I’m taking this course on Elliptic curves and I’m struggling a bit trying not to lose sight of the bigger picture. We’re following Silverman and Tate’s, Rational Points on Elliptic Curves, and even though the professor teaching it is great, I can’t shake away the feeling that some core intuition is missing. I’m fine with just following the book, understanding the proofs and attempting the excercise problems, but I rarely see the beauty in all of it.

What was something that you read/did that helped you put your understanding of elliptic curves into perspective?

Edit: I’ve already scoured the internet looking for recourse on my own, but I don’t think I’ve stumbled upon many helpful things. It feels like studying elliptic curves the same way I study the rest of math I do, isn’t proving of much worth. Should I be looking more into applications and finding meaning in that? Or its connections to other branches of math?

1. The halting problem states that any computer eventually stops working, which is a problem.
2. Hall's marriage problem asks how to recognize if two dating profiles are compatible.
3. In probability theory, Kolmogorov's zero–one law states that anything either happens or it doesn't.
4. The four color theorem states that you can print any image using cyan, magenta, yellow, and black.
5. 3-SAT is how you get into 3-college.
6. Lagrange's four-square theorem says 4 is a perfect square.
7. The orbit–stabilizer theorem states that the orbits of the solar system are stable.
8. Quadratic reciprocity states that the solutions to ax²+bx+c=0 are the reciprocals of the solutions to cx²+bx+a=0.
9. The Riemann mapping theorem states that one cannot portray the Earth using a flat map without distortion.
10. Hilbert's basis theorem states that any vector space has a basis.
11. The fundamental theorem of algebra says that if pⁿ divides the order of a group, then there is a subgroup of order pⁿ.
12. K-theory is the study of K-means clustering and K-nearest neighbors.
13. Field theory the study of vector fields.
14. Cryptography is the archeological study of crypts.
15. The Jordan normal form is when you write a matrix normally, that is, as an array of numbers.
16. Wilson's theorem states that p is prime iff p divides p factorial.
17. The Cook–Levin theorem states that P≠NP.
18. Skolem's paradox is the observation that, according to set theory, the reals are uncountable, but Thoralf Skolem swears he counted them once in 1922.
19. The Baire category theorem and Morley's categoricity theorem are alternate names for the Yoneda lemma.
20. The word problem is another name for semiology.
21. A Turing degree is a doctoral degree in computer science.
22. The Jacobi triple product is another name for the cube of a number.
23. The pentagramma mirificum is used to summon demons.
24. The axiom of choice says that the universe allows for free will. The decision problem arises as a consequence.
25. The 2-factor theorem states that you have to get a one-time passcode before you can be allowed to do graph theory.
26. The handshake lemma states that you must be polite to graph theorists.
27. Extremal graph theory is like graph theory, except you have to wear a helmet because of how extreme it is.
28. The law of the unconscious statistician says that assaulting a statistician is a federal offense.
29. The cut-elimination theorem states that using scissors in a boxing match is grounds for disqualification.
30. The homicidal chauffeur problem asks for the best way to kill mathematicians working on thinly-disguised missile defense problems.
31. Error correction and elimination theory are both euphemisms for murder.
32. Tarski's theorem on the undefinability of truth was a creative way to get out of jury duty.
33. Topos is a slur for topologists.
34. Arrow's impossibility theorem says that politicians cannot keep all campaign promises simultaneously.
35. The Nash embedding theorem states that John Nash cannot be embedded in Rⁿ for any finite n.
36. The Riesz representation theorem states that there's no Riesz taxation without Riesz representation.
37. The Curry-Howard correspondence was a series of trash talk between basketball players Steph Curry and Dwight Howard.
38. The Levi-Civita connection is the hyphen between Levi and Civita.
39. Stokes' theorem states that everyone will misplace that damn apostrophe.
40. Cauchy's residue theorem states that Cauchy was very sticky.
41. Gram–Schmidt states that Gram crackers taste like Schmidt.
42. The Leibniz rule is that Newton was not the inventor of calculus. Newton's method is to tell Leibniz to shut up.
43. Legendre's duplication formula has been patched by the devs in the last update.
44. The Entscheidungsproblem asks if it is possible for non-Germans to pronounce Entscheidungsproblem.
45. The spectral theorem states that those who study functional analysis are likely to be on the spectrum.
46. The lonely runner conjecture states that it's a lot more fun to do math than exercise.
47. Cantor dust is the street name for PCP.
48. The Thue–Morse sequence is - .... ..- .
49. A Gray code is hospital slang for a combative patient.
50. Moser's worm problem could be solved using over-the-counter medicines nowadays.
51. A character table is a ranking of your favorite anime characters.
52. The Jordan curve theorem is about that weird angle on the Jordan–Saudi Arabia border.
53. Shear stress is what fuels students.
54. Löb's theorem states that löb is greater than hãtę.
55. The optimal stopping theorem says that this is a good place to stop. (This is frequently used by Michael Penn.)
56. The no-communication theorem states that

