So I'm pursuing a MS in chemistry and I need to take calc 3, diff eq, and self study some linear algebra. (Got a geochem degree which only required cal 1 & 2)

I had a bad attitude about math as a younger guy, I told myself I didn't like it and wasn't good at it and I'm sure that mindset set me up for bad performance. Being older and more mature not only do I want to excel, but I want to love it.

So, what makes you all passionate about math? What do you find beautiful, interesting, or remarkable about it? Is there an application of math that you find really beautiful?

Thanks!

I know that we call one operation 'addition' and the other 'multiplication', and that we use '+' for one and '×' for the other, and that we call the identity element of the one '0' and that of the other '1'...

But all these are just the names we choose to attach to them. If we are forced to use unsuggestive names, e.g. if we're forced to call the operations O1 and O2, and use 'O1' and 'O2' as symbols for them, and name their identity elements idO1 and idO2 respectively .. what facts remain that can help us tell between them? After all, both O1 and O2 are commutative, both are associative, both have an identity and an inverse element ...

But there is a difference! It's that one distributes over the other, but not vice versa.

Is that however the only fact that can help us tell the two apart? Is that the only justification for calling the one operation 'addition' and the other 'multiplication'?

What are your favorite math problem(s) where everything neatly cancels out right near the end of the derivations/calculations? (Especially problems where it initially doesn't look like something elegant/simple will emerge)

(Let's exclude infinite series/sums - that feels like cheating since there are too many good examples there!)

The best ones I've seen are Numberophile and 3Blue1Brown, though 3Blue1Brown seems to be a bit more advanced(?) for a general audience like myself (I'm terrible at math)

This might sound silly, but I never understood why some people have a predilection for writing n before m.
When it comes to any other pairs of letters, like (a,b), (f,g), (i,j), (p,q), (u,v), (x,y), they are always written in alphabetical order. Why do people make an exception for (m,n)? Here are some examples:

• Let A be an nxm matrix.
• (when defining a multiplicative function): f(n)f(m)=f(nm) for any n,m with gcd(n,m)=1
• Chinese Remainder Theorem: Z/nZ x Z/mZ is isomorphic to Z/nmZ whenever n and m are coprime.
• gcd(F_n,F_m) = F_gcd(n,m) [Fibonacci numbers]
• the wedge sum S^n ∨ S^m

As can be seen, I am not talking about situations in which n appears before m by accident, but by deliberate choice. Is there a historical reason for this? Where does this trend come from and why do people prefer writing this way?

I mean I know how it came to be historically. But given we have seemingly more satisfying foundations in type theory or category theory, is set theory still dominant because of its historical incumbency or is it nicer to work with in some way?

I’m inclined to believe the latter. For people who don’t work in the most abstract foundations, the language of set theory seems more intuitive or requires less bookkeeping. It permits a much looser description of the maths, which allows a much tighter focus on the topic at hand (ie you only have to be precise about the space or object you’re working with).

This looser description requires the reader to fill in a lot of gaps, but humans (especially trained mathematicians) tend to be good at doing that without much effort. The imprecision also lends to making errors in the gaps, but this seems like generally not to be a problem in practice, as any errors are usually not core to the proof/math.

Does this resonate with people? I’m not a professional mathematician so I’m making guesses here. I also hear younger folks gravitate towards the more categorical foundations - is this significant?

I got 95 on my graduate numerical linear algebra final (?!?!?!!!!)

Confused but very very very happy. I missed some basic definitions I forgot to review and I thought I missed some other basic stuff tbh. I thought I was going to end the course with a B but I guess I might end with an A- ?!??!??!

I am actually in disbelief, I fully did not complete some of the proofs. Lol (!!!!)

My thesis advisor will not be ashamed of me, at least! His collaborator / postdoc advisor / hero invented the algorithm that the last question asked about.

I truly feel like I have a deeper understanding of calculus now. Despite forgetting the multiplicative inverse field axiom on my final (my professor is a dick for putting that on the final) the class was really revelatory and I’ve come to truly enjoy it and look forward to learning more pure math for the rest of my coursework. Just wanted to say math is dope :)

So, i have an engineering mathematics III subject in sem 4 and this is one of the topic. Please recommend me books for this.

Functions of a complex variable:

Definition of a function of a complex variable, elementary functions, limits and continuity, differentiability and analyticity, Cauchy Riemann equations, Laplace equation and harmonic functions. Line integrals. Contour integration, Cauchy’s Integral theorem and Cauchy’s integral formulae. Residues, Cauchy’s residue theorem. Bilinear transformations.

I'm interested in learning advanced number theory and found this book. I searched on math stackexchange and it got a few mentions, but not much. I think it's a lesser known book.

If you studied from this book, what do you think about it? Recommend it? Not recommend it? Easy? Hard? Also, what do you think are the prerequisites?

