Yang Mills Existence part of "Yang Mills Existence and Mass Gap" has been proved and formalized in Lean!

These last two years in my BSc program have been trying for me. I just did a Measure Theory midterm today, and I feel defeated.

There’s nothing that I love more than doing math. It has become a great part of my life and I cannot imagine my life doing something other than math.

That said, I’m starting to think I’m not as good as I thought I was. I’m not a fast thinker. I’m a slow, methodical thinker that has to have a grasp on every small detail to make sense of the whole situation. I can’t quickly come up with original counterexamples and get working with new definitions in a 2 hour test.

As I got to advanced courses, I started getting worse grades. I don’t feel like I’m a bad student. I get some A(+)’s, mostly B’s, and occasional C’s. The more advanced the courses, the more effort, time and stress I have to put up. I got to really dread upcoming tests.

I love when I get to work through a textbook myself and when doing my own personal research, but I dislike having the pressure of a course workload and the dread of the tests I have to take. It takes a toll on me.

For this Measure Theory course, we have a main textbook and a secondary one. I worked through almost all the exercises of the main textbook, but I didn’t do many exercises from the secondary one.

The midterm consisted almost entirely of exercises from the second textbook. I feel like I put more a lot of effort just to get a lowly D.

Now, while I love math, I don’t want to have my whole day to day be consumed in math. I know the answer is to put more effort in, but I struggle justifying disrupting my current study-life balance.

I want to pursue a PhD in math, but part of me thinks: “If I’m struggling this much at the BSc level, will I even make it to the PhD?”.

Now, as everything in life, there are people who are very good at what they do and some that are not so good.

Do you consider yourself to be a “mediocre” mathematician? How do you deal with higher level math? How do you reconcile being “mediocre” with your self-worth? Are there any “mediocre” mathematicians doing well on their MSc/PhD here?

Is the answer just to put more effort in? I’d like some advice, if possible. Thanks.

Hello. I'm a 30+ adult & I would like to relearn math from scratch. I used to struggle with it when I was in school & something happened recently (too personal to share) that ignited the idea of me relearning math. I'm serious about this. How can I approach this? How and where do I start? And what advice can you give me to be able to stay consistent in learning? Any resources are welcome too. Thank you all in advance.

This might be a little personal, but it’s still related to maths.

I honestly don’t feel like I’m fit to be a math major. For starters, I struggle to form a train of thought when solving math problems. It’s like my brain refuses to focus on problem, and I just end up staring at it. If it’s not immediately obvious, then the chances of me coming up with a solution are dim.

This happened today in class. We were learning about double integrals, and we got to a problem involving trigonometry. I sat there and tried solving the problem, before I did something wrong. I didn’t know what exactly I did wrong, but I just knew something wasn’t right. I tried scanning the problem for any mistakes, but I couldn’t form any type of thought process on the problem. I eventually gave up and waited until my professor solved it.

I also have pretty crappy memory, which is unfortunate because I relied on memory for the entirety of precalculus - calculus 3. I often times forget very elementary things. For example, when were doing double integrals, we got to a problem with exponents. It was like I forgot that e^r + 2s can be separated into e^r • e^2s. I also still don’t remember the cosine and sine of basic trigonometry angles. I still keep a formula sheet for that.

Another example pertains to my exam. I spent all of yesterday cramming, and a part of this morning as well, just to open up the practice test, and forget how to solve the first question. I had to look at the example and got it immediately. So now I’m going through the entire test of 37 questions, and will probably have to relearn everything.

Stuff like this is so demoralizing. I wanted to major in pure mathematics because I thought I liked it and it was the only thing I’m semi good at. Is there just a problem in my thought process or is it something else?

I also have ADHD (I am medicated, but it’s a really low dose because I just started it.), but I’m sure many people with ADHD (and more) still do amazing in math.

I recently finished Music Of the Primes and loved it. I'm an A level student doing Maths and FM, which other books on number theory (or some other topic) would everyone recommend?

I'm at chapter 7 of the book, in which Tao introduces the notion of series, I'll introduce 2 definitions below (as context) before getting to the problem.

First, Tao defines a finite series on a sequence as follows: (See [1] in the attached image)

And then he defines summations on a finite set X and a function f: X -> R as such: (See [2] in the attached image)

[It's also proven later that this definition is well-defined and gives off the same value for every bijective g]

So far so good, the problem is that on a lemma, he claims the following: (See [3] in the attached image)

I get the maths Tao is trying to convey, but shouldn't the left hand side of the equation be syntactically invalid? Regarding the inner sum, y in Y isn't a valid input for the function f(x,y) as it isn't an ordered pair (x,y).

I think a way to resolve this would be to define double-summation on a function through another 'intermediary' function (f_x)(y) (See [4] in the attached image)

and then we can proceed with proving the proposition by using the original definition of summations on a finite set.

Edit: Also the x subscript in f_x is (meant to be) purely notational, it has nothing to do with the actual input x. Note also that [3] is not a definition, it's presented as a lemma (so the fact that the LHS is syntactically invalid does matter here) and that the image above is not Tao's book, I only wrote the necessary definitions and the lemma (that I have a problem with) down for the sake of discussion.

In topology, a pair of pants is a surface with three boundary components: one waist opening and two leg openings.
Suppose we take two identical pairs of pants and sew them together by matching the two leg openings of the first pair with the corresponding two leg openings of the second pair. The two waist openings remain unsewn.
What is the genus of the resulting surface? Is it a torus with two boundary components (genus 1), or am I misunderstanding how the sewing affects the topology?
I’m particularly confused about whether the large opening in the middle, formed by joining the leg openings, counts as an additional hole.

Hello all, I just started my MSc in statistics, after a background in Economics where I desperately tried to bridge the gap through self study. In my thesis I had the chance to read some chapters in Optimal Transport and manifold theory. It was really a great opportunity but still my understanding was very limited cause I have never done even an analysis course (just basic calculus and linalg, and measure theory which I struggled with during the proofs and exercises).

Now I decided for first semester to follow mathematical statistics and probability theory, cause I really really wanted to follow maths-theoretical modules and also have the right foundation for further courses. I thought that taking only these two courses would be beneficial because I'd have more time to study them deeply, but I feel like I am just taking the wrong choices: I am so slow at understanding how to do the Prob theory exercises because I feel like they use analysis tools stuff which I am not familiar with, and also I am 'familiar' with the measure theoretical tools but I keep going back to re read stuff like continuity of measures, or the fubini theorem which I didn't even know the existence of. So yeah maybe not the brightest choice, I was thinking of going through abbott's understanding analysis WHILE taking these two courses but still it feels like I am so behind in terms of mathematical knowledge.

So my question is basically if maths should always be studied in a linear order, or if some things can be done while also taking more difficult courses; I understand that it makes sense to build knowledge starting from basic stuff like analysis and only after build up on it, or e.g. take probability theory only after having studied measure theory, but is it sometimes somehow possibile to learn some things backwards or in parallel with the courses that actually require them? I mean like taking a course we don't have the background of, like prob theory, and learn the analysis tools used in the proofs and exercises from it? Or is it just more correct to give up and take certain courses only after making sure that we have the prerequisites?

i know the basel problem proof with the sin(x)/x product but it feels like magic that a circle constant shows up in a sum of integers. is there a more intuitive reason or is it just that sin has those zeros at multiples of pi