I have created a draft for a sequence to be submitted into OEIS, it got some comments for changes, which I have resolved. But after a few days I have realized that I have made slight calculation error, so both the data, and formula are incorrect. Do I just fix these, or should I delete the draft and start from scratch? I would also need to fix comments, and few other lines. Thanks.

Maybe i got something wrong but all the videos i can find seems to show the generalized integral with respect to a lebesgue mesure so if i have not misunderstood , we would have under the integral f(x)F(dx)=f(x)dx , but how do i visualize If F(x) Is actually not a lebesgue measure? (Would be even more helpfull if someone can answer considering as example a probability ,non uniform , measure )

I’m an upper level real analysis and complex analysis class in undergrad, and the class is entirely proof based. I find that whenever I am reading the textbook, I feel always under-prepared in what I read in the chapter to answer the practise problems.

Most of the time the questions feel so abstract and obfuscated I just get overwhelmed and don’t even know where to start from or if I’m doing the steps correct.

Or when I see sample solutions, I have trouble understanding what’s going on to recreate it or have no idea what’s going on. I have taken senior level physics and computer science classes and do very well, but I find myself always struggling with proofs and the poor teaching structures in place.

What can I do to get better, as I find myself completely overwhelmed in almost all practise questions and dont usually know how to start to finish a proof. I have taken easier proof based math classes with discrete and linear, but even then I have struggled, but my upper level math classes are overwhelming and with proofs in general

I was trying to register it, but I couldn’t find the link where I could register. What happened to the competition? If it has vanished, is there a math competition for adults other than Alibaba’s?

Hello, Im a freshman majoring in math and I started sending out emails to profs/PhD students whose research interested me to ask about opportunities in research. Out of the emails that I sent, 2 responded. They both wanted to meet on zoom, but I’m not exactly sure what to expect from the call. Is it similar to an interview? What are some small tips that I can keep in mind to make sure that I dont screw anything up? Thanks!

This might be a really stupid question and this might be the wrong subreddit to ask this but I recently had an epiphany about the method of characteristics despite learning it a few semesters ago and suddenly everything clicked. Now I'm trying to see how far I can take this idea. One thing that I thought about is the Euler equation. It's first order and hyperbolic so I began to wonder if the method of characteristics can be used for it. I assume it can't since we would otherwise have an explicit solution for it but as far as I know that hasn't been discovered yet. On the other hand, I tried searching around and saw a lot of work being done investigating shocks in the compressible Euler equation.

Are the Euler equations solvable using the method of characteristics? If so, how do you deal with the equations having two unknown functions (pressure and velocity) instead of just one? If not, why not and how do people use characteristics to do analysis if you can't solve for them?

This may be an uninformed question but the issue with the axiom of choice is it allows many funky behaviors to be proven (banach tarski paradox). Yet we recognize it as fundamental to quite a lot of mathematics. Rather than opting in or out of accepting the axiom of choice, is there some sort of limiting factor on what we can apply it to found at the very core of quantum mechanics? Or some unknown rule for how the universe works which renders what seems theoretically possible in certain situations void? I’m assuming this half step has been explored and was wondering in what way?

I've always enjoyed math in school but it was never anything more to me than fun and useful. I am a practicing scientist in a field in which mathematics is not widely taught or used (with exceptions of course), so I never took much math courses during my studies - a single semester intro to calculus and basic linear algebra were it. Although I learned the basics of those two, I never truly understood them at a level deeper than just algebraic manipulation of symbols. In the years since I've taught myself the math I need here and there as I explored more topics in statistics, modeling and probability related to my research.