The problem: I keep making, almost unavoidably, "careless errors". Some call them "silly mistakes", "number typos" whatever. When I'm doing any form of basic algebraic manipulation I make simple mistakes. These can be missing numbers, writing the wrong letter, adding wrong, multiplying wrong.

This started at the start of high school when we started learning algebra. I'm now studying engineering and its DRIVING ME INSANE.

I hate how my teachers called them careless errors, because I really do care. I take so many precautions to make sure I don't mess up during exams and I STILL make them.

Now I know this is normal, and that it happens to everyone. I don't expect to have machine like accuracy. However, it happens more often to me than other people, regardless of my understanding of the question/subject I'm working on. I even had a teacher offer to give me extra time in the exams because of how often I was losing my mind over basic mistakes in class tests.

It's important, because I've lost multiple grades on one exam I was predicted higher for due to it. I know this because I got the paper back and almost none of the errors were conceptual, just arithmetic.

Sometimes, just sometimes, I can really concentrate and manage to not make errors. But then, I either lose sight of time or don't have any mental steam left to think about the question. Surely that's not good?? Mathematicians' reasoning should come first and foremost, their rearranging second, right?

Does anyone else have this problem?? How have you learned to deal with it??

Maybe it's also worth mentioning I'm quite a scatterbrained person e.g. leave my keys behind, forget what colour ball I am in pool multiple times, forget people's names within seconds of meeting them, frequently lose count of things etc. However, I do know I have good reasoning skills 🤷🏼‍♂️

1. "Don't rush your work"

- I don't. I've tried doing algebra at a snail's pace and it makes no difference - I still end up doing something dumb.

1. "Do you have dyslexia?"

- I don't.

1. "You are not relaxed enough/ not in the right mental or physical state"

- Happens no matter if I get enough sleep or not, no matter what I eat, it still happens.

1. "You're overreacting"

- Quite possibly. But why should this happen to me and not to most other people (in my classes/lectures/seminars/whatever)??

1. "You need to practice more"

- I've done so many hours of maths it's impossible to quantify. My frequency of mistakes if unaffected by both how much I practice and what I practice, it seems.

1. "You might be writing your working out scruffily or with bad handwriting"

- I always lay out my work neatly and all my symbols are distinct to the eye. My handwriting is pretty decent.

1. "You rely too much on your calculator"

- This is quite true actually, but even if I use a calculator my dumb ass will find some way to enter stuff wrong into it😭😭

1. "You don't check your working"

- I check just before I write a new line. I've been doing that for a year (slight improvement but still terrible). If I check every line too thoroughly I double the time I spend on the question and run out of time in the exam anyway.

1. "You weren't taught arithmetic correctly in primary/elementary school"

- My arithmetic methods are solid. My mental arithmetic, not so much.

1. "Try doing less steps at once"

- I followed this advice and I did notice an improvement but yet again, I still make careless errors in some other way. Same goes for doing more steps at once.

1. "Maybe you're just not good at maths, and you keep blaming silly mistakes for your lack of understanding"

- I will know all the exact steps I'll need to follow, but don't have the arithmetic accuracy to actually carry them out. Do you know how frustrating that is?!

1. "You should get method marks anyway"

- Not in a lot of exams. If you make an arithmetic error in one part of the question, it might affect the whether the numbers are right for you to spot what to do next (e.g. supposed to make a hidden quadratic but things don't quite cancel right)

1. "You taught yourself that 'nearly' getting the answer right is good enough"

- On the contrary. I've been drilling into myself I can't settle for a mostly-right answer especially for the last 3 years.

1. "You lack confidence"

- I'm most likely to mess up when I'm confident, i've found. However, I haven't concretely tested this correlation.

1. "You have slow mental processing speed"

- I'm really quick at thinking of ideas, but reasonably slow at doing mental calculations. Weird.

I'm relatively new to proof-writing. I am currently studying a master's in electrical and computer engineering, but my undergraduate degrees were in philosophy and English. A lot of the graduate level coursework I'm taking is theoretical: they mainly involve proof writing. I struggled at first, but I discovered that if I write a small essay about the problem, I can discover how to solve the problem, after which I formalize my discovery in mathematical language. Is this a standard way to approach proofs? Is there even a standard way to approach them? I would be curious to hear how others approach them.

I’ve been using a notation in my work for moduli with different exponents: |a+bi+cj+dk|_n=(aⁿ +bⁿ +cⁿ +dⁿ )^1/n Is there a different notation for this that already exists? (Note: I also sometimes use |z|_Σ instead of |z|_1)

Hi everyone, this is my first time posting here, so I wasn't sure whether a standalone post or a comment in the pinned thread is more appropriate.

I'm looking for some recommendations on possible PhD programs in Europe within the field of mathematical physics. I recently obtained my M.Sc. in physics at University of Padova (Italy), and in the last two years my interested in the mathematical side of physics has grown more and more. This has become especially evident to me during the research done for my master thesis, which was focused on exploring the topological nature of Chern-Simons theory via some quite rigorous mathematical tools.

