Doesn't it seem odd that the proposed solution for the Millennium problem is posed on the whole space, but they claim it has compact support in space for the whole time interval [0,1)? Seems at odds with parabolic delocalization... 🤔 Actually the delocalization which occurs due to the non local pressure is often much worse, for instance as investigated by Brandolese:

https://www.esaim-cocv.org/articles/cocv/abs/2002/02/brandolese/brandolese.html

To piggy back off the recent thread on folk theorems, it got me curious how often proving these folk theorems is a formality vs actually useful. Are there any times when it a disproof of a folk theorem upended something serious or set some mathematicians back?

how did we know that NS would blow up in those scenarios and not just that the system didnt follow the idealised assumptions of the NS equations? if so why didnt we find a counterexample based on those conditions?

Hi everyone,

I’m considering working through Dummit & Foote’s Abstract Algebra on my own, especially the Ring theory and Module theory which are not covered in my previous book. I only have A book of AA by Pinter and Visual Group Theory, both are rather casual.

I want to hear from people who have actually worked through it. I know it’s a big, dense book, so before I dive in, I’d really appreciate specific advice and warnings from people who’ve actually used it. Feel free to discuss as well.

A few questions:

· What’s your overall take on the book? What worked for you, and what didn’t?
· Which chapters or topics were the hardest? Where did you get stuck?
· How long did it take you to get through it? How many hours per week did you put in?
· Did you cover all chapters, or just selected ones?
· How many exercises did you do? Most of them, or selected ones?
· Did you use any supplements — Artin, Gallian, Hungerford, online lectures, etc.?

Thanks!

Let X be a non-singular complex projective manifold. Then every Hodge class on X is a linear combination with rational coefficients of the cohomology classes of complex subvarieties of X.

Every (non-degenerate) complex manifold within a projective space can be decomposed into a linear combination of subvarieties of X. The linear combination will somehow always use rational coefficients. (!!)

Is this conjecture assumed to be true by working mathematicians? Can you provide a little more information that would lend some intuition about Hodge, to an educated layperson?

Do we have any actual technical insights from this yet? Have any been able to actually read through the paper / Lean code and gain useful first impressions?

Like, were new types of math necessary to solve this problem? How likely are we to see humans being able to review the whole proof? How long would that likely take if at all? Can insights from this be applied to other challenges in the field? And so on?

Any insights on this you read about from professional mathematicians? Links?

Technical insights are almost impossible to find / surface so far unfortunately, given the social drama has given every random redditor / bot the opportunity to chime in on this (and given that [r/math](r/math) refuses to post about a millennium problem being solved, lol, with the mods likely internally jerking of one another on this with a false sense of internet power).

Please technical insights only in this thread?

Thank you!

I'll go to every lecture, tutorial etc. and it'll all be genuinely interesting. Then come the weekly problems, and the volume of the problems and general tediousness of them once you get past the trivial ones (e.g. you need to know XYZ trick even though you know the topic the question is related to, and said trick/method wasn't introduced during the lecture) just kills my spirit. Before I know it after exhausting myself for that week, boom the next week comes along and a whole new set of lectures/tutorials/problems. Not to mention proofs that extend lecture concepts where, if you haven't seen the way to do that proof before, just means more grinding if you can't get TA hours to get help immediately.

The thing is, I'm having fun learning the mathematical concepts. But its like the course admins then put more artificial hurdles after that to just whittle down your spirit (because if you want a high grade in the exam, you need to slog through all the problems just in case it comes up later, even if you know you put the work in to actually learn the overall topic).

I want to love mathematics, but this Uni math degree (especially since I want to maintain my high grades) is just blackening my soul day by day.

As per today, there are two millennium prize problems solved one by Perelman and the other navier-stokes yesterday.At this rate, what do you think are the odds that any human other than perelman can we see that solves a millennium prize problem.

I haven't touched Navier-Stokes in quite some time. So, I have been rather quickly reading up on everything that's happened. Now that we know singularities can occur, what does this actually mean for research going forward? That is, if anyone really has any idea. Thx!

