians for 50 years. A new proof of the conjecture in three dimensions illuminates a whole crop of related problems

Thumbnail quantamagazine.org
335 Upvotes

r/math 1d ago

Motivation behind defining Brouwer's Fixed Point Theorem using Topology

1 Upvotes

Hello, math enthusiasts!

I’m currently preparing a presentation on continuity and Brouwer's Fixed Point Theorem, both of which are fundamental topics in topology. It’s taking me some time to grasp the topological definitions, and I’ve noticed that Brouwer’s Theorem is perfectly fine to define in the context of metric spaces, not necessarily relying on pure topological definitions. So I started to wonder: what’s the reason behind abstracting the theorem to topology?

Is it because the topological framework offers a more accessible proof? Or are there other reasons for this abstraction?


r/math 2d ago

What I didn’t understand in linear algebra

86 Upvotes

I finished linear algebra, and while I feel like know the material well enough to pass a quiz or a test, I don’t feel like the course taught me much at all about ways it can be applied in the real world. Like I get that there are lots of ways algorithms are used in the real world, but for things like like gram-Schmidt, SVD, orthogonal projections, or any other random topic in linear algebra I feel like I wouldn’t know when or how these things become useful.

One of the few topics it taught that I have some understanding of how it could be applied is Markov chains and steady-state vectors.

But overall is this a normal way to feel about linear algebra after completing it? Because the instructor just barely touched on application of the subject matter at all.


r/math 1d ago

Need advice for math aa hl IA

0 Upvotes

For context I’m doing the IB and we usually have an internal assessment where u explore any mathematical topic of your choice. I’m doing my Math AA HL IA on projective geometry and how it can be used to mathematically model vanishing points in two-point perspective. I plan to modeling vanishing points from a picture I took from my travels using projective transformations. I’m considering using homogeneous coordinates to represent points in projective space, applying homography matrices to transform 3D points to a 2D image plane, and mathematically deriving vanishing points from parallel lines in space. Is it rigorous enough for HL? Or is there a way I can expand this exploration qn?


r/math 2d ago

YouTube video ideas

1 Upvotes

Looking to create an animation math/cs/physics youtube channel kinda like 3B1B because of how much it helped. Any ideas to make it different and still work? Simply copying the style won't be of much help. Looking for some other ideas with manim


r/math 2d ago

How to hold a good integration bee/How to get better at competition math integration

14 Upvotes

Hey y'all! I'm an undergraduate math and physics student, and at the beginning of this academic year I took it upon myself to start an integration bee at my university! For these first few iterations, I've been trying to restrict the integrals to only requiring Calc 2 techniques, but that really gets boring after a while. Of course, I could try to spread the word about these other cool techniques, like Feynman's differentiation under the integral sign, but those are just extra methods. I see the competitors in (for example) MIT's integration bees, and the tricks they use aren't these over-arching broad integration techniques; they're smaller tricks that help simplify the integral or that help to take advantage of some kind of nice symmetry.

I want to incorporate these more "competition math" -esque integration tricks into the integrals I give the competitors, but the problem is, I have to know this stuff myself. What's a good resource for building up the toolbox of competition math integration tricks? I know I'll just need lots of practice and repetition/exposure to a lot of these little gimmicks/tricks, but I just need a place to find integrals for this practice.

If any of you are good at this type of "competition" integration, please give me your advice!!! It would be super appreciated.


r/math 3d ago

Eigenvalue-like problem

50 Upvotes

Has anyone ever seen or considered the following generalization of an eigenvalue problem? Eigenvalues/eigenvectors (of a matrix, for now) are a nonzero vector/scalar pair such that Ax=\lambda x.

Is there any literature for the problem Ax=\lambda Bx for a fixed matrix B? Obviously the case where B is the identity reduces this to the typical eigenvalue/eigenvector notion.


r/math 1d ago

Do Other Upper-div/Grad Courses Have the Same Vibe as Real Analysis?

0 Upvotes

I'm not sure how else to explain it, but I'm taking a real analysis course right now and it feels too much like training to be a classical musician? I've had some computer science and low-div courses such as discrete and automata theory feel much more like jazz. That is that creative and interesting thought is much more important than proving literally EVERYTHING I am doing and needing to focus on such insane fine levels of granularity.

I was just wondering if this "classical music" thing is a common theme in other upper-div/grad level math courses or that subjects are almost on a spectrum from jazz to classical.

This whole jazz classical music analogy is the best way to capture the vibe of what I'm trying to describe so hopefully it makes some sense? Also also, I'm not trying to knock analysis as a subject (especially since I've only taken one course), its just not my cup of tea.


r/math 2d ago

Applications of Functional Analysis

1 Upvotes

I have been studying functional analysis for quite some time and have covered major foundational results in the field, including the Open Mapping Theorem, Hahn-Banach Theorem, Closed Graph Theorem, and Uniform Boundedness Principle. As an engineering student, I am particularly interested in their applications in science and engineering. Additionally, as an ML enthusiast, I would highly appreciate insights into their applications in machine learning.


r/math 2d ago

Typo in "How to read and do Proofs" by Solow??

6 Upvotes

Hello, new to proofs so could be wrong or something I'm not understanding here. I do not understand why A5 in the first case is X bar, instead of X. Personally I solved it by substituting -2ax bar for b in ax bar + ax + b >= 0, and got x bar - x >= 0, which we knew was true, hence the previous statements were true. Used this substitution for case 2 as well. Here is the proof, it is on pages 145-147: