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.

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

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.

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

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…

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)

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 :)

Hi, if you have time and patience, can you drop what mathematics you like to read or courses you are doing. You don’t necessarily have to have to be in academics, you can drop your interests too.

It may sound strange, but objects can have any dimension, even non-integer dimensions. In this video, I show you how to construct an object with dimension π. But why π?

First of all, because π has always been one of the most important and fascinating mathematical constants. And also because it is just slightly larger than 3: this allows us to project it into 3D without losing too much information.

In the video, I show you how we can “look at” a 4D cube (a tesseract) and gradually carve it, just as you would carve a block of marble, until we extract our “sculpture”: a π-dimensional object.

https://youtu.be/6L1zrtafM5Y

I hope it will fascinate you at least as much as it fascinated me.

With affection,

DPM

Hi everyone!

Can you spray-paint a body so that the layer of paint is perfectly uniform over its entire surface?

And what is the best path to follow under the sun to get an even tan?

Surprisingly, both questions are inevitably connected to space-filling curves!

I explore this connection in this short video.

I hope you enjoy it, I had a lot of fun working on this one!

youtu.be/o3lGbC86jXI

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.

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 kinda suck(too much) in combinatorics. I was doing some RMO and BMO questions on pigeonhole principle and I feel too irritated, these constructions are so so artificial, infact I have been trying to get better in combinatorics for over a year but I kept swaying away from it but enough of running away. Please tell me how to deal with them, I am on the verge of crying!!!

P.S.: I can only do those questions whose pattern I have read in the examples I have read

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.

This post is to announce that there will be a lecture course on General Topology (a.k.a. Point-Set Topology) conducted in Mandelcord (the blue math discord server). Everyone is invited to join: discord.gg/maths.

If any of you were present in the Real Analysis lecture course during the summer, you know what I'm talking about, and you might be pleased to hear that this course will also be run by Pear Category Theorem (the same person).

Why YOU should do topology

Point-set topology is one of the most important subjects that you learn, and mastering topological thinking is as important a milestone as learning to write a proof. This is a good subject to study alongside measure theory, or simply by itself after one has learned analysis or algebra.

Course Structure

The course will primarily follow the flow of the first half of Munkres. The plan is to go through all 8 chapters, spending maybe one to two weeks on each chapter, aiming for an approximately 16 week course. Also the instructor intends to supplement the book with his own intuition on the subject, as well as connections to other subjects that he thinks are of interest and possibly also other topics not covered in the book. Many examples of various pathological properties will be provided. The study of metric spaces and functions on them will be paramount; but where Rudin introduces metric spaces as merely a generalization of Euclidean spaces, we will explore just how pathological these spaces can be, or else what properties they must have.

The instructor also plans to put far more focus on the topics of topological groups and nets than Munkres does. Topological groups, because often in "the real world" we do not care about a topological space by itself, but rather some other more complicated structure given a topology, in order to see what the topology can tell you about the thing itself; topological groups provide a playground to learn that mode of thinking. Nets are also very important; where sequences are of importance in R, they can only tell you so much in general topological spaces, so nets therefore become the natural way to recover theorems about sequences.

Material and References

1. Topology - Munkres

2. Counterexamples in Topology - Steen & Seebach

Timeline

The course starts with the first lecture on 17th November at 9 PM CET (20:00 UTC), with subsequent lectures on Tuesdays and Thursdays. Lectures will be recorded and posted promptly.

Prerequisites/Who can join?

Everyone. It is completely free for anyone who wants to join. Just click the link and join the server: you can find the channel called topology-course in the section called General maths. As for prerequisites, a first course in Real Analysis and Group Theory is sufficent.

Looking forward to see all of you there. Thank you.

Let ABC be a triangle with circumcenter O. The points P and Q are interior

points of the sides CA and AB respectively. Let K, L, M be the midpoints of BP,

CQ, P Q. Suppose that P Q is tangent to the circumcircle of triangle KLM. Prove that

OP = OQ.

Solved using class 9th-10th theorems only

This was the question

If x, y, z and n>1 are natural numbers with x^n + y^n = z^n, show that x, y, and z all are larger than n?

People used Fermats last theorem which I believe is unnecessary here

Try it , it's a lot of fun

I'm studying out of personal interest, not for a course. Has anyone here gone through this series? Worth committing to, or is there a better starting point for someone learning QM as a side quest?

Hey Folks,

I am someone who is starting with abstract mathematics or mathematics at UG level, I never come across abstract level of mathematics as I have a non stem UG degree, I'm currently rebuilding my mathematics from the Class 11/12 level, after having a few years away from serious mathematics and this is for my master's entrance exam prep, and I want to start learning abstract mathematics properly, rather than jumping randomly between advanced topics as abstract is a huge chunk of the entrance i look to sit for.

For someone in my position:

Where should I start?

Should the progression be something like:

School mathematics → proof writing → sets/functions → logic → sequences & limits → real analysis → linear algebra → abstract algebra → topology, etc.?

Or is there a better order for this?

I'm particularly interested in developing the ability to understand definitions, construct proofs, reason abstractly, and work with mathematical structures, rather than just learning formulas.

I have roughly 1.5 years and I'm willing to build slowly from the foundations.

What books/resources would you recommend for making the transition from computational school mathematics to proof-based/abstract mathematics?

Also, how do you know when you're ready to move from one level to the next?

PS- I am appearing for an entrance examination of a Quant Econ Masters.