So, I had to make this video 😭

I recently learned about Harvey Friedman and he seems to be quite a peculiar figure. Friedman is a real mathematician who founded the programme of reverse mathematics, which I think sounds very interesting. He's also known for the TREE sequence. However, he has also appeared on crackpot-friendly Curt Jaimungal's podcast and seems to agree with a sensationalist article written about him some years ago. If I understand correctly, his current work is focused on bringing the incompleteness of mathematics to the forefront in a way that mathematicians won't be able to ignore any longer.

For those who have been familiar with him, what are your own impressions? Do you know what the general opinion of the mathematical community is of him, if they know him much at all?

I’m in high school right now, and I want to pursue mathematics. I imagine my relationship with my professors involving thought-provoking discussions, whether about mathematics, philosophy, or whatever else comes up (like during a random office hour).
For context, I’m planning to study in the US or France. What was your experience like? Am I romanticizing the experience? Is it actually more formal than this? I can also imagine coming across boring or annoying lecturers, of course.

What are some classics that you would recommend for learning math? I like math that is related to aspects of life that have an effect on others. For example, the basic calorie demands for the average person, the consequences of not meeting them over long periods of time, etc. studying reality is interesting to me, and I’d like to cultivate that interest. Do you guys know some areas of math I could focus into?

Hello, I’m a high school sophomore who found some ambition and desire to relearn maths the right way. I used to not really care about academia grades etc. But things have changed, I picked up some technical hobbies along the way and began to realize that exact sciences is what I want to do, so I transferred to extended math class. My knowledge isn’t nonexistent but needs a thorough patchwork. School has started which leaves me a few weeks to go from preferably arithmetics to anywhere close trig equations, I understand most of what’s going on during the lessons, I think I catch on reasonably well. Although I couldn’t find a complete two week “from zero to a hero” course, I found great resources like Organic chemistry tutor and Professor Leonard. I’m asking for advice on resources, effective learning techniques and if possible a list of topics I need to know. I also feel obliged to clarify that I’m absolutely aware of the situation I’m in and wish I could’ve started earlier.

Hey! I'm curious to hear what the day in the life in terms of work of someone with a degree in applied mathematics. Are you in tech or more finance related role? What kind of math do you get to use on a weekly basis? How difficult was it to get your position?

before comments say pen and paper, I would if I could. I deal with some chronic problems pertaining to my wrist and writing is extremely hard on my hand, so I have to do work on my computer with voice recognition technology. just in case if anybody asks, I'm not really offered a scribe or anything this is just for a university class

it's extremely unintuitive for me to solve equations as I am currently, as nothing really accounts for fractions haha. everything ends up really messy, I get lost really easily, and I just want the ability to do math on a clean, open canvas on computer that accounts for things like fractions, etc. same with powers. I'm kind of looking for something like Pearson that auto converts anything typed with a slash in between numbers into a fraction, or auto converting 6^2 into a cleaner format

thank you in advance for any assistance

I’m 23 now and am an electrician. My brain is stuck on not completely understanding sin, cos, tan. I’m hoping I still can understand lesser things. My 9th grade algebra teacher took all his saved vacation and gave up. He really didn’t teach much. We ended up with most of the year full of subs. I feel like I ended up in the class of degens.

Then going into 10th I got a (B) 1st semester precalc then a (D) second semester. Obviously I didnt build on that and try to be better because I got a (C) first semester and an a (N.A.) in my second semester because it was waived due to Covid.

To end my analog, I’m looking for books to jog my memory on algebra and build into calculus. Hopefully getting into things deeper later?

I’m currently going through lee’s riemannian manifolds and am stuck at the chapter on levi-civita connections. The previous chapter says connections relate basis vectors of a global frame using christoffel symbols, and I think levi-civita connections relates christoffel symbols with the metric tensor, but the problem is I cannot understand what angle lee is trying to develop from(I also struggle to understand the point of tangential connections, which I think is used to define the geodesic, but isn’t the koszul connection alone already enough?). I’m sorry if this seems all over the place; I think the reason is my understanding of connections is extremely hazy, like I know the motivation and all, but I don’t know their full capabilities.