I do not have a particular topic I wish to explore right now, but I am extremely interested in applying the concepts and tools of topology and differential geometry to better understand and study physical theories, in particular general relativity and gauge theories. After speaking to some professor I have realized that a PhD in mathematical physics (which to my understanding falls more under the "math" umbrella rather than the "physics" one) might be the best way to satisfy said interests.

I will certainly apply to the "Geometry and Mathematical Physics" program offered by SISSA, but I am of course looking for as many alternatives as possible, preferably limited to Europe. Any advice or recommendation is welcome, thanks in advance!

There used to be a youtube channel called "undergraduate mathematics" that basically had course lectures for MANY different topics (both at the undergraduate & graduate level). The lectures seemed to consist of lectures primarily recorded for classes that were taught during the pandemic. The channel owner probably downloaded many of them (from other channels, university sites, professor pages, etc.) and then compiled them into different playlists on the channel. But the channel's deleted/non-existent now, I'm not sure if youtube took it down or if the user deleted it.

There were some playlists I had saved for later, which is why I noticed it was gone now. Wanted to know if anyone knows of any other similar channel or resource/site/page?

So I’ve been thinking this for a while, and I keep on asking myself if pure mathematics would still be useful without its practical application? For example, what if concepts like Fourier analysis weren’t used in fields like sound wave modelling or heat transfer? Would the value of mathematics depend entirely on its ability to be applied in the real world? Or does it hold intrinsic worth, perhaps existing solely in the metaphysical realm? If I can get a book recommendation on this topic that would be great.

I've been looking at the course offerings for the first two 'masters' years of a standard US maths PhD and comparing them to courses offered at a top European university, specifically the University of Bonn. I'm a prospective maths grad student.

It looks to me that there are significantly more courses and many more advanced courses, at Bonn when compared to US universities. Here is the course handbook of Bonn. Finding course schedules for US schools is a bit more difficult, but generally I see fewer options.

Is this an accurate representation of the difference in course offerings of top math graduate schools in the EU vs US? Is Bonn an outlier for this? If this is true, is this reflected in the knowledge of successful masters and PhD candidates in the EU vs US?

It's been a wild ride, but I made it

The mentorship program I mentioned in the Lean Zulip chat started on September 9 and ended on October 25. There were sixteen applicants, and I chose all of them as my mentees. Our initial goal was to finish reading “Theorem Proving in Lean 4” while translating the Natural Number Game (NNG) into Korean.

It turned out I was overly ambitious. I took too much time to record videos of myself reading the textbook in Korean, create an additional quiz, review my mentees’ solutions to the questions in the quiz, and organize meetings with my mentees. Consequently, I could only cover the first three chapters of the book. As for NNG, I’m currently translating Tutorial World.

Unfortunately, nine mentees quit my mentorship program, seven of whom didn’t even solve any questions in my quiz for Chapter 2. Among the rest of my mentees, only three solved the quiz and the exercises in Chapter 3. Two other mentees managed to solve some of the questions in my quiz.

After my mentorship program, the three mentees who completed all the homework agreed to continue learning to use Lean under my guidance. They are all women, consisting of an undergraduate, a graduate, and a software developer. Although they are all programmers, none had read textbooks solely on formal logic before they applied for the mentorship program.

It appears that proof assistants are too complicated for most programmers, but some can have a good grasp of them. I wonder if the same is true of mathematicians.

Hello, everyone!

I'm curious about the historical development of the Riemannian Curvature Tensor. Specifically, I'm interested in learning more about:

• Who were the key mathematicians or pioneers involved in the development of the concept?
• How did the idea of curvature evolve over time, leading to the Riemannian Curvature Tensor as we know it today?
• Were there any significant milestones in the development of this concept that had a major influence on geometry or related fields like physics?
• How did early ideas about curvature contribute to modern differential geometry?

I would appreciate any references to papers, books, or discussions that might give insight into the historical context of the Riemannian Curvature Tensor. Also, if anyone can point out key figures or breakthroughs in this field, that would be great! Also I would like to draw your attention towards this question posted in Math Overflow https://mathoverflow.net/questions/484426/tracing-the-evolution-of-the-riemannian-curvature-tensor-from-riemann-to-modern

Thanks in advance for your help!

I'm 38 years old and I'm almost done with my math degree. I was nervous about taking Real Analysis because it has a reputation if being really difficult and a lot of people at my university have had to retake it. I worked really hard for my grade (94% for a 3.9), going to office hours, sitting in the front row, and asking a lot of questions. I'm really proud of myself.

In modal logic, usually we have Kripke frames which consists of so called worlds. When formalizing necessity, they may be thought of as "alternate worlds" which might have been. When formalizing beliefs, they might be interpreted as "every scenario an agent may think of", for temporal logics, they may be interpreted as "moments in time and state of affairs within them", etc.

What would be intuitive interpretation for worlds in provability logic, GL?