A year and a half ago I became obsessed with a problem about a novel statistical distribution. I quickly realized I am way over my head and started buying tons of math books and started teaching myself more and more math. After months of struggle and many sleepless night I was eventually able to solve it and speed up the estimation of my distribution by many orders of magnitude. But more importantly, that experience made me fall in love with math. Over the past year I've had many moments when things finally connected. Like, I vividly remember the moment I realized that matrices are just functions, that matrix multiplications is function composition, that you can represent operators like derivatives as matrices, and so on - so much of different parts of math suddenly felt connected. Suddenly things like taking the exponential of a matrix or an operator made perfect sense, when coupled with Taylor series expansions. Or when I understood how you can construct the natural numbers from the null set and successor operations - it opened up a huge realization about what it means for something to be a symbol and to have semantics. What it means for something to be a mathematical object. Learning about the history of complex numbers as rotations, the n-th roots of unity, Euler's equation and so on, I had one moment when the connection between trigonometric functions, hyberbolic functions and exp() suddenly clicked and brought me so much joy.

The more I learn, the more beautiful and addicting I find math as a whole. I've been studying it in a incredibly haphazard and chaotic way - I don't think I've worked through a single textbook in linear order. I jump from calculus to combinatorics to algebra to set theory to category theory topics as my questions arise from one topic to another. In some ways that has been frustrating since, especially in the beginning it was difficult to find sources at my desired level - when I had a particular question, I would end up on a rabbit hole where the sources I find to address it presumed too much prior knowledge, but the more beginner sources that would give me that background I found to be incredibly dull. At the same time, it has been very rewarding, since my learning has been entirely driven by the need to understand something specific at a particular moment to solve a particular problem (either practical, or just because I was trying to solve some puzzle from prior learning).

For example, I've been exploring combinatorics in the last few months, and I've become obsessed with understanding things like Sterling numbers, various transforms of sequences, and so on. It's funny, but I care (at this moment) almost 0 about the combinatorial interpretations but I am just fascinated with polynomial structures and generating functions as mathematical objects for some reason. Last year I read Generatingfunctionology and the opening line "A generating function is a clothesline on which we hang up a sequence of numbers for display" blew my mind and made me appreciate polynomial sequences immensely. Yesterday I suddenly realized that two-element recurrence relations like those for binomial coefficients and Stirling numbers can be represented as infinite matrices with two diagonals filled in (and then quickly found out that I basically reinvented production matrices as defined in this paper). That you can get any binomial/stirling coefficient row n by raising these matrices to n-th degree and just use the resulting matrix to multiply the initial [1,0,0,...] starting vector. the And suddenly I felt like I truly understood the objects that binomial coefficients and Stirling numbers represent, and various relations between binomial and stirling transforms of sequences.

Anyway, long-story short, I just wanted to do the opposite of venting and express my excitement and growing love for math. I'd love to hear others' stories - do you remember what made you fall in love with math? What are your current obsessions?

April Fools! I've been waiting month to post this.

Now in a serious attempt to spark discussion, do you think certain long proofs have much simpler ways of solving them that we haven't figured out yet? It might not seems useful to find another proof for something that has already been solved but it's interesting nonetheless like those highschoolers who found a proof for Pythagoras' Theorem using calculus.

Hi everyone,

I’m currently looking for a reference on PDEs to delve deeper into the subject. From what my professors have told me, there are two schools of thought in PDEs:

1.  Those who like and use functional analysis whenever they can, and try to turn PDE problems into problems of functional analysis (or Fourier analysis).
2.  Those who don’t really like to use it and prefer to compute things ‘by hand.’

I really like the first school of thought and I don’t like at all Evan’s presentation in his book. Moreover, I already know about Brezis book.

Does someone know about a rigourous book about PDEs that uses a lot of functional analysis (or Fourier analysis) in their treatment of PDEs ?

Thank you.

What are the prerequisites for the book by Saunders Mac Lane, "Categories for the Working Mathematician"?

I have been working on an alternative number system for a while and have just finished writing up the main results here. The results are pretty interesting and include some new lattices and Heyting algebras but I'm struggling to find any applications. I'm looking for people with more number theory expertise to help explore some new directions.

