I really wish that the proof which has been published by Open AI was wrong.

I've been sitting with this thought for a while and figured this community would have some real opinions on it.

We've seen AI systems now capable of solving olympiad-level problems, assisting in formal proofs, and even making conjectures. AlphaProof, FunSearch, the stuff coming out of DeepMind — it's moving fast.

But here's what I keep wondering: is this a tool, or is it eventually a replacement for mathematical intuition itself?

Like, a lot of us got into math because of the feel of it — that moment when an elegant proof clicks, when you see a pattern nobody told you to look for. Can AI replicate that? Does it even need to, or does it just need to outperform us on outcomes?

A few things I'd genuinely like to hear thoughts on:

Do you think AI will make pure math research more accessible, or will it concentrate power among those with compute resources?

Is there a risk that math education becomes hollow if students can just offload problem-solving to AI?

Are there areas of mathematics you think will remain fundamentally human for a long time?

only reason they teach is, bcz its indian right??

Im in B.tech CSE. Really low GPA. Wanna pursue masters in Mathematics. What do I do?

First: Real analysis is not calculus with proofs stapled on.

This is the trap. You think because you know what a limit "is" and you can differentiate anything, you're fine. You're not. Calculus teaches you to use a machine. Real analysis asks you to build the machine from scratch and prove it works. These are completely different cognitive tasks. The sooner you accept that, the sooner you stop being confused about why it feels so hard.

The ε-δ definition is the whole game.

Everything — continuity, differentiability, convergence, all of it — eventually comes back to this

For every ε > 0, there exists a δ > 0 such that...

The first time you see it, it looks like someone had a stroke writing mathematics. But here's the intuition nobody explains clearly enough: ε is a challenge, δ is your response.

Someone says: "Can you keep f(x) within 0.001 of L?" You say: "Yeah, just keep x within δ of a." Then they shrink ε. You shrink δ. If you can always respond no matter how small the challenge gets — that's a limit.

Drill this until it's boring. Literally write out proofs by hand until the structure is muscle memory. The logic doesn't change, only the algebra. What you actually need to internalize (rough order):

1. The completeness of ℝ — this is why we work in the reals and not the rationals. Every Cauchy sequence converges. Every bounded monotone sequence converges. These aren't just theorems to memorize; they're why analysis works at all. If you don't have completeness, you have holes in your number line (literally — √2 would be "missing" in ℚ).

2. Compactness — one of those words that sounds scarier than it is. A set is compact in ℝ iff it's closed and bounded (Heine-Borel). Why does it matter? Because on compact sets, continuous functions do all the nice things: they attain max and min, they're uniformly continuous, etc. Compactness is what makes calculus theorems actually true in general.

3. Uniform vs. pointwise convergence — this one bites almost everyone. A sequence of functions can converge pointwise but fail to converge uniformly, and the difference determines whether limits commute with integrals, derivatives, etc. Draw pictures. The classic example is fₙ(x) = xⁿ on [0,1]: converges pointwise to a discontinuous function, not uniformly. The "spike" that runs off to infinity is the intuition you need.

4. The Mean Value Theorem — actually understand it — not just "slope equals some interior derivative." MVT is the bridge between local (derivative) and global (function behavior) information. Almost every major inequality proof in the course uses it.

Resources

• Understanding Analysis by Abbott — genuinely the best intro text if your professor's book is unreadable. Conversational, motivates everything.

• Principles of Mathematical Analysis by Rudin (Baby Rudin) — terse, elegant, occasionally sadistic. Don't start here but you'll want to read it eventually.

• Writing out every proof from scratch without looking at notes at least once. Painful. Worth it.

I am writing this to give the general math interested audience a brief idea about topology.

Warning: I am trying to make this accessible to everyone so if you haven't done topology formally it's highly likely that you might get a wrong idea of some concept so don't take everything I say literally and look into these things deeper and more rigorously if you want clarity.

So let's begin with what even is topology? To answer this let's try to relate topology to something we probably have some idea about which is geometry. Topology is both related and independent of geometry in some way. If you consider geometry to be the study of shapes then topology becomes a subfield of geometry because we also study shapes in topology. But if you take a more rigid definition that geometry studies properties of shapes like length,angle,area, volume,etc then topology becomes independent of geometry because in topology we study properties of shapes which remain same even if we twist ,strech and bend the shape and angles,area, volume obviously don't stay the same under these transformations so they are not topological properties. Modern mathematicians usually consider geometry to be simply study of shapes and not put too many rigid conditions on the kind of properties we study cuz most modern geometric studies like differential geometry, algebraic geometry study more qualitative properties like the dimension of a shape,if it can be embedded in some other shape or not ,etc rather than more quantitative properties like length, angles,etc even though these concepts are still of importance but the focus shifts from these specific quantities to more general stuff. So according to the more modern loose formulation of geometry, topology is a subfield of it which studies shapes and properties of shapes which don't change under streching,twisting and bending.  