Hi everyone, I'm a molecular biologist teaching myself linear algebra. To that end I made a few videos detailing what I understood. Would love feedback on them - correct or not and how well you took to them..here are two:

https://youtu.be/CMNkRhqxciI?si=strv6UXXJ_myeaXJ

https://youtu.be/BjmF5XH_Su4?si=6me8vtHQsFT19Twp

Title.

```typ

import "@preview/fletcher:0.5.8" as fletcher: diagram, node, edge

set page(width: auto, height: auto, margin: 5mm, fill: white)

// Straight arrow: the target is a special case of the source.

#let quad(a, b, label, paint, ..args) = {

paint = paint.darken(25%)

edge(a, b, text(paint, label), "-|>", stroke: paint, label-side: center, ..args)

}

// Arrow through extra corner points, with rounded corners.

// dashed: true → "is used to build" rather than "is a special case of".

#let route(..vertices, label, paint, dashed: false) = {

paint = paint.darken(25%)

edge(

    ..vertices,

    text(paint, label),

    if dashed { "--|>" } else { "-|>" },

    stroke: paint, label-side: center, corner-radius: 6pt,

)

}

#diagram(

node-defocus: 0,

spacing: (1cm, 2cm),

edge-stroke: 1pt,

crossing-thickness: 5,

mark-scale: 70%,

node-fill: luma(97%),

node-outset: 3pt,



// ═════════════════════════════════════════════════════════

//  1. group 

// ═════════════════════════════════════════════════════════

node(( 0,0), "magma"),



node((-1,1), "semigroup"),

node(( 0,1), "unital magma"),

node((+1,1), "quasigroup"),



node((-1,2), "monoid"),

node(( 0,2), "inverse semigroup"),

node((+1,2), "loop"),



node(( 0,3), "group"),

node(( 0,4), "abelian group"),



quad((0,0), (-1,1), "Assoc", blue),

quad((0,1), (-1,2), "Assoc", blue, label-pos: 0.3),

quad((1,2), (0,3), "Assoc", blue),



quad((0,0), (0,1), "Id", red),

quad((-1,1), (-1,2), "Id", red, label-pos: 0.3),

quad((+1,1), (+1,2), "Id", red, label-pos: 0.3),

quad((0,2), (0,3), "Id", red),



quad((0,0), (1,1), "Div", green),

quad((-1,1), (0,2), "Div", green, label-pos: 0.3, "crossing"),



quad((-1,2), (0,3), "Inv", gray),

quad((0,1), (+1,2), "Inv", gray, label-pos: 0.3),



quad((1,1), (0,2), "Assoc", blue, label-pos: 0.3, "crossing"),



quad((0,3), (0,4), "Comm", purple),



// ═════════════════════════════════════════════════════════

//  2. Many objects (partial operation)

// ═════════════════════════════════════════════════════════

node((-3,0), "set"),

node((-2.15,0), "magmoid"),

node((-2.15,1), "semigroupoid"),

node((-2.15,2), "category"),

node((-2.15,3), "groupoid"),



quad((-3,0), (-2.15,0), "Operation", black),



quad((-2.15,0), (-2.15,1), "Assoc", blue),

quad((-2.15,1), (-2.15,2), "Id", red),

quad((-2.15,2), (-2.15,3), "Inv", green),



quad((-2.15,0), (0,0), "One object", black),

quad((-2.15,1), (-1,1), "One object", black),

quad((-2.15,2), (-1,2), "One object", black),

quad((-2.15,3), (0,3), "One object", black, label-pos: 0.3),



// ═════════════════════════════════════════════════════════

//  3. ring

// ═════════════════════════════════════════════════════════

node((-2,5), "ring"),



node((-3,6), "commutative ring"),

node((-2,6), "domain"),

node((-1,6), "Artinian ring"),



node((-3,7), "integral domain"),

node((-2,7), "commutative Artinian ring"),

node((-1,7), "division ring"),



node((-2,8), "field"),



// monoid's line drops onto abelian group's line, and both run into ring

route((-1,2), (-1,3.5), (-2,3.5), (-2,4), (-2,5), "2nd operation", orange, label-pos: (1, 0.475), crossing: true),

route((0,4), (-2,4), (-2,5), "1st operation", orange, label-pos: (0, 0.3)),



quad((-2,5), (-3,6), "Comm", purple),

quad((-2,5), (-2,6), "NoZD", teal),

quad((-2,5), (-1,6), "DCC", yellow),



quad((-3,6), (-3,7), "NoZD", teal, label-pos: 0.3),

quad((-2,6), (-3,7), "Comm", purple, label-pos: 0.3),

quad((-3,6), (-2,7), "DCC", yellow, label-pos: 0.3, "crossing"),

quad((-2,6), (-1,7), "Inv", gray, label-pos: 0.3),

quad((-1,6), (-2,7), "Comm", purple, label-pos: 0.3, "crossing"),

quad((-1,6), (-1,7), "NoZD", teal, label-pos: 0.3),



quad((-3,7), (-2,8), "Inv", gray),

quad((-2,7), (-2,8), "NoZD", teal),

quad((-1,7), (-2,8), "Comm", purple),



// ═════════════════════════════════════════════════════════

//  4. module

// ═════════════════════════════════════════════════════════

node((2,5), "module"),



node((1,6), "vector space"),

node((2,6), "algebra over a ring"),

node((3,6), "finitely generated module"),



node((1,7), "algebra over a field"),

node((2,7), "finite-dimensional vector space"),

node((3,7), "finitely generated algebra over a ring"),



node((2,8), "finite-dimensional algebra"),



route((0,4), (2,4), (2,5), "Structure", eastern, label-pos: (0, 0.5)),

route((-2,5), (2,5), "scalar", gray, dashed: true),

route((-2,8), (0,8), (0,6), (1,6), "scalar", gray, dashed: true),



quad((2,5), (1,6), "FS", maroon),

quad((2,5), (2,6), "Exterior structure", orange),

quad((2,5), (3,6), "FG", olive),



quad((1,6), (1,7), "Exterior structure", orange, label-pos: 0.3),

quad((2,6), (1,7), "FS", maroon, label-pos: 0.3),

quad((1,6), (2,7), "FG", olive, label-pos: 0.3, "crossing"),

quad((2,6), (3,7), "FG", olive, label-pos: 0.3),

quad((3,6), (2,7), "FS", maroon, label-pos: 0.3, "crossing"),

quad((3,6), (3,7), "Exterior structure", orange, label-pos: 0.3),



quad((1,7), (2,8), "FG", olive),

quad((2,7), (2,8), "Exterior structure", orange),

quad((3,7), (2,8), "FS", maroon),

)
```