The main idea of productive numbers (aka prods) is to represent a natural number as a recursive list of its exponents. So 24 = [3,1] = [[0, 1], 1] = [[0, []], []] ([] is a shorthand for [0] = 2^0 = 1). This works for any number and is unique (up to padding with zeros) by fundamental theorem of arithmetic.

Usual arithmetic operations don't work but I've found some new (recursive) ones that do and kind of look like lcm/gcd. These are what form lattices - example for 24 (written as a tree) below.

This link contains all the formal definitions, results and interesting proofs. As well as exploring new directions, I'd also love some help formalizing the proofs in lean. If any of this is interesting to you - please let me know!

Edit: fixed image

Last semester, I didn’t do that well in my discrete math course. I’d never been exposed to that kind of math before, and while I did try to follow the lectures and read the notes/textbook, I still didn’t perform well on exams. At the time, I felt like I had a decent grasp of the formulas and ideas on the page, but I wasn’t able to apply them well under exam conditions.

Looking back, I’ve realized a few things. I think I was reading everything too literally -- just trying to memorize the formulas and understand the logic as it was presented, without taking a step back to think about the big picture. I didn’t reflect on how the concepts connected to each other, or how to build intuition for solving problems from scratch. On top of that, during exams, I didn’t really try in the way I should’ve. I just wrote down whatever I remembered or recognized, instead of actively thinking and problem-solving. I was more passive than I realized at the time.

Because of this experience, I came away thinking maybe I’m just not cut out for math. Like maybe I lack the “raw talent” that others have -- the kind of intuition or natural ability that helps people succeed in these kinds of classes, even with minimal prep. But now that I’m a bit removed from that semester, I’m starting to question that narrative.

This semester, I’m taking linear algebra and a programming course, and I’ve been doing better. Sure, these courses might be considered “easier” by some, but I’ve also made a conscious shift in how I study. I think more deeply about the why behind the concepts, how ideas fit together, and how to build up solutions logically. I’m more engaged, and I challenge myself to understand rather than just review.

So now I’m wondering: was my poor performance in discrete math really a reflection of my abilities? Or was it more about the mindset I had back then -- the lack of active engagement, the passive studying, the exam mentality of “just write what you know”? Could it be that I do have what it takes, and that I just hadn’t developed the right approach yet?

I’d really appreciate honest and objective feedback. I’m not looking for reassurance -- I want to understand the reality of my situation. If someone truly talented would’ve done better under the same circumstances, I can accept that. But I also want to know if mindset and strategy might have been the bigger factors here.

Thanks for reading.

So im currently a senior in college going to graduate with a double major in computational biology and statistics. Through my majors I've been able to take into calc courses up to diff eq, linear algebra, 2 math bio courses, stat inference, probability theory, bayesian statistics, 2 linear regression courses, and a good mix of CS and data mining courses with regards to math and a mix of biology courses as well. Most of my research in undergrad has been in bioinformatics and doing a lot of data and statistical analysis on cancer genetic data. Now im getting a lot more interesting in math modeling of biological systems and im wondering if there are any other areas of math I should study before jumping all into the research im hoping to do (im going to grad school for a PhD in comp bio in the fall btw). Any advice would be really appreciated :D

You struggle for months, nearly lose your sanity, and finally - FINALLY - prove the result. You submit, expecting applause. The response? “Too trivial.” So you generalize it. Submit again. Now it’s “too complicated.” Meanwhile, someone else proves a worse version and gets published. Mathematicians, we suffer in silence.

Title. I will be going through this process soon, and I would love to hear any stories or advice people have!

I know some math forums, but they all seem to be organized in a Q&A format. I’m wondering if there’s a platform focused on sharing notes and articles.

This is a question I made up myself. Consider a simple closed curve C in R^2. We say that C is self similar somewhere if there exist two continuous curves A,B subset of C such that A≠B (but A and B may coincide at some points) and A is similar to B in other words scaling A by some positive constant 'c' will make the scaled version of A isometric to B. Also note that A,B can't be single points . The question is 'is every simple closed curve self similar somewhere'. For example this holds for circles, polygons and symmetric curves. I don't know the answer

Recently I started learning about algebraic structures and I created a very basic one (specifically a commutative magma) for fun, would you say this is useless/pointless or not? also why or why not?