The pathway
Six stages. At each one: what it is for, what it covers, and exactly which material on this site belongs to it.
Stage 1 · Class 8 – 10
Where the habit of proof begins.
Number, ratio, mensuration, elementary geometry and the first real algebra. The aim at this stage is not speed but the discovery that a mathematical claim can be argued, not merely asserted.
Written material is in preparation for this stage. The Intermediate and school courses are taught in person for now — see about & contact.
Stage 2 · IPE · EAMCET · JEE
The five pillars of the senior-secondary syllabus.
The classical Intermediate course, taught the way it is actually used later: calculus as a theory of change rather than a table of rules, vectors as geometry rather than bookkeeping, probability as measure-in-miniature.
Written material is in preparation for this stage. The Intermediate and school courses are taught in person for now — see about & contact.
Stage 3 · CUET · IISER IAT · ISI · CMI
Selection papers, treated as mathematics.
Competitive papers reward structure, not tricks. The material here works each question from first principles and names the principle used, so that practice compounds into understanding instead of pattern-matching.
Part I builds the grammar of counting from the ground to JEE Advanced. Part II turns to structure and symmetry for ISI, CMI, IAT, CSIR–NET, NBHM and olympiad work.
What the practice exam returns when you finish it: score, subject and topic breakdown, and a full worked solution for every one of the 125 questions.
A full mock in exam conditions: 75 mathematics, 25 physics, 25 chemistry, live timer, question navigator, and a step-by-step worked solution for every item afterwards.
Coverage tracking across both papers at once, since their syllabi overlap far more than the coaching material admits.
Stage 4 · B.Sc. · B.Tech · B.Stat
The first encounter with rigour.
Epsilons and deltas, groups and rings, linear maps and their canonical forms. This is where mathematics stops being computation with symbols and becomes the study of structure.
A self-contained course in the mathematics of combining ordered lists — from Borda and Condorcet through Arrow’s theorem to the fusion rules used in modern retrieval.
The notation f⁻¹ is overloaded: it names two entirely different objects depending on context, and conflating them leads to real errors. A few pages to fix it permanently.
Stage 5 · CSIR-NET · GATE MA · JAM MA · NBHM
Breadth, at speed, with proof.
The national qualifying papers span the whole of the M.Sc. syllabus. The Manifold approach is to build a spine — one worked commentary per paper, every item developed from first principles — rather than a bank of shortcuts.
All one hundred and twenty items, treated from first principles. Every concept developed in full — no step assumed, no result quoted without its reason.
A working desk for postgraduate revision: topic-by-topic study state, spaced revisits and progress across the three national papers.
The full syllabus of all three papers as a single checkable tree, so you can see coverage rather than guess at it.
Turns the CSIR NET Mathematical Sciences syllabus into a dated plan — what to study, in what order, against the calendar you actually have.
Stage 6 · Research
Where the syllabus runs out.
Past coursework there is no list to finish. The bridge monographs are written for exactly this crossing: they start at a picture a second-year student can hold and end at the discretisation schemes of a current research paper, without a gap in between.
A complete course in sequential decision making: dynamic programming, temporal-difference learning, policy gradients, deep RL, offline methods, and policy optimisation for large language models.
From first pictures to the research frontier: weak solutions, finite elements, operator semigroups, stochastic calculus, Gaussian measures and SPDE discretisation. Prepared as a self-study spine.