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?

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 :)

10th grader here who's trying to learn some pre-calc rn. Whenever I search up a topic, the first video that comes up is often some one-shot of PW or the many other JEE channels, I have also explored the lectures in paid batches, and often find myself completely unsatisfied with the way they teach.

Its like every 5 minutes they just have to crack an unnecessary and irrelevant joke or start randomly giving motivation. The beauty and geometrical intuition behind identities and formulas are almost never taught (even though it is SO much easier when you think of trig in unit circles and a coordinate system). Oh and how can I forget, the way they just ASSUME you're a guy by referring to the audience actively with male pronouns. I know these things might be considered a non-issue by some, and I can see where you are coming from,

But is it worth it to keep watching these lectures even after all of this and even though books provide significantly higher amounts of information? Personally, no. If the quality of these one shots and lectures were truly unparalleled, I would stick around.

But let me know your opinions!

https://preview.redd.it/5szm6itfvx0h1.png?width=720&format=png&auto=webp&s=1f017c47337227d85d5d7b0ed5c956c115685b5e

Hi everyone,

I’m looking for a second-hand copy of Calculus by Michael Spivak.

The original edition is quite expensive for me right now, so I was hoping someone here might have a used copy they’d be willing to sell, lend, or share.

I’d really appreciate any help. Thanks!

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.

india's math olympiad performance has been quietly getting really good. 4th at IMO 2024 — four gold medals — which is the best we've done since we started participating in 1989. that's not luck, that's years of building a selection and training pipeline through HBCSE. and then 7th again in 2025 with a national record score. these are high school kids. the talent is clearly there.

TIFR's school of mathematics in mumbai does work that genuinely competes internationally. i'm not being patriotic here — i mean people there publish in annals, inventiones, JAMS. number theory, algebraic geometry, ergodic theory — the faculty are serious. IMSc in chennai is excellent. ISI kolkata has a history going back decades. CMI's undergraduate programme produces students who regularly get into top 10 global phd programmes.

the IISERs were a genuinely good decision by whoever made that call in 2006. seven institutes, proper research culture from the undergrad level, students who actually read papers before graduating. compared to what the situation was 20 years ago it's a real improvement.

the money situation is embarrassing. TIFR postdoc pays ₹47,000–54,000 a month. that is your salary if you have a phd and you're doing research at what is supposed to be our flagship math institute. in mumbai. have you tried renting in mumbai on ₹47k? a phd stipend at the best institutes is ₹31,000–37,000. meanwhile india's R&D spending as a share of GDP has actually gone DOWN — from around 0.9% in 2008 to 0.64% in 2021. china is at 2.4%. south korea is nearly 5%. we are going the wrong direction.

i'm not saying this to be dramatic. i'm saying this because i have watched extremely talented people — people who genuinely love mathematics, who would have been happy to stay — do the calculation (ironically) and leave. a us postdoc in math pays around $60k a year. that's roughly 50 lakh rupees. the gap isn't bridgeable by "passion for the subject."

there's a study that found over 73% of indian researchers who move abroad never come back. never. and the ones who leave aren't random — they're disproportionately from the good institutions, the ones we spent public money training. it's not brain drain as a metaphor. it's a literal, measurable, ongoing transfer of human capital that we funded and then gifted to the west.

the structural problems that don't get talked about enough

most IIT math departments are service departments. their primary job, implicitly or explicitly, is to make sure engineering students pass calculus. the faculty are evaluated on teaching loads that would make it very hard for anyone to do deep research. this isn't anyone's fault individually — it's how the system is set up. but it means that "math research at IITs" is often a very different thing from math research at TIFR or IMSc, and we shouldn't pretend otherwise.

the postdoc to faculty pipeline is basically a bottleneck. TIFR hires a few people a year. ISI a few more. IMSc a few. the IITs and IISERs have more positions but the hiring process is slow, political in the usual ways, and the positions aren't always in pure math areas. a person finishing a strong phd in, say, analytic number theory or low-dimensional topology faces a genuinely bleak domestic market. the options are: leave for a foreign postdoc (and probably not come back), take a position somewhere where you'll spend 18 hours a week teaching and maybe get two hours of research time if lucky, or just leave academia.

and then there's the JEE thing. i'll probably get flak for this but i'll say it anyway: years of JEE prep does something to how people think about math. JEE trains you to be fast, to pattern-match, to know which trick applies to which problem type. that's a specific skill. it's not the same skill as sitting with a problem for three weeks and not knowing if you're on the right track. a lot of students arrive at IISERs and CMI genuinely shocked that math can involve extended confusion and that this is normal and fine. the olympiad pipeline is a partial corrective but it reaches maybe a few hundred students seriously at the national level. JEE reaches millions.

the weird paradox that nobody wants to say out loud

india is good at producing mathematicians. like, actually good. the olympiad results, the quality of graduates from CMI and IISERs, the names — ramanujan obviously, but also harish-chandra, c.s. seshadri, m.s. narasimhan — these aren't flukes. the country has mathematical culture in the real sense.

what we're bad at is keeping them. we build the pipeline and then we don't finish the job. it's like spending years growing a plant and then not watering it when it's about to bear fruit. the people who could build india's mathematical future are making tenure decisions in chicago and cambridge and paris right now, and a non-trivial number of them would have stayed, or come back, if the conditions were different.

what i actually think would help (not a policy paper, just common sense)

• postdoc and phd stipends need to double minimum. ₹37k in 2025 is not serious. the PM research fellowship (₹70k) is a good idea — expand it massively and stop restricting it so heavily.
• NBHM (the national board for higher mathematics) does useful work but is chronically underfunded. triple its budget. it's not a lot of money in the scheme of things.
• we need more permanent positions at the serious research institutes, not just more IIT expansion. TIFR and IMSc are the crown jewels. act like it.
• there are indian diaspora mathematicians at good western universities who would genuinely consider coming back for the right package. make that package exist. even 20-30 of these people returning would be transformative.
• protect the olympiad ecosystem. HBCSE does heroic work on not enough money. the selection camps and training are a public good.

Everyone talks about the "prestige" and the "funding," but nobody tells you that actually using that money is like trying to win a fight with a brick wall. If you’re a research aspirant, you probably think you’ll spend your time doing, you know... research. In reality, Its 40% researcher and 60% clerk/accountant.

The "L1" .... If you need to buy a specific sensor for your setup, you can’t just buy the one that works. Because of government "L1" rules, the institute is basically forced to buy from the lowest bidder. So, you end up with some cheap, knock-off version from a random vendor in Noida who has no idea how to support the tech. It breaks in two weeks, and you’re back to square one.

I have a bachelor's degree in Electronics and Telecommunications Engineering where I studied Maths from Alan Oppenheim's course available on YouTube, Went through his book on Signals and Systems

I also went through Rosen's book on Discrete Maths during my undergraduate

However I've spent the last 6 years working in the industry as a Software Engineer and am going back to Academia again with a degree in Electrical and Computer Engineering with an AI specialization

I want to brush my concepts again and become competent enough to understand maths and express ideas mathematically

Is there any specific path that I can follow for the same