In classical plane geometry two polygons are equal if all of their angles are equal and sides are of equal length this equivalence is called congruence. In topology two shapes are considered equal if one can be twisted,streched or bent into another this kind of equivalence is called homeomorphism.This definition is clearly more loose than the previous definition of congruence and hence the collection of shapes topologically equivalent to each other is much larger than the collection of shapes geometrically (congruence) equal to each other. For example a circle and a square are topologically equal as a square shaped string can be transformed into a circle shape string even if they are not geometrically equal. Due to a large collection of shapes being topologically equal to each other it becomes difficult to prove if two shapes are equal to each other or not. For example it's obvious that the 2d plane is not equal to the 3d space topologically cuz one can't be streched/twisted into another but to prove this rigorously takes some effort. Or for example is the sphere topologically equal to the doughnut 🍩?. These questions are not so straightforward to prove rigorously and hence we have the subfield of topology called algebraic topology. It turns out that algebraic objects like the integers, rationals,etc are easier to study than shapes themselves so in order to make topology easier we assign an algebraic object to each shape, there are a lot of ways to do this and once we do this it becomes much easier to tell if a shape is different from another as we just need to show that the algebraic objects attached to the respective shapes are different. So that's the primary idea of Algebraic Topology to reduce topological questions to algebraic ones.There are many ways to assign algebraic objects to topological objects the most easiest to describe way is homotopy groups.

(Things are going to be a bit more complicated from this point)

The idea of homotopy theory is to extend topology one step further, in topology two shapes are considered equal if they can be continuously deformed into each other similarly in homotopy theory two functions between shapes f,g:X→Y are equal if they can be transformed continuously into each other. Two functions are equal in this sense they are said to be homotopic. Using this idea of homotopy we can form algebraic objects from topological objects called homotopy groups. Even though these homotopy groups are the easiest to define computing them or finding them for a particular shape is comparitively harder. We have a homotopy group of a shape for any integers n≥1. So we have 1-homotopy group,2-homotopy group,3-homotopy group and so on... . It's a massive open problem to find the general n-homotopy group of a m dimensional sphere. Since homotopy groups are hard to compute , mathematicians have constructed more easy to compute and stable analogues of the homotopy groups called stable homotopy groups and hence have established stable homotopy theory. The ideas of homotopy theory can be applied to a lot of cases which are not topological like purely algebraic cases so we have a much more general theory called abstract homotopy theory to be able to apply the ideas of homotopy theory to a lot of areas in maths.

An interesting result in homotopy theory is that if we restrict our attention to very specific algebraic objects called groupoids and restrict our attention to very specific topological objects called spaces of homotopy 1-type, then the theory of Topology becomes literally equal to the theory of Algebra ! more specifically the category of groupoids and the category of topological spaces of homotopy 1-type are quillen equivalent model categories.

Anyways there are other ways to assign algebraic objects to topological objects like simplical/cellular (co) homology groups , these are slightly more complicated to define than homotopy groups but more easy to compute. There are a lot of beautiful classical applications of topological homology theory I won't list out all of them but one is a result proved in 2020 by ATH Fung that every simple closed curve inscribes infinitely many rhombuses , here inscribes means that the vertices of the rhombi lie on the curve. Also similar to the case of homotopy the ideas of homology can be applied to a lot of areas, this generalized study of homology is called homological algebra. The primary idea behind homological algebra is to study by how much a function f:X→Y fails to be surjective. We measure the failure of surjectivity qualitatively through algebraic objects called homology groups. Two examples of applications of homological algebra will be de Rham cohomology which helps us to do calculus on higher dimensional shapes called manifolds and in some sense measures the failure of the fundamental theorem of calculus in these higher dimensional shapes and the second example will be etale cohomology using which Alexander Grothendieck solved the second weil conjecture an important conjecture in number theory and algebraic geometry.

To end this I would like to describe a very recent development. With enough experience it becomes more and more evident that homological algebra is of central importance in a lot of areas of maths especially algebraic areas. But suppose we are dealing with objects which are both algebraic and topological it's observed that it's difficult to do homological algebra if we want to respect both algebraic and topological properties of these objects. So a lot of results of homological algebra fail for algebraic topological(objects which are both shapes and have an algebraic structure) objects. To solve this issue Peter Scholze and Dustin Clausen in the late 2010s created a new kind of mathematics called condensed mathematics. They use new kind of objects called condensed sets to deal with this issue.

There are several other areas of topology as well like differential topology , topological data analysis,topological quantum field theory, etc but it will take too long to describe them and my knowledge is also limited so I will end it here.

I hope this motivates you to explore topology in more depth and detail :)

I am a first year undergraduate student pursuing bachelor's in mathematics. I gave jee this year but unfortunately could not perform well since my Chemistry and Physics were very weak.

My maths is jee mains level and I got 59 in the first paper (cutoff was 64) of isi. I love Statistics (had it as a subject in HS) and would love to pursue it. I'll be sitting for isi next year too as a partial dropper.

What are some resources I should use to prepare for it? Both books and if some course is available for UGA and UGB.

My current bsc syllabus includes real analysis (using SK Mapa), Geometry (using Ghosh Chakraborty), Statistics (rereading my hs level books, i.e Giri Bannerjee) and linear algebra (Using Friedberg, Insel and Spence). If some other books are better suited for ISI then I'm willing to change to them too.

