The backbone of the entire ISI M.Math exam. Sequences, series, continuity, differentiability, and Riemann integration form the core. Problems frequently combine epsilon-delta arguments with deeper structural reasoning. Without strong Real Analysis, no other topic fully makes sense.

Analytic functions, Cauchy's theorem, residue calculus, conformal mappings. ISI frequently tests whether students understand the geometric meaning behind complex-analytic concepts, not just the formal machinery. Integration in the complex plane appears regularly in Paper II.

Group Theory, Ring Theory, and Field Theory are tested in depth. ISI problems go well beyond standard BSc-level questions — you will need to construct and verify homomorphisms, work with quotient structures, and understand field extensions and Galois theory at a preliminary level.

Vector spaces, linear transformations, eigenvalues, inner product spaces, and canonical forms. Often combined with Real Analysis in ISI problems. Linear Algebra is one of the more approachable areas — but ISI tests it at a depth that goes well beyond matrix computation into abstract structural reasoning.

Topological spaces, open and closed sets, compactness, connectedness, continuity in topological terms, separation axioms. This is where many students struggle most — because Topology requires a level of abstract intuition that cannot be built through formula-based study. It demands deep engagement with definitions.

ODEs and their analytical solutions, Sturm-Liouville theory, and basic PDE concepts. Appears less frequently than core analysis, but when it does appear in Paper II, it typically requires a proof-based answer — not computational solving.

Elementary number theory, divisibility, congruences, and combinatorial reasoning. These tend to appear in Paper I MCQ sections. Though classified as "easier" areas, ISI frames combinatorics problems with surprising twists that catch overconfident candidates.