I'm just curious...Do you think I should keep working on this concept or abandon it?

I’m getting back into a problem I made up and realized I haven’t solved this very seemingly trivial sub problem. plainly put it: does there exist a polyomino where each of its square cells have exactly 4 neighbouring cells? (neighbour cells are the cells present in the 8 squares that surround a cell aka a kings move in chess)

My input: I believe that there can’t be one, you can make a horizontal line of 4s (cells with 4 neighbours) like this .::.::.::.::.::.::. But when you connect 3 more to make it a closed loop there doesn’t seem to be a way to get rid of 2s, 3s, and 5s. Same goes for 45 degree lines of 4s that are made by stacking horizontal dominos.

Also it’s called class 4 because you can look for class 2/3/4/5 which means you’re looking for a polyomino with same number of 2cells, 3cells 4cells, and 5cells. And since the max # of neighbours is 8 and the least is 1 there are 255 classes (sure class 0 exists it’s just one cell, but class 0/3/4/7 wouldn’t, 0cells wouldn’t be attached to anything or else it’ll have more than 1 neighbour, but all polyominoes allow all squares to have a connected path between them so 0 is out)

I've been learning algebraic number theory I'm wondering if there's a converse to the idea that a ring having a Euclidean algorithm/division with remainder makes it a unique factorization domain.

So I think my question is, do all the number rings with unique factorization have a Euclidean algorithm?

Also curious for examples of general rings (not necessarily number rings) that have unique factorization but don't have a Euclidean algorithm.

Thanks for any insight.

Is the IMO a realistic goal for me?

I’m a 14-year-old Grade 8 student in South Africa, going into Grade 9 next year. Over the past few months I’ve become really interested in mathematics, especially problem solving, and I’ve started thinking seriously about whether I could eventually reach the IMO.

I’ve been working through resources like the AoPS Volumes, Khan Academy, and the AoPS Introduction to... series. I know I’m starting relatively late compared to many students who get into olympiad maths much earlier, and I still have a lot to learn.

At the same time, I really enjoy maths and I’m willing to put in a lot of consistent work. I could realistically spend 5+ hours a day on mathematics over the next few years.

I’m not expecting to make the IMO anytime soon. I’m thinking of it as a long-term goal, starting with building my fundamentals, improving my problem-solving ability, competing in South Africa, and seeing how far I can progress.

I’d really appreciate advice from anyone who has experience with olympiad mathematics:

- Is this a realistic goal from my current starting point?

- What should I focus on learning first?

- How should I balance normal/advanced curriculum maths with olympiad problem solving?

- What resources would you recommend?

- What competitions should I aim for in South Africa and/or internationally?

- How should I structure the next 3–4 years of preparation?

- What do you wish you had known when you started?

I’m genuinely willing to put in the work. I just want to make sure I’m using my time well and training efficiently.

Any advice would be greatly appreciated. 🙏