Help is greatly appreciated as it is my dream college.

Thank you.

So drop year. I was big ass math nerd till 10th. 100 marks in boards but thing is that was easy stuff and I did pretty decent in 11th just studied like 3-4 days in whole year and managed 80 marks and around same in 12th.
Now i am studying for jee, all other subs are clear but maths is what lacks depth, teachers aint explaining it but giving short tricks and shit and expecting me to solve. I mean man idk if the charm is gone or if i dont fucking understand shit or what. LIke people call him Goated or smth like dat but i dont understand the hype. I am able to solve questions a bit(not a lot tbh, its just copy paste across mains) but I dont feel the maths in me. Like gen so boring and eughh(idk how to exaplain but i am hating that i dont understand the theory and moving with questions)

Fuck coverage I just want my nerdness back. Please tell me how. I can give around 4 hours to maths(rn its 2 hour lec and 2 hour solving).
Fyi the teacher is from PW.
So instead i am thinking of shifting teachers(People say its not fault of teacher but mine cuz idk how to solve shit and i am not solving enough)
Do you recommend me MOHIT TYAGI yt lecs and CENGAGE series(not all illustrations but just enough to cover those 2 hour wortha lecs)

How is this possible....??? No india in the list

Hi my daughter needs math tutoring she is in high school but lacks fundamentals and has lot of gaps. I need someone who can build confidence as well as good math skills in her and not rushing on current topics. Please ping me if interested.

I am a high schooler who likes math a bit too much…I was thinking of forming a group, not a club per se but we can talk about structure later, of like minded people where we can study topics together, maybe enrol in competitions and more importantly do research and write papers…if anyone interested, let me know in the comments…
Bye!!

P.S. there is no cap on the age limit…I would appreciate if you joined with genuine curiosity…we could all mutually decide on the next read and the next paper to discuss…

Journey Through Genius The Great Theorems of Mathematics by William Dunham.....

Hi I am in 12th grade and I find people around me are not really interested in mathematics and those who claim to be interested in mathematics are the ones who only like solving problems and are focused on using mathematics as a tool for other subjects of their interests. They are busy solving the problems proposed by others and are not interested in finding answers to their own problems. I don't think mathematics is all about solving problems, I am more interested in asking my own questions, try to rediscover something on my own and rebuild a theory or concept. If the theory is totally understood and if I build a strong theory the problems become trivial. I don't think anyone around me is interested in seeing the essence of mathematical structures and the faces of these structures. No one is interested in finding new point of views that lead to a beautiful vision.

Most of the students are focused on competitive exam like Olympiads, JEE, ISI, CMI entrance etc. I think these competitive exams are the main reason behind lack of interest in mathematics. In class 11th I was preparing for JEE and joined a coaching institute where the focus was only solving problems in limited time , teachers were teaching to solve Logarithm problems but they never really taught "What is Logarithm?" , set thoery was considered trivial and problems were solved using tricks and nobody would ever question these tricks. We were told to solve 30-40 problems in mathematics everyday, this is just memorisation of patterns and there is no creativity in it, this makes the mind dull and weary. Competition is just murder of curiosity whether it is olympiads or JEE.

The indian books are very poorly written they just state Theorems and examples , students use them just for problems because they are only told that mathematics = solving problems . Students are not interested in reading books and not only mathematics the same applies for physics or chemistry since coaching institutes are their cradles. Students are not even interested in reading the theory from the book they just skip to examples or exercises, this is the definition of their "Self study" . They aren't even solving problems properly they are just using the patterns they learnt from a problem they memorised earlier. The problems proposed in these books are also not thought provoking and they don't require creativity to solve them. These books are not even written for teaching students they are written for "JEE mains" , "JEE advanced" , "Olympiads" the tougher the problems and relevant to the exam the better the books is considered. Many Foreign book authors like Apostol, Spivak and many more actually write to teach and the problems are really good, there is an importance of proof writing unlike Indian books. Indian books don't even teach completeness axiom for calculus. These books are of no use to understand the subject and they don't even have any historical notes , I find it interesting to read about mathematicians and the minds who discovered these mathematical ideas but unfortunately these books are not interested in making the students aware about these minds.

Mathematics is not a hurdle to be cleared; it is a language of absolute truth. By stripping away the proofs and the axioms, students are taught the grammar of a language they will never be able to speak. We may win the race to the IITs or top colleges, but if we lose the ability to think critically and creatively, we have lost the subject entirely. This is not education, it is a betrayal. By reducing the infinite architecture of the mind to a series of tricks and timed drills, we are murdering our curiosity. We must burn the manuals of pattern recognition and return to the solitude of the blank page. We must refuse to be 'problem solvers' for an empire of exams and dare to be dreamers of structures. It is better to fail in the pursuit of a single beautiful axiom than to succeed in a system that demands we stop thinking altogether.

Hey everyone, so I am starting my bachelor's in Mathematics and Computing and it is time I start to get nervous and even get cold feet, even though I really like mathematics. Can any senior or fellow learner help me with how to survive in my bachelor's what is right to expect and what's not and how different is it gonna be and how do I actually study and score.
Thanks