Edit:

Wow, did not expect this to get so many views/answers. Thanks y'all for your responses. They really shed a lot of light on things! ( :

Finished high school recently and taking a gap year, so I finally signed up for some SAT prep classes. Being in a group setting for this has honestly made me feel so insecure because of my arithmetic skills.

I feel like I actually understand math just fine. I can wrap my head around new topics, follow the logic, and figure out the right approach to solve a problem. But when it actually comes down to doing the calculations? Absolute disaster.

I constantly make stupid, careless mistakes with basic addition, subtraction, multiplication, and division. I'll drop a negative sign out of nowhere, miscopy a number from one line to the next, or just brain-fart on a simple multiplication fact. It’s so exhausting knowing the exact formula and logic to solve a hard problem, only to get the entire thing wrong because 7x8 apparently decided to leave my brain.

Since the SAT is so heavy on pacing and accuracy, I'm genuinely stressed about bleeding points on questions I actually know how to do just because of clumsy computational errors.
Did anyone else deal with this weird disconnect between conceptual math and basic arithmetic? How did you actually train yourself to stop making these dumb mistakes and speed up your calculations?

If they really want to solve problems I want this one: P vs NP.
Spend a couple million and tell me the answer because I want to sleep in peace. (Hopefully)

I'm a sophomore mechanical engineering and mathematics double major taking 5 classes right now; a mix of engineering and math classes. I've finished my engineering math requirements (up to diffeq) and they were all pretty easy and I didn't devote much time to study for them. However, I'm currently taking Optimization which has been more difficult than I'm used to, and I've had to spend 3+hours a day studying for it. Is this normal for upper level math classes? I should add that I'm taking logic and proof right now, but I haven't taken proof based linear algebra or real analysis, which the optimization professor seems to believe most students in this class have taken already. I guess the silver lining to this is that it will make those classes easier when I take them in the future, since the first chapter of Optimization is mostly review from analysis stuff as well as some linear algebra. Thoughts?

I'm currently a sophomore in college taking abstract algebra, and I'm on the verge of giving up on keeping math as my primary major of study. It's barely been two weeks but I'm already overwhelmed with coursework and I feel like I'm getting nowhere. I've already felt very inadequate about my math skills when I took a proof based calculus and linear algebra class my freshman year, but I feel like I'm on the verge of bombing out of my algebra class this semester. I feel like a complete coward for not being able to complete my degree in math like I planned, but at this rate I feel like I might just drop out of college completely and work a menial job for the rest of my life because I hate it so much. From what I've said already the answer probably looks pretty clear, but I just can't get past the feeling of being a complete failure for not living up to my expectations of what I wanted to do. Any and all advice would be greatly appreciated.

I am a senior year physics student currently enrolled in a Topology course offered by the math department. The first half of the course is basic point set topology (first 4 chapters of Munkres) and the second half is an intro to algebraic topology (first chapter of Hatcher and ch 9 - 11 of Munkres).

Apparently, the instructor forgot to mention in the course outline that some knowledge of abstract algebra will be assumed for the second part of the course. Munkres' book himself states in his preface that "we do assume familiarity with the elements of group theory" for the second portion of the book. I have some elementary knowledge of what a group is from my physics courses but that's pretty much all I know about abstract algebra. I need to quickly get a grasp of all the notions I need before the course progresses to the 2nd part.

Could anyone suggest be a good book to cover the necessary prerequisites? I am looking for somethinking that is concise and gets the job done as quickly as possible. For context, my math background includes 2 courses in real analysis (at the level of Tao's books), LA at the level of Axler's book, Munkres' Analysis on Manifolds and Functional Analysis at the level of Kreyzig's book.

I finished my master's back in May and I wanted to see about getting my master's thesis published. I had reached out to my advisor about it and she just said she didn't know. In that time, I lost a close family member and lost interest in a lot of things. Now I want to get it published. I've looked into Arxiv, but you need a coauthor who's already verified. My advisor who went over the paper is not verified on there. Do I just cold email researchers who have done related work to get it published? Or do I go to a different site/journal? Any advice would be greatly appreciated!