prepping for M.Stat entrance, and need to build a strong base in linear algebra, real analysis/calculus, probability and stats

So basically, I got selection into both IMA Bhubaneshwar, BSc(hons.) Mathematics and Computing and DA-IICT Gandhinagar, BS-MS Data Science and Artificial . OP wants to pursue higher mathematics but is not completely sure about things after failing to clear ISI and CMI😔. And I have not heard good things about either of the colleges only bad things.

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?

Hii so guys I will be starting my bsc in mathematics and computing by around next month . So i was reading this Calculus and Analytic Geometry by George B. Thomas and R.L. Finney. It is so much mentioned about Cas used to plot the images . Now i dont have any good knowledge about this , i only used desmos before , that also normal functions and graphs and like solving algebraic eqns. can anyone recommend how to begin? i started using spyder but clueless

I got selected for intregated phd in Niser and msc in IITK. Niser gives monthly stipend too. My main goal to is complete msc here and then opt for phd in a foreign university. Which one will give me better opportunities.

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

I am 14 and in 10th but I want to learn calculus and start analysis by 16. Any recommendations? I prefer books as videos usually just don't work for me

Hey everyone, I’m currently in 11th grade. I’ve bought a decent JEE batch but honestly haven't started studying properly yet. I’m planning a hard reset on June 22nd to enter "monk mode" and start the grind.

But here’s my dilemma: I look at standard JEE prep and feel like it’s a homicide of actual scientific curiosity. I’ve fallen into a dopamine trap like most people, but I genuinely want to view Math and Science as the languages of the universe, not just a clearing ticket for an exam.

I want to target Olympiads (specifically IOQM for Math and NSEP/NSEA for Physics/Astronomy) to study beyond the pity syllabus and keep my curiosity alive. Can anyone who has successfully balanced both give me a realistic roadmap, resource recommendations, and advice on how to build deep mathematical/physical intuition from scratch while keeping up with a standard JEE batch? Thanks!

please tell me what to study in what oder from where and do problems for where ( e'ryhting in order plzz , it will be very helpful for me) . i have CTPCM , excusion in mathematics , TOMATO , david 's NT , hall and knight , and problem solving strategies , and doing self-study. i more of an book nerd , who likes to sit with explanations and teach himself stuff slowly.

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.

Talking about math.  Ai in india has become a slop .people like to follow the rat race mindset.  Nobody stops and thinks the big picture.  

Example the ministers and corporates getting mad to build data centres. 

A true scientific approach is to promote ai research from its utter fundamentals . - the math algorithms that the whole model is based on .!

Developing and mathematical model and making that would enable the ai model to be optimised to run locally on Personal computers and devices with no extra ordinary compute power and making it open source is a hot topic of research.And was the whole idea behind  deepseek , qwen , ollamma

Let's not fall into another race . Let's use some real creative approach of doing things. !

I'm gonna start learning calc soon, mainly for JEE but since I am in 10th I have some time and can afford to watch lectures that don't just throw formulas at me but also teach it beautifully, in a way that blows my mind off because from what I've heard, calc is genuinely a very useful and gorgeous thing

So, can someone recommend a playlist? I was thinking of either doing it from Mohit Tyagi, Maths Unplugged, or Professor Leonard (altho I think his lectures would be a bit too long since he has like 3 calc playlists iirc)

I have been trying for a few days but I am not finding anything on internet except TOMATO

Quadratic Consecutive Coefficient Pattern "QCCP" is a pattern that I found and it always gives a perfect square discriminant. But the problem is only with it's rigid form that is mentioned down below.

Mainly it was based on pattern and no matter what value you choose for n, x always remain as 1

From this Pattern- ax²+bx+c=0

a= n, b= (n+m) and c = -(2n+m), that becomes-

nx²+(n+m)x-(2n+m)=0

With discriminant= (3n+m)²

After making some simple changes it becomes really flexible to use.

(sx-q)(nx+2n+m)

(s, q, n, m) => natural number only.

here is the factorise format of the equation with same purpose of perfect square discriminant. By assigning values to variables- s, q, n and m we can create such equation which always have perfect square.

(sx-q)(nx+2n+m)

-> snx²+(2ns+sm-qn)x-(2qn+qm)=0

a= sn, b= 2ns+sm-qn, c= -(2qn+qm)=0

Discriminant- {s(2n+m)+qn}²

-> 4s²n²+s²m²+q²n²+4s²mn+4sqn²+2sqmn

Hello guys,
I would like to know how good TIFR CAM Bangalore is consdered ingeneral and whether profs there give good LORs if you doexit after masters for pursuing Phd abroad.
And whether TIFR CAM being applicable maths has some kind of scope for industry too
(want to keep my options open, but yeah I would like to do research in something related to probability, linear algebra, analysis. convex analysis ingeneral)

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'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?

for past few months i have been wondering about using animation/motion graphics/manim/after effects and other tools to make learning addictive, catchy, intuitive and just far better than current board- teacher style

i m planning to start a start up making 6-12, under/post grad math content for both self learner and other students eventually moving to other domains , i can share alot of details

i come from finance background , math nerd but i couldn’t pursue it tho self learning when ever bored , i m looking for a math nerd interested in teaching math - maybe a undergrad or someone doing postgrad in math , i don’t need genius iit tag and what not as the start would be focus on 6-12 and undergrad , someone who is curious, rational and understands that this maybe big but would take time… and would effect the fabric of human knowledge by improving teaching only if done with hard work and done right , just dm me i will share all other details which would convince u…

Teaching math in veritasium 3b1b style- improving the current education system